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Page 1: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

wwwmonasheduau

Non-standard Computational Methods in Numerical Relativity

Leo Brewin

School of Mathematical Sciences

Monash University

Experimental gravity

Experimental gravity

Experimental gravity

Rules of the game

Numerical relativity

Construct discrete solutions of Einsteinrsquos equations

Sounds simple but

Non-standard methods

Numerical Relativity

Regge calculus

Multiquadrics

Spectral methods

Smooth lattices

Tetrad methods Buchman amp BardeenPhysRevD67(2003)084017 van Putten PhysRevD55(1997)4705

Discrete differential forms Frauendiener CQG23(2006)S369 Richter amp Frauendiener CQG24(2007)433

Finite volumes Alic etalPhysRevD76(2007)104007

Finite elements Korobkin etalCQG26(2009)145007 ZambuschCQG26(2009)175011

Multiquadrics

VOL 76 NO 8 7OURNAL OF GEOPHYSICAL RESEARCH MARCH 10 1bull71

Multiquadric Equations of Topography and Other Irregular Surfaces

ROLLAbullD L

Department o] Civil Engineering and Engineering Research Institute Iowa State University Ames 50010

A new analytical method of representing irregular surfaces that involves the summation of equations of quadric surfaces having unknown coefficients is described The quadric surfaces are located at significant points throughout the region to be mapped Procedures are given for solving multiquadric equations of topography that are based on coordinate data Con- toured multiquadric surfaces are compared with topography and other irregular surfaces from which the multiquadric equation was derived

Topography can be represented by various analytical numerical and digital methods in

addition to the classical contour map The

extremes in generalization or detail that result

from use of these methods are perhaps demon-

strated best by Lee and Kaula [1967] and by

Gilbert [1968] Lee and Kaula described the

topography of the whole earth in the form of

thirty-sixth-degree spherical harmonics Gilbert

reported the magnetic tape storage of more

than six million increments of height informa-

tion in digital form measured or interpolated

from one ordinary map sheet In Lee and Kaulas work we have an ex-

treme generalization of existing topographic

information over a wide area by highly analyti-

cal methods whereas Gilberts work is extremely

detailed but scarcely analytical As valuable

as these techniques are in certain cases they

are related more to map utilization than to map

making Basically the problem they solve is

given continuous topographic information in a

certain region reduce it to an equivalent set of

discrete data eg spherical harmonic coeffi-

cients or digital terrain increments

Other investigators including myself are

concerned with a procedural inverse of the

above problem namely given a set of discrete

data on a topographic surface reduce it to a

satisfactory continuous function representing

the topographic surface Practical solutions to

this problem will tend to eliminate the classical

Copyright 1971 by the American Geophysical Union

contour map as the first step in representing terrain information

An equation of topography can be evaluated

digitally or analytically without its having

been reduced to graphical form The same equa-

tion can be treated analytically for the auto-

matic production of contoured maps Automatic

contouring can become a computer-plotter

problem in analytical geometry ie to deter-

mine and plot the intercept equations of hori- zontal planes passed through a three-dimen-

sional equation of topography This approach could also lead to reconsideration of the need

for digitized map data Problems involving map use such as determining unobstructed

lines of sight areas of deftlade volumes of

earth and minimum length of surface curves

may involve the more direct application of

analytical geometry and calculus to the inter-

relationship of these parameters with a mathe-

matical surface of topography For these rea-

sons the question of representing a topographic surface in detail by unique equations deserves increased consideration

bullNUMERICAL SURFACE TECbullNmUES

Fourier and polynomial series approximations have been applied in recent years to both sur- face and subsurface trend analysis in geological

mapping Investigators in this field have been

Krumbein [1966] Mandelbaum [1963] James

[1966] and Merriam and Sheath [1966] There

has been a natural tendency to apply these

trend surface techniques to the problem of

1905

Multiquadrics

Originally used for interpolation on scattered data

Adapted by Kansa to solve ODEs

Given compute such that

Multiquadrics

Computers Math Appfic Vol 19 No 89 pp 127-145 1990 0097-494390 $300 + 000 Printed in Great Britain Pergamon Press plc

M U L T I Q U A D R I C S - - A S C A T T E R E D D A T A

A P P R O X I M A T I O N S C H E M E W I T H A P P L I C A T I O N S T O

C O M P U T A T I O N A L F L U I D - D Y N A M I C S - - I

SURFACE APPROXIMATIONS AND PARTIAL DERIVATIVE ESTIMATES

E J KANSA

Lawrence Livermore National Laboratory L-200 PO Box 808 Livermore CA 94550 USA

A~traet - -We present a powerful enhanced multiquadrics (MQ) scheme developed for spatial approxi- mations MQ is a true scattered data grid free scheme for representing surfaces and bodies in an arbitrary number of dimensions It is continuously differentiable and integrable and is capable of representing functions with steep gradients with very high accuracy Monotonicity and convexity are observed properties as a result of such high accuracy

Numerical results show that our modified MQ scheme is an excellent method not only for very accurate interpolation but also for partial derivative estimates MQ is applied to a higher order arbitrary Lagrangian-Eulerian (ALE) rezoning In the second paper of this series MQ is applied to parabolic hyperbolic and elliptic partial differential equations The parabolic problem uses an implicit time-marching scheme whereas the hyperbolic problem uses an explicit time-marching scheme We show that MQ is also exceptionally accurate and efficient The theory of Madych and Nelson shows that the MQ interpolant belongs to a space of functions which minimizes a semi-norm and gives credence to our results

1 BACKGROUND

The study of arbitrarily shaped curves surfaces and bodies having arbitrary data orderings has

immediate application to computational fluid-dynamics The governing equations not only include

source terms but gradients divergences and Laplacians In addition many physical processes occur

over a wide range of length scales To obtain quantitatively accurate approximations of the physics

quantitatively accurate estimates of the spatial variations of such variables are required In two

and three dimensions the range of such quantitatively accurate problems possible on current

multiprocessing super computers using standard finite difference or finite element codes is limited

The question is whether there exist alternative techniques or combinations of techniques which can

broaden the scope of problems to be solved by permitting steep gradients to be modelled using

fewer data points Toward that goal our study consists of two parts The first part will investigate

a new numerical technique of curve surface and body approximations of exceptional accuracy over

an arbitrary data arrangement The second part of this study will use such techniques to improve

parabolic hyperbolic or elliptic partial differential equations We will demonstrate that the study

of function approximations has a definite advantage to computational methods for partial

differential equations

One very important use of computers is the simulation of multidimensional spatial processes

In this paper we assumed that some finite physical quantity F is piecewise continuous in some

finite domain In many applications F is known only at a finite number of locations

xk k = 1 2 N where xk = x~ for a univariate problem and Xk = (x~yk )X for the

multivariate problem

From a finite amount of information regarding F we seek the best approximation which can

not only supply accurate estimates of F at arbitrary locations on the domain but will also provide

accurate estimates of the partial derivatives and definite integrals of F anywhere on the domain

The domain of F will consist of points xk of arbitrary ordering and sub-clustering A rectangular

grid is a very special case of a data ordering

Let us assume that an interpolation function f approximates F in the sense that

f(Xk)=F(Xk) k = l 2 N (1)

127

Example

solve linear system for

uniform amp random

choose

reconstruct interpolant

Example

linear system for

Bad news

is dense

is almost singular condition number typically

Must use Singular Value Decomposition expensive

Convergence is very rapid approx exponential

Good news

Bowen-York initial data

Bowen-York initial data

Bowen-York initial data

Pros and cons

Free to place nodes wherever we like

Exponential convergence

Must solve exceedingly ill-conditioned system

Requires care for asymptoticly flat geometries

Only two papers with results volunteers most welcome

Very accurate for very steep functions

Living Rev Relativity 12 (2009) 1httpwwwlivingreviewsorglrr-2009-1

Spectral Methods for Numerical Relativity

Philippe GrandclementLaboratoire Univers et Theories

UMR 8102 du CNRS Observatoire de ParisF-92195 Meudon Cedex France

email PhilippeGrandclementobspmfrhttpwwwluthobspmfrminisitephpnom=Grandclement

Jerome NovakLaboratoire Univers et Theories

UMR 8102 du CNRS Observatoire de ParisF-92195 Meudon Cedex Franceemail JeromeNovakobspmfr

httpwwwluthobspmfrminisitephpnom=Novak

Living Reviews in RelativityISSN 1433-8351

Accepted on 23 October 2008Published on 9 January 2009

Abstract

Equations arising in general relativity are usually too complicated to be solved analyticallyand one must rely on numerical methods to solve sets of coupled partial dierential equationsAmong the possible choices this paper focuses on a class called spectral methods in whichtypically the various functions are expanded in sets of orthogonal polynomials or functionsFirst a theoretical introduction of spectral expansion is given with a particular emphasis on thefast convergence of the spectral approximation We then present dierent approaches to solvingpartial dierential equations first limiting ourselves to the one-dimensional case with one ormore domains Generalization to more dimensions is then discussed In particular the case oftime evolutions is carefully studied and the stability of such evolutions investigated We thenpresent results obtained by various groups in the field of general relativity by means of spectralmethods Work which does not involve explicit time-evolutions is discussed going fromrapidly-rotating strange stars to the computation of black-holendashbinary initial data Finallythe evolution of various systems of astrophysical interest are presented from supernovae corecollapse to black-holendashbinary mergers

This review is licensed under a Creative CommonsAttribution-Non-Commercial-NoDerivs 30 Germany Licensehttpcreativecommonsorglicensesby-nc-nd30de

Spectral methods

A gentle introduction

Spectral interpolants

is a function

is a polynomial approximation of degree to

Convergence3

-1 -05 0 05 1x

-15

-1

-05

0

05

1

u = cos3(x2) - (x+1)

38

Pu

Iu

N=4

-1 -05 0 05 1x

-15

-1

-05

0

05

1

u = cos3(x2) - (x+1)

38

Pu

Iu

N=8

FIG 2 Same as Fig 1 with also the interpolant of u The collocation points are denoted by the circles

3 Interpolation

If one applies the Gauss quadratures to approximate the coecient of the expansion one obtains

un =1

n

N

j=0

u (xj) pn (xj)wj with n =N

j=0

p2n (xj)wj (5)

Let us precise that this is not exact in the sense that un = un However the computation of u only requires toevaluate u at the N + 1 collocation points The interpolant of u is then defined as the following polynomial

INu =N

n=0

unpn (x) (6)

The dierence between Inu and Pnu is called the aliasing error The interpolant of u is the spectral approximate ofu and one can show that it is the only polynomials of degree N that coincides with u at each collocation point

[Inu] (xi) = u (xi) i N (7)

Figure 2 shows the same function as Fig1 but the interpolant is also plotted One can see that indeed INu coincideswith u at the collocation points that are indicated by the circles Once again even with as few points as N = 8 nodierence can be seen between the various functions

Figure 3 shows the maximum dierence between INu and u on as a function of the degree of the approximation N We can observe the very general feature of spectral methods that the error decreases exponentially until one reachesthe machine accuracy (here 1014 the computation being done in double precision) This very fast convergenceexplains why spectral methods are so ecient especially compared to finite dierence ones where the error followsonly a power-law in terms of N We will later be more quantitative about the convergence properties of the spectralapproximation

Let us note that a function u can be described either by its value at each collocation point u (xi) or by the coecientsof the interpolant ui If the values at collocation points are known one is working in the configuration space and inthe coecient space if u is given in terms of its coecients There is a bijection between the two descriptions and onesimply goes from one space to another by using

bull un =1

n

N

j=0

u (xj) pn (xj) wj (configuration $ coecient)

bull u (xn) =N

j=0

ujpj (xn) (coecient $ configuration)

Depending on the operation one has to perform one choice of space is usually more suited than the other Forinstance let us assume that one wants to compute the derivative of u This is easily done if u is known in the

Exponential convergence 4

0 5 10 15 20 25 30 35

Number of coefficients

1e-15

1e-12

1e-09

1e-06

1e-03

1e+00m

ax |I N

u -

u|

FIG 3 Maximum dierence between INu and u as a function of the degree of the approximation N

-1 -05 0 05 1x

-3

-2

-1

0

1

2

ursquoIN

(ursquo)

(IN

u)rsquo

N=4

-1 -05 0 05 1x

-3

-2

-1

0

1

2

ursquoIN

(ursquo)

(IN

u)rsquo

N=8

FIG 4 First derivative of u = cos3 (x2) (x + 1)3 8 interpolant of the derivative and derivative of the interpolant forN = 4 and N = 8

coecient space Indeed one can simply approximate u by the derivative of the interpolant

u (x) [INu] =N

n=0

unpn (x) (8)

Such an approximation only requires the knowledge of the coecients of u and how the basis polynomials are derivedLet us note that the obtained polynomial even if it is a good approximation of u is not the interpolant of u Inother terms the interpolation and the derivation are two operations that do not commute (INu) = IN (u) This isclearly illustrated on Fig 4 where the derivative of u = cos3 (x2) (x + 1)3 8 is plotted along with the functions(INu) and IN (u) In particular one can note that the functions used to represent u ie (INu) does not coincidewith u at the collocation points However even with N = 8 only the three functions can not be distinguished byeye

The maximum dierence between u and (INu) as a function of N is shown on Fig 5 Once again the convergenceis exponential

Gibbs phenomena9

-1 -05 0 05 1x

-02

0

02

04

06

08

1

FIG 8 Step function and some interpolants for N = 7 N = 15 and N = 31 As N increases the maximum dierence staysconstant and the number of oscillations increases

4 5 6 7 8 9 10 20 30 40 50 60 70 8090

N

10-2

10-1

100

-1

1 |I N

u -

u| dx

FIG 9

1

1|INu u| dx as a function of N u being a step function The convergence obeys a power-law which decays slower

than N1

B A test problem

We propose to solve the equation

d2u

dx2 4

du

dx+ 4u = exp (x) + C (25)

with x [1 1] and C = 4e

1 + e2

As boundary conditions we simply ask that the solution is zero at theboundaries

u (1) = 0 and u (1) = 0 (26)

Representations

can be represented by either or

for algebraic operators use

for differential operators use

Differential equations

solve for then recover

Spectral methods in Numerical Relativity

Black hole evolution by spectral methods

Lawrence E Kidder Mark A Scheel and Saul A TeukolskyCenter for Radiophysics and Space Research Cornell University Ithaca New York 14853

Eric D Carlson and Gregory B CookDepartment of Physics Wake Forest University Winston-Salem North Carolina 27109

Received 15 May 2000 published 26 September 2000

Current methods of evolving a spacetime containing one or more black holes are plagued by instabilities that

prohibit long-term evolution Some of these instabilities may be due to the numerical method used tradition-

ally finite differencing In this paper we explore the use of a pseudospectral collocation PSC method for theevolution of a spherically symmetric black hole spacetime in one dimension using a hyperbolic formulation of

Einsteinrsquos equations We demonstrate that our PSC method is able to evolve a spherically symmetric black

hole spacetime forever without enforcing constraints even if we add dynamics via a Klein-Gordon scalar field

We find that in contrast with finite-differencing methods black hole excision is a trivial operation using PSC

applied to a hyperbolic formulation of Einsteinrsquos equations We discuss the extension of this method to three

spatial dimensions

PACS numbers 0425Dm 0270Hm

I INTRODUCTION AND SUMMARY

A major thrust of research in classical general relativity in

the past decade has been to devise algorithms to solve Ein-

steinrsquos equations numerically Despite advances in our ana-

lytic understanding of general relativity we still do not knowwhat all the features of the theory really are Numerical so-lutions will continue to provide fresh insights into the theoryas they have in the past for example the critical behavior inblack hole formation 1$ and the formation of toroidal blackholes 2$New urgency has been injected into numerical relativity

by the imminent deployment of the Laser InterferometricGravitational Wave Observatory LIGO The prime targetfor LIGO is coalescence of binary neutron star and blackhole systems The waveform is reasonably well predicted bythe post-Newtonian approximation when the binary compo-nents are at large separation However extracting the mostimportant physics requires us to be able to deal with fullynon-linear general relativity as the system spirals togetherand coalesces Moreover a number of people believe thatthere is a significant event rate for the coalescence of mas-sive black hole systems (20M) 3$ In this case LIGO ismost sensitive to waves emitted from the strong field regimeIndeed without some theoretical guidance as to what to ex-pect from this regime it is possible we may miss theseevents entirely 4$However the goal of developing a general algorithm that

can solve Einsteinrsquos equations for two black holes has re-mained elusive All attempts to date have been plagued byinstabilities These instabilities are caused by an interplay ofthree factors 1 Einsteinrsquos equations are an overdeterminedsystem with the evolution equations subject to constraintsSo if for example you choose to solve only the evolutionequations then there can be unstable solutions that are in factsolutions of the evolution equations but do not satisfy theconstraints Small numerical errors may cause these solu-tions to appear and swamp the true solution lsquolsquoconstraint-

violating modesrsquorsquo 2 The coordinate freedom inherent in

the theory means that it is very easy to impose coordinate

conditions that lead to numerical instabilities lsquolsquogaugemodesrsquorsquo 3 Experience has shown that the kind of bound-ary conditions we choose and how we implement them can

affect the stability of an algorithm enormously

Similar instabilities have hampered efforts to solve therelated problem of binary neutron stars only very recently56$ has there been some success in finding stable algo-rithms However black hole evolutions face an additionalobstacle that is absent in the case of neutron stars for neu-tron stars the gravitational field is everywhere regular but forblack holes one must somehow deal with the physical singu-larity that lurks inside each holeThere are two main approaches for handling these singu-

larities The first is to use gauge conditions eg maximalslicing that avoid the singularities altogether Such condi-tions however lead to large gradients in the gravitationalfield variables near the horizon These grow exponentially intime and ultimately cannot be resolved by the numerical evo-lution causing the code to crash The alternative approach isto excise the region containing the singularity from the com-putational domain and evolve only the exterior region If theexcision boundary is placed inside the horizon of the blackhole causality assures us that we do not need to impose aphysical boundary condition thereHowever black hole excision is only known to be math-

ematically well posed if the evolution equations are hyper-bolic with characteristic speeds less than or equal to c In thiscase the structure of the equations guarantees that even un-physical modes present in the solution gauge modesconstraint-violating modes behave causally and cannotpropagate out of the horizon For many representations ofgeneral relativity such as the usual Arnowitt-Deser-MisnerADM 7$ formulation the evolution equations are of nomathematical type for which well-posedness has beenproved so the suitability of these formulations for black holeexcision must be determined empirically on a case-by-case

PHYSICAL REVIEW D VOLUME 62 084032

0556-2821200062808403220$1500 copy2000 The American Physical Society62 084032-1

Stable evolutions in 1+1 d

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

Pros and cons

No dissipation

Exponential convergence

Superb results for little effort

Gibbs phenomena

Discrete equations are fully coupled

Lack of dissipation allows high frequency errors to remain

Can not freely choose location of nodes

Regge calculus

Regge calculus

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is piecewise flat

Curvature is a distribution

Defect angle at each vertex

2d example

2d example

2d example

2d example

2d example

3d example

3d example

3d example

3d example

3d example

4d example

Field equations

Some questions

Can smooth metrics be accurately approximated by Regge lattices

Do the Regge equations reduce to the Einstein equations in some suitable limit

Do solutions of the Regge equations converge to solutions of the Einstein equations

Some answers

Regge

Reggesolutions

Yes

Yes as an average

Allendorfer Weyl

Cheeger Muller Schrader

Continuum

Einsteinsolutions

Applications

FRW cosmologies

Brill wave initial data

Quantum gravityKasner cosmologies

Misner initial data

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 2: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Experimental gravity

Experimental gravity

Experimental gravity

Rules of the game

Numerical relativity

Construct discrete solutions of Einsteinrsquos equations

Sounds simple but

Non-standard methods

Numerical Relativity

Regge calculus

Multiquadrics

Spectral methods

Smooth lattices

Tetrad methods Buchman amp BardeenPhysRevD67(2003)084017 van Putten PhysRevD55(1997)4705

Discrete differential forms Frauendiener CQG23(2006)S369 Richter amp Frauendiener CQG24(2007)433

Finite volumes Alic etalPhysRevD76(2007)104007

Finite elements Korobkin etalCQG26(2009)145007 ZambuschCQG26(2009)175011

Multiquadrics

VOL 76 NO 8 7OURNAL OF GEOPHYSICAL RESEARCH MARCH 10 1bull71

Multiquadric Equations of Topography and Other Irregular Surfaces

ROLLAbullD L

Department o] Civil Engineering and Engineering Research Institute Iowa State University Ames 50010

A new analytical method of representing irregular surfaces that involves the summation of equations of quadric surfaces having unknown coefficients is described The quadric surfaces are located at significant points throughout the region to be mapped Procedures are given for solving multiquadric equations of topography that are based on coordinate data Con- toured multiquadric surfaces are compared with topography and other irregular surfaces from which the multiquadric equation was derived

Topography can be represented by various analytical numerical and digital methods in

addition to the classical contour map The

extremes in generalization or detail that result

from use of these methods are perhaps demon-

strated best by Lee and Kaula [1967] and by

Gilbert [1968] Lee and Kaula described the

topography of the whole earth in the form of

thirty-sixth-degree spherical harmonics Gilbert

reported the magnetic tape storage of more

than six million increments of height informa-

tion in digital form measured or interpolated

from one ordinary map sheet In Lee and Kaulas work we have an ex-

treme generalization of existing topographic

information over a wide area by highly analyti-

cal methods whereas Gilberts work is extremely

detailed but scarcely analytical As valuable

as these techniques are in certain cases they

are related more to map utilization than to map

making Basically the problem they solve is

given continuous topographic information in a

certain region reduce it to an equivalent set of

discrete data eg spherical harmonic coeffi-

cients or digital terrain increments

Other investigators including myself are

concerned with a procedural inverse of the

above problem namely given a set of discrete

data on a topographic surface reduce it to a

satisfactory continuous function representing

the topographic surface Practical solutions to

this problem will tend to eliminate the classical

Copyright 1971 by the American Geophysical Union

contour map as the first step in representing terrain information

An equation of topography can be evaluated

digitally or analytically without its having

been reduced to graphical form The same equa-

tion can be treated analytically for the auto-

matic production of contoured maps Automatic

contouring can become a computer-plotter

problem in analytical geometry ie to deter-

mine and plot the intercept equations of hori- zontal planes passed through a three-dimen-

sional equation of topography This approach could also lead to reconsideration of the need

for digitized map data Problems involving map use such as determining unobstructed

lines of sight areas of deftlade volumes of

earth and minimum length of surface curves

may involve the more direct application of

analytical geometry and calculus to the inter-

relationship of these parameters with a mathe-

matical surface of topography For these rea-

sons the question of representing a topographic surface in detail by unique equations deserves increased consideration

bullNUMERICAL SURFACE TECbullNmUES

Fourier and polynomial series approximations have been applied in recent years to both sur- face and subsurface trend analysis in geological

mapping Investigators in this field have been

Krumbein [1966] Mandelbaum [1963] James

[1966] and Merriam and Sheath [1966] There

has been a natural tendency to apply these

trend surface techniques to the problem of

1905

Multiquadrics

Originally used for interpolation on scattered data

Adapted by Kansa to solve ODEs

Given compute such that

Multiquadrics

Computers Math Appfic Vol 19 No 89 pp 127-145 1990 0097-494390 $300 + 000 Printed in Great Britain Pergamon Press plc

M U L T I Q U A D R I C S - - A S C A T T E R E D D A T A

A P P R O X I M A T I O N S C H E M E W I T H A P P L I C A T I O N S T O

C O M P U T A T I O N A L F L U I D - D Y N A M I C S - - I

SURFACE APPROXIMATIONS AND PARTIAL DERIVATIVE ESTIMATES

E J KANSA

Lawrence Livermore National Laboratory L-200 PO Box 808 Livermore CA 94550 USA

A~traet - -We present a powerful enhanced multiquadrics (MQ) scheme developed for spatial approxi- mations MQ is a true scattered data grid free scheme for representing surfaces and bodies in an arbitrary number of dimensions It is continuously differentiable and integrable and is capable of representing functions with steep gradients with very high accuracy Monotonicity and convexity are observed properties as a result of such high accuracy

Numerical results show that our modified MQ scheme is an excellent method not only for very accurate interpolation but also for partial derivative estimates MQ is applied to a higher order arbitrary Lagrangian-Eulerian (ALE) rezoning In the second paper of this series MQ is applied to parabolic hyperbolic and elliptic partial differential equations The parabolic problem uses an implicit time-marching scheme whereas the hyperbolic problem uses an explicit time-marching scheme We show that MQ is also exceptionally accurate and efficient The theory of Madych and Nelson shows that the MQ interpolant belongs to a space of functions which minimizes a semi-norm and gives credence to our results

1 BACKGROUND

The study of arbitrarily shaped curves surfaces and bodies having arbitrary data orderings has

immediate application to computational fluid-dynamics The governing equations not only include

source terms but gradients divergences and Laplacians In addition many physical processes occur

over a wide range of length scales To obtain quantitatively accurate approximations of the physics

quantitatively accurate estimates of the spatial variations of such variables are required In two

and three dimensions the range of such quantitatively accurate problems possible on current

multiprocessing super computers using standard finite difference or finite element codes is limited

The question is whether there exist alternative techniques or combinations of techniques which can

broaden the scope of problems to be solved by permitting steep gradients to be modelled using

fewer data points Toward that goal our study consists of two parts The first part will investigate

a new numerical technique of curve surface and body approximations of exceptional accuracy over

an arbitrary data arrangement The second part of this study will use such techniques to improve

parabolic hyperbolic or elliptic partial differential equations We will demonstrate that the study

of function approximations has a definite advantage to computational methods for partial

differential equations

One very important use of computers is the simulation of multidimensional spatial processes

In this paper we assumed that some finite physical quantity F is piecewise continuous in some

finite domain In many applications F is known only at a finite number of locations

xk k = 1 2 N where xk = x~ for a univariate problem and Xk = (x~yk )X for the

multivariate problem

From a finite amount of information regarding F we seek the best approximation which can

not only supply accurate estimates of F at arbitrary locations on the domain but will also provide

accurate estimates of the partial derivatives and definite integrals of F anywhere on the domain

The domain of F will consist of points xk of arbitrary ordering and sub-clustering A rectangular

grid is a very special case of a data ordering

Let us assume that an interpolation function f approximates F in the sense that

f(Xk)=F(Xk) k = l 2 N (1)

127

Example

solve linear system for

uniform amp random

choose

reconstruct interpolant

Example

linear system for

Bad news

is dense

is almost singular condition number typically

Must use Singular Value Decomposition expensive

Convergence is very rapid approx exponential

Good news

Bowen-York initial data

Bowen-York initial data

Bowen-York initial data

Pros and cons

Free to place nodes wherever we like

Exponential convergence

Must solve exceedingly ill-conditioned system

Requires care for asymptoticly flat geometries

Only two papers with results volunteers most welcome

Very accurate for very steep functions

Living Rev Relativity 12 (2009) 1httpwwwlivingreviewsorglrr-2009-1

Spectral Methods for Numerical Relativity

Philippe GrandclementLaboratoire Univers et Theories

UMR 8102 du CNRS Observatoire de ParisF-92195 Meudon Cedex France

email PhilippeGrandclementobspmfrhttpwwwluthobspmfrminisitephpnom=Grandclement

Jerome NovakLaboratoire Univers et Theories

UMR 8102 du CNRS Observatoire de ParisF-92195 Meudon Cedex Franceemail JeromeNovakobspmfr

httpwwwluthobspmfrminisitephpnom=Novak

Living Reviews in RelativityISSN 1433-8351

Accepted on 23 October 2008Published on 9 January 2009

Abstract

Equations arising in general relativity are usually too complicated to be solved analyticallyand one must rely on numerical methods to solve sets of coupled partial dierential equationsAmong the possible choices this paper focuses on a class called spectral methods in whichtypically the various functions are expanded in sets of orthogonal polynomials or functionsFirst a theoretical introduction of spectral expansion is given with a particular emphasis on thefast convergence of the spectral approximation We then present dierent approaches to solvingpartial dierential equations first limiting ourselves to the one-dimensional case with one ormore domains Generalization to more dimensions is then discussed In particular the case oftime evolutions is carefully studied and the stability of such evolutions investigated We thenpresent results obtained by various groups in the field of general relativity by means of spectralmethods Work which does not involve explicit time-evolutions is discussed going fromrapidly-rotating strange stars to the computation of black-holendashbinary initial data Finallythe evolution of various systems of astrophysical interest are presented from supernovae corecollapse to black-holendashbinary mergers

This review is licensed under a Creative CommonsAttribution-Non-Commercial-NoDerivs 30 Germany Licensehttpcreativecommonsorglicensesby-nc-nd30de

Spectral methods

A gentle introduction

Spectral interpolants

is a function

is a polynomial approximation of degree to

Convergence3

-1 -05 0 05 1x

-15

-1

-05

0

05

1

u = cos3(x2) - (x+1)

38

Pu

Iu

N=4

-1 -05 0 05 1x

-15

-1

-05

0

05

1

u = cos3(x2) - (x+1)

38

Pu

Iu

N=8

FIG 2 Same as Fig 1 with also the interpolant of u The collocation points are denoted by the circles

3 Interpolation

If one applies the Gauss quadratures to approximate the coecient of the expansion one obtains

un =1

n

N

j=0

u (xj) pn (xj)wj with n =N

j=0

p2n (xj)wj (5)

Let us precise that this is not exact in the sense that un = un However the computation of u only requires toevaluate u at the N + 1 collocation points The interpolant of u is then defined as the following polynomial

INu =N

n=0

unpn (x) (6)

The dierence between Inu and Pnu is called the aliasing error The interpolant of u is the spectral approximate ofu and one can show that it is the only polynomials of degree N that coincides with u at each collocation point

[Inu] (xi) = u (xi) i N (7)

Figure 2 shows the same function as Fig1 but the interpolant is also plotted One can see that indeed INu coincideswith u at the collocation points that are indicated by the circles Once again even with as few points as N = 8 nodierence can be seen between the various functions

Figure 3 shows the maximum dierence between INu and u on as a function of the degree of the approximation N We can observe the very general feature of spectral methods that the error decreases exponentially until one reachesthe machine accuracy (here 1014 the computation being done in double precision) This very fast convergenceexplains why spectral methods are so ecient especially compared to finite dierence ones where the error followsonly a power-law in terms of N We will later be more quantitative about the convergence properties of the spectralapproximation

Let us note that a function u can be described either by its value at each collocation point u (xi) or by the coecientsof the interpolant ui If the values at collocation points are known one is working in the configuration space and inthe coecient space if u is given in terms of its coecients There is a bijection between the two descriptions and onesimply goes from one space to another by using

bull un =1

n

N

j=0

u (xj) pn (xj) wj (configuration $ coecient)

bull u (xn) =N

j=0

ujpj (xn) (coecient $ configuration)

Depending on the operation one has to perform one choice of space is usually more suited than the other Forinstance let us assume that one wants to compute the derivative of u This is easily done if u is known in the

Exponential convergence 4

0 5 10 15 20 25 30 35

Number of coefficients

1e-15

1e-12

1e-09

1e-06

1e-03

1e+00m

ax |I N

u -

u|

FIG 3 Maximum dierence between INu and u as a function of the degree of the approximation N

-1 -05 0 05 1x

-3

-2

-1

0

1

2

ursquoIN

(ursquo)

(IN

u)rsquo

N=4

-1 -05 0 05 1x

-3

-2

-1

0

1

2

ursquoIN

(ursquo)

(IN

u)rsquo

N=8

FIG 4 First derivative of u = cos3 (x2) (x + 1)3 8 interpolant of the derivative and derivative of the interpolant forN = 4 and N = 8

coecient space Indeed one can simply approximate u by the derivative of the interpolant

u (x) [INu] =N

n=0

unpn (x) (8)

Such an approximation only requires the knowledge of the coecients of u and how the basis polynomials are derivedLet us note that the obtained polynomial even if it is a good approximation of u is not the interpolant of u Inother terms the interpolation and the derivation are two operations that do not commute (INu) = IN (u) This isclearly illustrated on Fig 4 where the derivative of u = cos3 (x2) (x + 1)3 8 is plotted along with the functions(INu) and IN (u) In particular one can note that the functions used to represent u ie (INu) does not coincidewith u at the collocation points However even with N = 8 only the three functions can not be distinguished byeye

The maximum dierence between u and (INu) as a function of N is shown on Fig 5 Once again the convergenceis exponential

Gibbs phenomena9

-1 -05 0 05 1x

-02

0

02

04

06

08

1

FIG 8 Step function and some interpolants for N = 7 N = 15 and N = 31 As N increases the maximum dierence staysconstant and the number of oscillations increases

4 5 6 7 8 9 10 20 30 40 50 60 70 8090

N

10-2

10-1

100

-1

1 |I N

u -

u| dx

FIG 9

1

1|INu u| dx as a function of N u being a step function The convergence obeys a power-law which decays slower

than N1

B A test problem

We propose to solve the equation

d2u

dx2 4

du

dx+ 4u = exp (x) + C (25)

with x [1 1] and C = 4e

1 + e2

As boundary conditions we simply ask that the solution is zero at theboundaries

u (1) = 0 and u (1) = 0 (26)

Representations

can be represented by either or

for algebraic operators use

for differential operators use

Differential equations

solve for then recover

Spectral methods in Numerical Relativity

Black hole evolution by spectral methods

Lawrence E Kidder Mark A Scheel and Saul A TeukolskyCenter for Radiophysics and Space Research Cornell University Ithaca New York 14853

Eric D Carlson and Gregory B CookDepartment of Physics Wake Forest University Winston-Salem North Carolina 27109

Received 15 May 2000 published 26 September 2000

Current methods of evolving a spacetime containing one or more black holes are plagued by instabilities that

prohibit long-term evolution Some of these instabilities may be due to the numerical method used tradition-

ally finite differencing In this paper we explore the use of a pseudospectral collocation PSC method for theevolution of a spherically symmetric black hole spacetime in one dimension using a hyperbolic formulation of

Einsteinrsquos equations We demonstrate that our PSC method is able to evolve a spherically symmetric black

hole spacetime forever without enforcing constraints even if we add dynamics via a Klein-Gordon scalar field

We find that in contrast with finite-differencing methods black hole excision is a trivial operation using PSC

applied to a hyperbolic formulation of Einsteinrsquos equations We discuss the extension of this method to three

spatial dimensions

PACS numbers 0425Dm 0270Hm

I INTRODUCTION AND SUMMARY

A major thrust of research in classical general relativity in

the past decade has been to devise algorithms to solve Ein-

steinrsquos equations numerically Despite advances in our ana-

lytic understanding of general relativity we still do not knowwhat all the features of the theory really are Numerical so-lutions will continue to provide fresh insights into the theoryas they have in the past for example the critical behavior inblack hole formation 1$ and the formation of toroidal blackholes 2$New urgency has been injected into numerical relativity

by the imminent deployment of the Laser InterferometricGravitational Wave Observatory LIGO The prime targetfor LIGO is coalescence of binary neutron star and blackhole systems The waveform is reasonably well predicted bythe post-Newtonian approximation when the binary compo-nents are at large separation However extracting the mostimportant physics requires us to be able to deal with fullynon-linear general relativity as the system spirals togetherand coalesces Moreover a number of people believe thatthere is a significant event rate for the coalescence of mas-sive black hole systems (20M) 3$ In this case LIGO ismost sensitive to waves emitted from the strong field regimeIndeed without some theoretical guidance as to what to ex-pect from this regime it is possible we may miss theseevents entirely 4$However the goal of developing a general algorithm that

can solve Einsteinrsquos equations for two black holes has re-mained elusive All attempts to date have been plagued byinstabilities These instabilities are caused by an interplay ofthree factors 1 Einsteinrsquos equations are an overdeterminedsystem with the evolution equations subject to constraintsSo if for example you choose to solve only the evolutionequations then there can be unstable solutions that are in factsolutions of the evolution equations but do not satisfy theconstraints Small numerical errors may cause these solu-tions to appear and swamp the true solution lsquolsquoconstraint-

violating modesrsquorsquo 2 The coordinate freedom inherent in

the theory means that it is very easy to impose coordinate

conditions that lead to numerical instabilities lsquolsquogaugemodesrsquorsquo 3 Experience has shown that the kind of bound-ary conditions we choose and how we implement them can

affect the stability of an algorithm enormously

Similar instabilities have hampered efforts to solve therelated problem of binary neutron stars only very recently56$ has there been some success in finding stable algo-rithms However black hole evolutions face an additionalobstacle that is absent in the case of neutron stars for neu-tron stars the gravitational field is everywhere regular but forblack holes one must somehow deal with the physical singu-larity that lurks inside each holeThere are two main approaches for handling these singu-

larities The first is to use gauge conditions eg maximalslicing that avoid the singularities altogether Such condi-tions however lead to large gradients in the gravitationalfield variables near the horizon These grow exponentially intime and ultimately cannot be resolved by the numerical evo-lution causing the code to crash The alternative approach isto excise the region containing the singularity from the com-putational domain and evolve only the exterior region If theexcision boundary is placed inside the horizon of the blackhole causality assures us that we do not need to impose aphysical boundary condition thereHowever black hole excision is only known to be math-

ematically well posed if the evolution equations are hyper-bolic with characteristic speeds less than or equal to c In thiscase the structure of the equations guarantees that even un-physical modes present in the solution gauge modesconstraint-violating modes behave causally and cannotpropagate out of the horizon For many representations ofgeneral relativity such as the usual Arnowitt-Deser-MisnerADM 7$ formulation the evolution equations are of nomathematical type for which well-posedness has beenproved so the suitability of these formulations for black holeexcision must be determined empirically on a case-by-case

PHYSICAL REVIEW D VOLUME 62 084032

0556-2821200062808403220$1500 copy2000 The American Physical Society62 084032-1

Stable evolutions in 1+1 d

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

Pros and cons

No dissipation

Exponential convergence

Superb results for little effort

Gibbs phenomena

Discrete equations are fully coupled

Lack of dissipation allows high frequency errors to remain

Can not freely choose location of nodes

Regge calculus

Regge calculus

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is piecewise flat

Curvature is a distribution

Defect angle at each vertex

2d example

2d example

2d example

2d example

2d example

3d example

3d example

3d example

3d example

3d example

4d example

Field equations

Some questions

Can smooth metrics be accurately approximated by Regge lattices

Do the Regge equations reduce to the Einstein equations in some suitable limit

Do solutions of the Regge equations converge to solutions of the Einstein equations

Some answers

Regge

Reggesolutions

Yes

Yes as an average

Allendorfer Weyl

Cheeger Muller Schrader

Continuum

Einsteinsolutions

Applications

FRW cosmologies

Brill wave initial data

Quantum gravityKasner cosmologies

Misner initial data

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 3: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Experimental gravity

Experimental gravity

Rules of the game

Numerical relativity

Construct discrete solutions of Einsteinrsquos equations

Sounds simple but

Non-standard methods

Numerical Relativity

Regge calculus

Multiquadrics

Spectral methods

Smooth lattices

Tetrad methods Buchman amp BardeenPhysRevD67(2003)084017 van Putten PhysRevD55(1997)4705

Discrete differential forms Frauendiener CQG23(2006)S369 Richter amp Frauendiener CQG24(2007)433

Finite volumes Alic etalPhysRevD76(2007)104007

Finite elements Korobkin etalCQG26(2009)145007 ZambuschCQG26(2009)175011

Multiquadrics

VOL 76 NO 8 7OURNAL OF GEOPHYSICAL RESEARCH MARCH 10 1bull71

Multiquadric Equations of Topography and Other Irregular Surfaces

ROLLAbullD L

Department o] Civil Engineering and Engineering Research Institute Iowa State University Ames 50010

A new analytical method of representing irregular surfaces that involves the summation of equations of quadric surfaces having unknown coefficients is described The quadric surfaces are located at significant points throughout the region to be mapped Procedures are given for solving multiquadric equations of topography that are based on coordinate data Con- toured multiquadric surfaces are compared with topography and other irregular surfaces from which the multiquadric equation was derived

Topography can be represented by various analytical numerical and digital methods in

addition to the classical contour map The

extremes in generalization or detail that result

from use of these methods are perhaps demon-

strated best by Lee and Kaula [1967] and by

Gilbert [1968] Lee and Kaula described the

topography of the whole earth in the form of

thirty-sixth-degree spherical harmonics Gilbert

reported the magnetic tape storage of more

than six million increments of height informa-

tion in digital form measured or interpolated

from one ordinary map sheet In Lee and Kaulas work we have an ex-

treme generalization of existing topographic

information over a wide area by highly analyti-

cal methods whereas Gilberts work is extremely

detailed but scarcely analytical As valuable

as these techniques are in certain cases they

are related more to map utilization than to map

making Basically the problem they solve is

given continuous topographic information in a

certain region reduce it to an equivalent set of

discrete data eg spherical harmonic coeffi-

cients or digital terrain increments

Other investigators including myself are

concerned with a procedural inverse of the

above problem namely given a set of discrete

data on a topographic surface reduce it to a

satisfactory continuous function representing

the topographic surface Practical solutions to

this problem will tend to eliminate the classical

Copyright 1971 by the American Geophysical Union

contour map as the first step in representing terrain information

An equation of topography can be evaluated

digitally or analytically without its having

been reduced to graphical form The same equa-

tion can be treated analytically for the auto-

matic production of contoured maps Automatic

contouring can become a computer-plotter

problem in analytical geometry ie to deter-

mine and plot the intercept equations of hori- zontal planes passed through a three-dimen-

sional equation of topography This approach could also lead to reconsideration of the need

for digitized map data Problems involving map use such as determining unobstructed

lines of sight areas of deftlade volumes of

earth and minimum length of surface curves

may involve the more direct application of

analytical geometry and calculus to the inter-

relationship of these parameters with a mathe-

matical surface of topography For these rea-

sons the question of representing a topographic surface in detail by unique equations deserves increased consideration

bullNUMERICAL SURFACE TECbullNmUES

Fourier and polynomial series approximations have been applied in recent years to both sur- face and subsurface trend analysis in geological

mapping Investigators in this field have been

Krumbein [1966] Mandelbaum [1963] James

[1966] and Merriam and Sheath [1966] There

has been a natural tendency to apply these

trend surface techniques to the problem of

1905

Multiquadrics

Originally used for interpolation on scattered data

Adapted by Kansa to solve ODEs

Given compute such that

Multiquadrics

Computers Math Appfic Vol 19 No 89 pp 127-145 1990 0097-494390 $300 + 000 Printed in Great Britain Pergamon Press plc

M U L T I Q U A D R I C S - - A S C A T T E R E D D A T A

A P P R O X I M A T I O N S C H E M E W I T H A P P L I C A T I O N S T O

C O M P U T A T I O N A L F L U I D - D Y N A M I C S - - I

SURFACE APPROXIMATIONS AND PARTIAL DERIVATIVE ESTIMATES

E J KANSA

Lawrence Livermore National Laboratory L-200 PO Box 808 Livermore CA 94550 USA

A~traet - -We present a powerful enhanced multiquadrics (MQ) scheme developed for spatial approxi- mations MQ is a true scattered data grid free scheme for representing surfaces and bodies in an arbitrary number of dimensions It is continuously differentiable and integrable and is capable of representing functions with steep gradients with very high accuracy Monotonicity and convexity are observed properties as a result of such high accuracy

Numerical results show that our modified MQ scheme is an excellent method not only for very accurate interpolation but also for partial derivative estimates MQ is applied to a higher order arbitrary Lagrangian-Eulerian (ALE) rezoning In the second paper of this series MQ is applied to parabolic hyperbolic and elliptic partial differential equations The parabolic problem uses an implicit time-marching scheme whereas the hyperbolic problem uses an explicit time-marching scheme We show that MQ is also exceptionally accurate and efficient The theory of Madych and Nelson shows that the MQ interpolant belongs to a space of functions which minimizes a semi-norm and gives credence to our results

1 BACKGROUND

The study of arbitrarily shaped curves surfaces and bodies having arbitrary data orderings has

immediate application to computational fluid-dynamics The governing equations not only include

source terms but gradients divergences and Laplacians In addition many physical processes occur

over a wide range of length scales To obtain quantitatively accurate approximations of the physics

quantitatively accurate estimates of the spatial variations of such variables are required In two

and three dimensions the range of such quantitatively accurate problems possible on current

multiprocessing super computers using standard finite difference or finite element codes is limited

The question is whether there exist alternative techniques or combinations of techniques which can

broaden the scope of problems to be solved by permitting steep gradients to be modelled using

fewer data points Toward that goal our study consists of two parts The first part will investigate

a new numerical technique of curve surface and body approximations of exceptional accuracy over

an arbitrary data arrangement The second part of this study will use such techniques to improve

parabolic hyperbolic or elliptic partial differential equations We will demonstrate that the study

of function approximations has a definite advantage to computational methods for partial

differential equations

One very important use of computers is the simulation of multidimensional spatial processes

In this paper we assumed that some finite physical quantity F is piecewise continuous in some

finite domain In many applications F is known only at a finite number of locations

xk k = 1 2 N where xk = x~ for a univariate problem and Xk = (x~yk )X for the

multivariate problem

From a finite amount of information regarding F we seek the best approximation which can

not only supply accurate estimates of F at arbitrary locations on the domain but will also provide

accurate estimates of the partial derivatives and definite integrals of F anywhere on the domain

The domain of F will consist of points xk of arbitrary ordering and sub-clustering A rectangular

grid is a very special case of a data ordering

Let us assume that an interpolation function f approximates F in the sense that

f(Xk)=F(Xk) k = l 2 N (1)

127

Example

solve linear system for

uniform amp random

choose

reconstruct interpolant

Example

linear system for

Bad news

is dense

is almost singular condition number typically

Must use Singular Value Decomposition expensive

Convergence is very rapid approx exponential

Good news

Bowen-York initial data

Bowen-York initial data

Bowen-York initial data

Pros and cons

Free to place nodes wherever we like

Exponential convergence

Must solve exceedingly ill-conditioned system

Requires care for asymptoticly flat geometries

Only two papers with results volunteers most welcome

Very accurate for very steep functions

Living Rev Relativity 12 (2009) 1httpwwwlivingreviewsorglrr-2009-1

Spectral Methods for Numerical Relativity

Philippe GrandclementLaboratoire Univers et Theories

UMR 8102 du CNRS Observatoire de ParisF-92195 Meudon Cedex France

email PhilippeGrandclementobspmfrhttpwwwluthobspmfrminisitephpnom=Grandclement

Jerome NovakLaboratoire Univers et Theories

UMR 8102 du CNRS Observatoire de ParisF-92195 Meudon Cedex Franceemail JeromeNovakobspmfr

httpwwwluthobspmfrminisitephpnom=Novak

Living Reviews in RelativityISSN 1433-8351

Accepted on 23 October 2008Published on 9 January 2009

Abstract

Equations arising in general relativity are usually too complicated to be solved analyticallyand one must rely on numerical methods to solve sets of coupled partial dierential equationsAmong the possible choices this paper focuses on a class called spectral methods in whichtypically the various functions are expanded in sets of orthogonal polynomials or functionsFirst a theoretical introduction of spectral expansion is given with a particular emphasis on thefast convergence of the spectral approximation We then present dierent approaches to solvingpartial dierential equations first limiting ourselves to the one-dimensional case with one ormore domains Generalization to more dimensions is then discussed In particular the case oftime evolutions is carefully studied and the stability of such evolutions investigated We thenpresent results obtained by various groups in the field of general relativity by means of spectralmethods Work which does not involve explicit time-evolutions is discussed going fromrapidly-rotating strange stars to the computation of black-holendashbinary initial data Finallythe evolution of various systems of astrophysical interest are presented from supernovae corecollapse to black-holendashbinary mergers

This review is licensed under a Creative CommonsAttribution-Non-Commercial-NoDerivs 30 Germany Licensehttpcreativecommonsorglicensesby-nc-nd30de

Spectral methods

A gentle introduction

Spectral interpolants

is a function

is a polynomial approximation of degree to

Convergence3

-1 -05 0 05 1x

-15

-1

-05

0

05

1

u = cos3(x2) - (x+1)

38

Pu

Iu

N=4

-1 -05 0 05 1x

-15

-1

-05

0

05

1

u = cos3(x2) - (x+1)

38

Pu

Iu

N=8

FIG 2 Same as Fig 1 with also the interpolant of u The collocation points are denoted by the circles

3 Interpolation

If one applies the Gauss quadratures to approximate the coecient of the expansion one obtains

un =1

n

N

j=0

u (xj) pn (xj)wj with n =N

j=0

p2n (xj)wj (5)

Let us precise that this is not exact in the sense that un = un However the computation of u only requires toevaluate u at the N + 1 collocation points The interpolant of u is then defined as the following polynomial

INu =N

n=0

unpn (x) (6)

The dierence between Inu and Pnu is called the aliasing error The interpolant of u is the spectral approximate ofu and one can show that it is the only polynomials of degree N that coincides with u at each collocation point

[Inu] (xi) = u (xi) i N (7)

Figure 2 shows the same function as Fig1 but the interpolant is also plotted One can see that indeed INu coincideswith u at the collocation points that are indicated by the circles Once again even with as few points as N = 8 nodierence can be seen between the various functions

Figure 3 shows the maximum dierence between INu and u on as a function of the degree of the approximation N We can observe the very general feature of spectral methods that the error decreases exponentially until one reachesthe machine accuracy (here 1014 the computation being done in double precision) This very fast convergenceexplains why spectral methods are so ecient especially compared to finite dierence ones where the error followsonly a power-law in terms of N We will later be more quantitative about the convergence properties of the spectralapproximation

Let us note that a function u can be described either by its value at each collocation point u (xi) or by the coecientsof the interpolant ui If the values at collocation points are known one is working in the configuration space and inthe coecient space if u is given in terms of its coecients There is a bijection between the two descriptions and onesimply goes from one space to another by using

bull un =1

n

N

j=0

u (xj) pn (xj) wj (configuration $ coecient)

bull u (xn) =N

j=0

ujpj (xn) (coecient $ configuration)

Depending on the operation one has to perform one choice of space is usually more suited than the other Forinstance let us assume that one wants to compute the derivative of u This is easily done if u is known in the

Exponential convergence 4

0 5 10 15 20 25 30 35

Number of coefficients

1e-15

1e-12

1e-09

1e-06

1e-03

1e+00m

ax |I N

u -

u|

FIG 3 Maximum dierence between INu and u as a function of the degree of the approximation N

-1 -05 0 05 1x

-3

-2

-1

0

1

2

ursquoIN

(ursquo)

(IN

u)rsquo

N=4

-1 -05 0 05 1x

-3

-2

-1

0

1

2

ursquoIN

(ursquo)

(IN

u)rsquo

N=8

FIG 4 First derivative of u = cos3 (x2) (x + 1)3 8 interpolant of the derivative and derivative of the interpolant forN = 4 and N = 8

coecient space Indeed one can simply approximate u by the derivative of the interpolant

u (x) [INu] =N

n=0

unpn (x) (8)

Such an approximation only requires the knowledge of the coecients of u and how the basis polynomials are derivedLet us note that the obtained polynomial even if it is a good approximation of u is not the interpolant of u Inother terms the interpolation and the derivation are two operations that do not commute (INu) = IN (u) This isclearly illustrated on Fig 4 where the derivative of u = cos3 (x2) (x + 1)3 8 is plotted along with the functions(INu) and IN (u) In particular one can note that the functions used to represent u ie (INu) does not coincidewith u at the collocation points However even with N = 8 only the three functions can not be distinguished byeye

The maximum dierence between u and (INu) as a function of N is shown on Fig 5 Once again the convergenceis exponential

Gibbs phenomena9

-1 -05 0 05 1x

-02

0

02

04

06

08

1

FIG 8 Step function and some interpolants for N = 7 N = 15 and N = 31 As N increases the maximum dierence staysconstant and the number of oscillations increases

4 5 6 7 8 9 10 20 30 40 50 60 70 8090

N

10-2

10-1

100

-1

1 |I N

u -

u| dx

FIG 9

1

1|INu u| dx as a function of N u being a step function The convergence obeys a power-law which decays slower

than N1

B A test problem

We propose to solve the equation

d2u

dx2 4

du

dx+ 4u = exp (x) + C (25)

with x [1 1] and C = 4e

1 + e2

As boundary conditions we simply ask that the solution is zero at theboundaries

u (1) = 0 and u (1) = 0 (26)

Representations

can be represented by either or

for algebraic operators use

for differential operators use

Differential equations

solve for then recover

Spectral methods in Numerical Relativity

Black hole evolution by spectral methods

Lawrence E Kidder Mark A Scheel and Saul A TeukolskyCenter for Radiophysics and Space Research Cornell University Ithaca New York 14853

Eric D Carlson and Gregory B CookDepartment of Physics Wake Forest University Winston-Salem North Carolina 27109

Received 15 May 2000 published 26 September 2000

Current methods of evolving a spacetime containing one or more black holes are plagued by instabilities that

prohibit long-term evolution Some of these instabilities may be due to the numerical method used tradition-

ally finite differencing In this paper we explore the use of a pseudospectral collocation PSC method for theevolution of a spherically symmetric black hole spacetime in one dimension using a hyperbolic formulation of

Einsteinrsquos equations We demonstrate that our PSC method is able to evolve a spherically symmetric black

hole spacetime forever without enforcing constraints even if we add dynamics via a Klein-Gordon scalar field

We find that in contrast with finite-differencing methods black hole excision is a trivial operation using PSC

applied to a hyperbolic formulation of Einsteinrsquos equations We discuss the extension of this method to three

spatial dimensions

PACS numbers 0425Dm 0270Hm

I INTRODUCTION AND SUMMARY

A major thrust of research in classical general relativity in

the past decade has been to devise algorithms to solve Ein-

steinrsquos equations numerically Despite advances in our ana-

lytic understanding of general relativity we still do not knowwhat all the features of the theory really are Numerical so-lutions will continue to provide fresh insights into the theoryas they have in the past for example the critical behavior inblack hole formation 1$ and the formation of toroidal blackholes 2$New urgency has been injected into numerical relativity

by the imminent deployment of the Laser InterferometricGravitational Wave Observatory LIGO The prime targetfor LIGO is coalescence of binary neutron star and blackhole systems The waveform is reasonably well predicted bythe post-Newtonian approximation when the binary compo-nents are at large separation However extracting the mostimportant physics requires us to be able to deal with fullynon-linear general relativity as the system spirals togetherand coalesces Moreover a number of people believe thatthere is a significant event rate for the coalescence of mas-sive black hole systems (20M) 3$ In this case LIGO ismost sensitive to waves emitted from the strong field regimeIndeed without some theoretical guidance as to what to ex-pect from this regime it is possible we may miss theseevents entirely 4$However the goal of developing a general algorithm that

can solve Einsteinrsquos equations for two black holes has re-mained elusive All attempts to date have been plagued byinstabilities These instabilities are caused by an interplay ofthree factors 1 Einsteinrsquos equations are an overdeterminedsystem with the evolution equations subject to constraintsSo if for example you choose to solve only the evolutionequations then there can be unstable solutions that are in factsolutions of the evolution equations but do not satisfy theconstraints Small numerical errors may cause these solu-tions to appear and swamp the true solution lsquolsquoconstraint-

violating modesrsquorsquo 2 The coordinate freedom inherent in

the theory means that it is very easy to impose coordinate

conditions that lead to numerical instabilities lsquolsquogaugemodesrsquorsquo 3 Experience has shown that the kind of bound-ary conditions we choose and how we implement them can

affect the stability of an algorithm enormously

Similar instabilities have hampered efforts to solve therelated problem of binary neutron stars only very recently56$ has there been some success in finding stable algo-rithms However black hole evolutions face an additionalobstacle that is absent in the case of neutron stars for neu-tron stars the gravitational field is everywhere regular but forblack holes one must somehow deal with the physical singu-larity that lurks inside each holeThere are two main approaches for handling these singu-

larities The first is to use gauge conditions eg maximalslicing that avoid the singularities altogether Such condi-tions however lead to large gradients in the gravitationalfield variables near the horizon These grow exponentially intime and ultimately cannot be resolved by the numerical evo-lution causing the code to crash The alternative approach isto excise the region containing the singularity from the com-putational domain and evolve only the exterior region If theexcision boundary is placed inside the horizon of the blackhole causality assures us that we do not need to impose aphysical boundary condition thereHowever black hole excision is only known to be math-

ematically well posed if the evolution equations are hyper-bolic with characteristic speeds less than or equal to c In thiscase the structure of the equations guarantees that even un-physical modes present in the solution gauge modesconstraint-violating modes behave causally and cannotpropagate out of the horizon For many representations ofgeneral relativity such as the usual Arnowitt-Deser-MisnerADM 7$ formulation the evolution equations are of nomathematical type for which well-posedness has beenproved so the suitability of these formulations for black holeexcision must be determined empirically on a case-by-case

PHYSICAL REVIEW D VOLUME 62 084032

0556-2821200062808403220$1500 copy2000 The American Physical Society62 084032-1

Stable evolutions in 1+1 d

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

Pros and cons

No dissipation

Exponential convergence

Superb results for little effort

Gibbs phenomena

Discrete equations are fully coupled

Lack of dissipation allows high frequency errors to remain

Can not freely choose location of nodes

Regge calculus

Regge calculus

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is piecewise flat

Curvature is a distribution

Defect angle at each vertex

2d example

2d example

2d example

2d example

2d example

3d example

3d example

3d example

3d example

3d example

4d example

Field equations

Some questions

Can smooth metrics be accurately approximated by Regge lattices

Do the Regge equations reduce to the Einstein equations in some suitable limit

Do solutions of the Regge equations converge to solutions of the Einstein equations

Some answers

Regge

Reggesolutions

Yes

Yes as an average

Allendorfer Weyl

Cheeger Muller Schrader

Continuum

Einsteinsolutions

Applications

FRW cosmologies

Brill wave initial data

Quantum gravityKasner cosmologies

Misner initial data

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 4: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Experimental gravity

Rules of the game

Numerical relativity

Construct discrete solutions of Einsteinrsquos equations

Sounds simple but

Non-standard methods

Numerical Relativity

Regge calculus

Multiquadrics

Spectral methods

Smooth lattices

Tetrad methods Buchman amp BardeenPhysRevD67(2003)084017 van Putten PhysRevD55(1997)4705

Discrete differential forms Frauendiener CQG23(2006)S369 Richter amp Frauendiener CQG24(2007)433

Finite volumes Alic etalPhysRevD76(2007)104007

Finite elements Korobkin etalCQG26(2009)145007 ZambuschCQG26(2009)175011

Multiquadrics

VOL 76 NO 8 7OURNAL OF GEOPHYSICAL RESEARCH MARCH 10 1bull71

Multiquadric Equations of Topography and Other Irregular Surfaces

ROLLAbullD L

Department o] Civil Engineering and Engineering Research Institute Iowa State University Ames 50010

A new analytical method of representing irregular surfaces that involves the summation of equations of quadric surfaces having unknown coefficients is described The quadric surfaces are located at significant points throughout the region to be mapped Procedures are given for solving multiquadric equations of topography that are based on coordinate data Con- toured multiquadric surfaces are compared with topography and other irregular surfaces from which the multiquadric equation was derived

Topography can be represented by various analytical numerical and digital methods in

addition to the classical contour map The

extremes in generalization or detail that result

from use of these methods are perhaps demon-

strated best by Lee and Kaula [1967] and by

Gilbert [1968] Lee and Kaula described the

topography of the whole earth in the form of

thirty-sixth-degree spherical harmonics Gilbert

reported the magnetic tape storage of more

than six million increments of height informa-

tion in digital form measured or interpolated

from one ordinary map sheet In Lee and Kaulas work we have an ex-

treme generalization of existing topographic

information over a wide area by highly analyti-

cal methods whereas Gilberts work is extremely

detailed but scarcely analytical As valuable

as these techniques are in certain cases they

are related more to map utilization than to map

making Basically the problem they solve is

given continuous topographic information in a

certain region reduce it to an equivalent set of

discrete data eg spherical harmonic coeffi-

cients or digital terrain increments

Other investigators including myself are

concerned with a procedural inverse of the

above problem namely given a set of discrete

data on a topographic surface reduce it to a

satisfactory continuous function representing

the topographic surface Practical solutions to

this problem will tend to eliminate the classical

Copyright 1971 by the American Geophysical Union

contour map as the first step in representing terrain information

An equation of topography can be evaluated

digitally or analytically without its having

been reduced to graphical form The same equa-

tion can be treated analytically for the auto-

matic production of contoured maps Automatic

contouring can become a computer-plotter

problem in analytical geometry ie to deter-

mine and plot the intercept equations of hori- zontal planes passed through a three-dimen-

sional equation of topography This approach could also lead to reconsideration of the need

for digitized map data Problems involving map use such as determining unobstructed

lines of sight areas of deftlade volumes of

earth and minimum length of surface curves

may involve the more direct application of

analytical geometry and calculus to the inter-

relationship of these parameters with a mathe-

matical surface of topography For these rea-

sons the question of representing a topographic surface in detail by unique equations deserves increased consideration

bullNUMERICAL SURFACE TECbullNmUES

Fourier and polynomial series approximations have been applied in recent years to both sur- face and subsurface trend analysis in geological

mapping Investigators in this field have been

Krumbein [1966] Mandelbaum [1963] James

[1966] and Merriam and Sheath [1966] There

has been a natural tendency to apply these

trend surface techniques to the problem of

1905

Multiquadrics

Originally used for interpolation on scattered data

Adapted by Kansa to solve ODEs

Given compute such that

Multiquadrics

Computers Math Appfic Vol 19 No 89 pp 127-145 1990 0097-494390 $300 + 000 Printed in Great Britain Pergamon Press plc

M U L T I Q U A D R I C S - - A S C A T T E R E D D A T A

A P P R O X I M A T I O N S C H E M E W I T H A P P L I C A T I O N S T O

C O M P U T A T I O N A L F L U I D - D Y N A M I C S - - I

SURFACE APPROXIMATIONS AND PARTIAL DERIVATIVE ESTIMATES

E J KANSA

Lawrence Livermore National Laboratory L-200 PO Box 808 Livermore CA 94550 USA

A~traet - -We present a powerful enhanced multiquadrics (MQ) scheme developed for spatial approxi- mations MQ is a true scattered data grid free scheme for representing surfaces and bodies in an arbitrary number of dimensions It is continuously differentiable and integrable and is capable of representing functions with steep gradients with very high accuracy Monotonicity and convexity are observed properties as a result of such high accuracy

Numerical results show that our modified MQ scheme is an excellent method not only for very accurate interpolation but also for partial derivative estimates MQ is applied to a higher order arbitrary Lagrangian-Eulerian (ALE) rezoning In the second paper of this series MQ is applied to parabolic hyperbolic and elliptic partial differential equations The parabolic problem uses an implicit time-marching scheme whereas the hyperbolic problem uses an explicit time-marching scheme We show that MQ is also exceptionally accurate and efficient The theory of Madych and Nelson shows that the MQ interpolant belongs to a space of functions which minimizes a semi-norm and gives credence to our results

1 BACKGROUND

The study of arbitrarily shaped curves surfaces and bodies having arbitrary data orderings has

immediate application to computational fluid-dynamics The governing equations not only include

source terms but gradients divergences and Laplacians In addition many physical processes occur

over a wide range of length scales To obtain quantitatively accurate approximations of the physics

quantitatively accurate estimates of the spatial variations of such variables are required In two

and three dimensions the range of such quantitatively accurate problems possible on current

multiprocessing super computers using standard finite difference or finite element codes is limited

The question is whether there exist alternative techniques or combinations of techniques which can

broaden the scope of problems to be solved by permitting steep gradients to be modelled using

fewer data points Toward that goal our study consists of two parts The first part will investigate

a new numerical technique of curve surface and body approximations of exceptional accuracy over

an arbitrary data arrangement The second part of this study will use such techniques to improve

parabolic hyperbolic or elliptic partial differential equations We will demonstrate that the study

of function approximations has a definite advantage to computational methods for partial

differential equations

One very important use of computers is the simulation of multidimensional spatial processes

In this paper we assumed that some finite physical quantity F is piecewise continuous in some

finite domain In many applications F is known only at a finite number of locations

xk k = 1 2 N where xk = x~ for a univariate problem and Xk = (x~yk )X for the

multivariate problem

From a finite amount of information regarding F we seek the best approximation which can

not only supply accurate estimates of F at arbitrary locations on the domain but will also provide

accurate estimates of the partial derivatives and definite integrals of F anywhere on the domain

The domain of F will consist of points xk of arbitrary ordering and sub-clustering A rectangular

grid is a very special case of a data ordering

Let us assume that an interpolation function f approximates F in the sense that

f(Xk)=F(Xk) k = l 2 N (1)

127

Example

solve linear system for

uniform amp random

choose

reconstruct interpolant

Example

linear system for

Bad news

is dense

is almost singular condition number typically

Must use Singular Value Decomposition expensive

Convergence is very rapid approx exponential

Good news

Bowen-York initial data

Bowen-York initial data

Bowen-York initial data

Pros and cons

Free to place nodes wherever we like

Exponential convergence

Must solve exceedingly ill-conditioned system

Requires care for asymptoticly flat geometries

Only two papers with results volunteers most welcome

Very accurate for very steep functions

Living Rev Relativity 12 (2009) 1httpwwwlivingreviewsorglrr-2009-1

Spectral Methods for Numerical Relativity

Philippe GrandclementLaboratoire Univers et Theories

UMR 8102 du CNRS Observatoire de ParisF-92195 Meudon Cedex France

email PhilippeGrandclementobspmfrhttpwwwluthobspmfrminisitephpnom=Grandclement

Jerome NovakLaboratoire Univers et Theories

UMR 8102 du CNRS Observatoire de ParisF-92195 Meudon Cedex Franceemail JeromeNovakobspmfr

httpwwwluthobspmfrminisitephpnom=Novak

Living Reviews in RelativityISSN 1433-8351

Accepted on 23 October 2008Published on 9 January 2009

Abstract

Equations arising in general relativity are usually too complicated to be solved analyticallyand one must rely on numerical methods to solve sets of coupled partial dierential equationsAmong the possible choices this paper focuses on a class called spectral methods in whichtypically the various functions are expanded in sets of orthogonal polynomials or functionsFirst a theoretical introduction of spectral expansion is given with a particular emphasis on thefast convergence of the spectral approximation We then present dierent approaches to solvingpartial dierential equations first limiting ourselves to the one-dimensional case with one ormore domains Generalization to more dimensions is then discussed In particular the case oftime evolutions is carefully studied and the stability of such evolutions investigated We thenpresent results obtained by various groups in the field of general relativity by means of spectralmethods Work which does not involve explicit time-evolutions is discussed going fromrapidly-rotating strange stars to the computation of black-holendashbinary initial data Finallythe evolution of various systems of astrophysical interest are presented from supernovae corecollapse to black-holendashbinary mergers

This review is licensed under a Creative CommonsAttribution-Non-Commercial-NoDerivs 30 Germany Licensehttpcreativecommonsorglicensesby-nc-nd30de

Spectral methods

A gentle introduction

Spectral interpolants

is a function

is a polynomial approximation of degree to

Convergence3

-1 -05 0 05 1x

-15

-1

-05

0

05

1

u = cos3(x2) - (x+1)

38

Pu

Iu

N=4

-1 -05 0 05 1x

-15

-1

-05

0

05

1

u = cos3(x2) - (x+1)

38

Pu

Iu

N=8

FIG 2 Same as Fig 1 with also the interpolant of u The collocation points are denoted by the circles

3 Interpolation

If one applies the Gauss quadratures to approximate the coecient of the expansion one obtains

un =1

n

N

j=0

u (xj) pn (xj)wj with n =N

j=0

p2n (xj)wj (5)

Let us precise that this is not exact in the sense that un = un However the computation of u only requires toevaluate u at the N + 1 collocation points The interpolant of u is then defined as the following polynomial

INu =N

n=0

unpn (x) (6)

The dierence between Inu and Pnu is called the aliasing error The interpolant of u is the spectral approximate ofu and one can show that it is the only polynomials of degree N that coincides with u at each collocation point

[Inu] (xi) = u (xi) i N (7)

Figure 2 shows the same function as Fig1 but the interpolant is also plotted One can see that indeed INu coincideswith u at the collocation points that are indicated by the circles Once again even with as few points as N = 8 nodierence can be seen between the various functions

Figure 3 shows the maximum dierence between INu and u on as a function of the degree of the approximation N We can observe the very general feature of spectral methods that the error decreases exponentially until one reachesthe machine accuracy (here 1014 the computation being done in double precision) This very fast convergenceexplains why spectral methods are so ecient especially compared to finite dierence ones where the error followsonly a power-law in terms of N We will later be more quantitative about the convergence properties of the spectralapproximation

Let us note that a function u can be described either by its value at each collocation point u (xi) or by the coecientsof the interpolant ui If the values at collocation points are known one is working in the configuration space and inthe coecient space if u is given in terms of its coecients There is a bijection between the two descriptions and onesimply goes from one space to another by using

bull un =1

n

N

j=0

u (xj) pn (xj) wj (configuration $ coecient)

bull u (xn) =N

j=0

ujpj (xn) (coecient $ configuration)

Depending on the operation one has to perform one choice of space is usually more suited than the other Forinstance let us assume that one wants to compute the derivative of u This is easily done if u is known in the

Exponential convergence 4

0 5 10 15 20 25 30 35

Number of coefficients

1e-15

1e-12

1e-09

1e-06

1e-03

1e+00m

ax |I N

u -

u|

FIG 3 Maximum dierence between INu and u as a function of the degree of the approximation N

-1 -05 0 05 1x

-3

-2

-1

0

1

2

ursquoIN

(ursquo)

(IN

u)rsquo

N=4

-1 -05 0 05 1x

-3

-2

-1

0

1

2

ursquoIN

(ursquo)

(IN

u)rsquo

N=8

FIG 4 First derivative of u = cos3 (x2) (x + 1)3 8 interpolant of the derivative and derivative of the interpolant forN = 4 and N = 8

coecient space Indeed one can simply approximate u by the derivative of the interpolant

u (x) [INu] =N

n=0

unpn (x) (8)

Such an approximation only requires the knowledge of the coecients of u and how the basis polynomials are derivedLet us note that the obtained polynomial even if it is a good approximation of u is not the interpolant of u Inother terms the interpolation and the derivation are two operations that do not commute (INu) = IN (u) This isclearly illustrated on Fig 4 where the derivative of u = cos3 (x2) (x + 1)3 8 is plotted along with the functions(INu) and IN (u) In particular one can note that the functions used to represent u ie (INu) does not coincidewith u at the collocation points However even with N = 8 only the three functions can not be distinguished byeye

The maximum dierence between u and (INu) as a function of N is shown on Fig 5 Once again the convergenceis exponential

Gibbs phenomena9

-1 -05 0 05 1x

-02

0

02

04

06

08

1

FIG 8 Step function and some interpolants for N = 7 N = 15 and N = 31 As N increases the maximum dierence staysconstant and the number of oscillations increases

4 5 6 7 8 9 10 20 30 40 50 60 70 8090

N

10-2

10-1

100

-1

1 |I N

u -

u| dx

FIG 9

1

1|INu u| dx as a function of N u being a step function The convergence obeys a power-law which decays slower

than N1

B A test problem

We propose to solve the equation

d2u

dx2 4

du

dx+ 4u = exp (x) + C (25)

with x [1 1] and C = 4e

1 + e2

As boundary conditions we simply ask that the solution is zero at theboundaries

u (1) = 0 and u (1) = 0 (26)

Representations

can be represented by either or

for algebraic operators use

for differential operators use

Differential equations

solve for then recover

Spectral methods in Numerical Relativity

Black hole evolution by spectral methods

Lawrence E Kidder Mark A Scheel and Saul A TeukolskyCenter for Radiophysics and Space Research Cornell University Ithaca New York 14853

Eric D Carlson and Gregory B CookDepartment of Physics Wake Forest University Winston-Salem North Carolina 27109

Received 15 May 2000 published 26 September 2000

Current methods of evolving a spacetime containing one or more black holes are plagued by instabilities that

prohibit long-term evolution Some of these instabilities may be due to the numerical method used tradition-

ally finite differencing In this paper we explore the use of a pseudospectral collocation PSC method for theevolution of a spherically symmetric black hole spacetime in one dimension using a hyperbolic formulation of

Einsteinrsquos equations We demonstrate that our PSC method is able to evolve a spherically symmetric black

hole spacetime forever without enforcing constraints even if we add dynamics via a Klein-Gordon scalar field

We find that in contrast with finite-differencing methods black hole excision is a trivial operation using PSC

applied to a hyperbolic formulation of Einsteinrsquos equations We discuss the extension of this method to three

spatial dimensions

PACS numbers 0425Dm 0270Hm

I INTRODUCTION AND SUMMARY

A major thrust of research in classical general relativity in

the past decade has been to devise algorithms to solve Ein-

steinrsquos equations numerically Despite advances in our ana-

lytic understanding of general relativity we still do not knowwhat all the features of the theory really are Numerical so-lutions will continue to provide fresh insights into the theoryas they have in the past for example the critical behavior inblack hole formation 1$ and the formation of toroidal blackholes 2$New urgency has been injected into numerical relativity

by the imminent deployment of the Laser InterferometricGravitational Wave Observatory LIGO The prime targetfor LIGO is coalescence of binary neutron star and blackhole systems The waveform is reasonably well predicted bythe post-Newtonian approximation when the binary compo-nents are at large separation However extracting the mostimportant physics requires us to be able to deal with fullynon-linear general relativity as the system spirals togetherand coalesces Moreover a number of people believe thatthere is a significant event rate for the coalescence of mas-sive black hole systems (20M) 3$ In this case LIGO ismost sensitive to waves emitted from the strong field regimeIndeed without some theoretical guidance as to what to ex-pect from this regime it is possible we may miss theseevents entirely 4$However the goal of developing a general algorithm that

can solve Einsteinrsquos equations for two black holes has re-mained elusive All attempts to date have been plagued byinstabilities These instabilities are caused by an interplay ofthree factors 1 Einsteinrsquos equations are an overdeterminedsystem with the evolution equations subject to constraintsSo if for example you choose to solve only the evolutionequations then there can be unstable solutions that are in factsolutions of the evolution equations but do not satisfy theconstraints Small numerical errors may cause these solu-tions to appear and swamp the true solution lsquolsquoconstraint-

violating modesrsquorsquo 2 The coordinate freedom inherent in

the theory means that it is very easy to impose coordinate

conditions that lead to numerical instabilities lsquolsquogaugemodesrsquorsquo 3 Experience has shown that the kind of bound-ary conditions we choose and how we implement them can

affect the stability of an algorithm enormously

Similar instabilities have hampered efforts to solve therelated problem of binary neutron stars only very recently56$ has there been some success in finding stable algo-rithms However black hole evolutions face an additionalobstacle that is absent in the case of neutron stars for neu-tron stars the gravitational field is everywhere regular but forblack holes one must somehow deal with the physical singu-larity that lurks inside each holeThere are two main approaches for handling these singu-

larities The first is to use gauge conditions eg maximalslicing that avoid the singularities altogether Such condi-tions however lead to large gradients in the gravitationalfield variables near the horizon These grow exponentially intime and ultimately cannot be resolved by the numerical evo-lution causing the code to crash The alternative approach isto excise the region containing the singularity from the com-putational domain and evolve only the exterior region If theexcision boundary is placed inside the horizon of the blackhole causality assures us that we do not need to impose aphysical boundary condition thereHowever black hole excision is only known to be math-

ematically well posed if the evolution equations are hyper-bolic with characteristic speeds less than or equal to c In thiscase the structure of the equations guarantees that even un-physical modes present in the solution gauge modesconstraint-violating modes behave causally and cannotpropagate out of the horizon For many representations ofgeneral relativity such as the usual Arnowitt-Deser-MisnerADM 7$ formulation the evolution equations are of nomathematical type for which well-posedness has beenproved so the suitability of these formulations for black holeexcision must be determined empirically on a case-by-case

PHYSICAL REVIEW D VOLUME 62 084032

0556-2821200062808403220$1500 copy2000 The American Physical Society62 084032-1

Stable evolutions in 1+1 d

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

Pros and cons

No dissipation

Exponential convergence

Superb results for little effort

Gibbs phenomena

Discrete equations are fully coupled

Lack of dissipation allows high frequency errors to remain

Can not freely choose location of nodes

Regge calculus

Regge calculus

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is piecewise flat

Curvature is a distribution

Defect angle at each vertex

2d example

2d example

2d example

2d example

2d example

3d example

3d example

3d example

3d example

3d example

4d example

Field equations

Some questions

Can smooth metrics be accurately approximated by Regge lattices

Do the Regge equations reduce to the Einstein equations in some suitable limit

Do solutions of the Regge equations converge to solutions of the Einstein equations

Some answers

Regge

Reggesolutions

Yes

Yes as an average

Allendorfer Weyl

Cheeger Muller Schrader

Continuum

Einsteinsolutions

Applications

FRW cosmologies

Brill wave initial data

Quantum gravityKasner cosmologies

Misner initial data

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 5: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Rules of the game

Numerical relativity

Construct discrete solutions of Einsteinrsquos equations

Sounds simple but

Non-standard methods

Numerical Relativity

Regge calculus

Multiquadrics

Spectral methods

Smooth lattices

Tetrad methods Buchman amp BardeenPhysRevD67(2003)084017 van Putten PhysRevD55(1997)4705

Discrete differential forms Frauendiener CQG23(2006)S369 Richter amp Frauendiener CQG24(2007)433

Finite volumes Alic etalPhysRevD76(2007)104007

Finite elements Korobkin etalCQG26(2009)145007 ZambuschCQG26(2009)175011

Multiquadrics

VOL 76 NO 8 7OURNAL OF GEOPHYSICAL RESEARCH MARCH 10 1bull71

Multiquadric Equations of Topography and Other Irregular Surfaces

ROLLAbullD L

Department o] Civil Engineering and Engineering Research Institute Iowa State University Ames 50010

A new analytical method of representing irregular surfaces that involves the summation of equations of quadric surfaces having unknown coefficients is described The quadric surfaces are located at significant points throughout the region to be mapped Procedures are given for solving multiquadric equations of topography that are based on coordinate data Con- toured multiquadric surfaces are compared with topography and other irregular surfaces from which the multiquadric equation was derived

Topography can be represented by various analytical numerical and digital methods in

addition to the classical contour map The

extremes in generalization or detail that result

from use of these methods are perhaps demon-

strated best by Lee and Kaula [1967] and by

Gilbert [1968] Lee and Kaula described the

topography of the whole earth in the form of

thirty-sixth-degree spherical harmonics Gilbert

reported the magnetic tape storage of more

than six million increments of height informa-

tion in digital form measured or interpolated

from one ordinary map sheet In Lee and Kaulas work we have an ex-

treme generalization of existing topographic

information over a wide area by highly analyti-

cal methods whereas Gilberts work is extremely

detailed but scarcely analytical As valuable

as these techniques are in certain cases they

are related more to map utilization than to map

making Basically the problem they solve is

given continuous topographic information in a

certain region reduce it to an equivalent set of

discrete data eg spherical harmonic coeffi-

cients or digital terrain increments

Other investigators including myself are

concerned with a procedural inverse of the

above problem namely given a set of discrete

data on a topographic surface reduce it to a

satisfactory continuous function representing

the topographic surface Practical solutions to

this problem will tend to eliminate the classical

Copyright 1971 by the American Geophysical Union

contour map as the first step in representing terrain information

An equation of topography can be evaluated

digitally or analytically without its having

been reduced to graphical form The same equa-

tion can be treated analytically for the auto-

matic production of contoured maps Automatic

contouring can become a computer-plotter

problem in analytical geometry ie to deter-

mine and plot the intercept equations of hori- zontal planes passed through a three-dimen-

sional equation of topography This approach could also lead to reconsideration of the need

for digitized map data Problems involving map use such as determining unobstructed

lines of sight areas of deftlade volumes of

earth and minimum length of surface curves

may involve the more direct application of

analytical geometry and calculus to the inter-

relationship of these parameters with a mathe-

matical surface of topography For these rea-

sons the question of representing a topographic surface in detail by unique equations deserves increased consideration

bullNUMERICAL SURFACE TECbullNmUES

Fourier and polynomial series approximations have been applied in recent years to both sur- face and subsurface trend analysis in geological

mapping Investigators in this field have been

Krumbein [1966] Mandelbaum [1963] James

[1966] and Merriam and Sheath [1966] There

has been a natural tendency to apply these

trend surface techniques to the problem of

1905

Multiquadrics

Originally used for interpolation on scattered data

Adapted by Kansa to solve ODEs

Given compute such that

Multiquadrics

Computers Math Appfic Vol 19 No 89 pp 127-145 1990 0097-494390 $300 + 000 Printed in Great Britain Pergamon Press plc

M U L T I Q U A D R I C S - - A S C A T T E R E D D A T A

A P P R O X I M A T I O N S C H E M E W I T H A P P L I C A T I O N S T O

C O M P U T A T I O N A L F L U I D - D Y N A M I C S - - I

SURFACE APPROXIMATIONS AND PARTIAL DERIVATIVE ESTIMATES

E J KANSA

Lawrence Livermore National Laboratory L-200 PO Box 808 Livermore CA 94550 USA

A~traet - -We present a powerful enhanced multiquadrics (MQ) scheme developed for spatial approxi- mations MQ is a true scattered data grid free scheme for representing surfaces and bodies in an arbitrary number of dimensions It is continuously differentiable and integrable and is capable of representing functions with steep gradients with very high accuracy Monotonicity and convexity are observed properties as a result of such high accuracy

Numerical results show that our modified MQ scheme is an excellent method not only for very accurate interpolation but also for partial derivative estimates MQ is applied to a higher order arbitrary Lagrangian-Eulerian (ALE) rezoning In the second paper of this series MQ is applied to parabolic hyperbolic and elliptic partial differential equations The parabolic problem uses an implicit time-marching scheme whereas the hyperbolic problem uses an explicit time-marching scheme We show that MQ is also exceptionally accurate and efficient The theory of Madych and Nelson shows that the MQ interpolant belongs to a space of functions which minimizes a semi-norm and gives credence to our results

1 BACKGROUND

The study of arbitrarily shaped curves surfaces and bodies having arbitrary data orderings has

immediate application to computational fluid-dynamics The governing equations not only include

source terms but gradients divergences and Laplacians In addition many physical processes occur

over a wide range of length scales To obtain quantitatively accurate approximations of the physics

quantitatively accurate estimates of the spatial variations of such variables are required In two

and three dimensions the range of such quantitatively accurate problems possible on current

multiprocessing super computers using standard finite difference or finite element codes is limited

The question is whether there exist alternative techniques or combinations of techniques which can

broaden the scope of problems to be solved by permitting steep gradients to be modelled using

fewer data points Toward that goal our study consists of two parts The first part will investigate

a new numerical technique of curve surface and body approximations of exceptional accuracy over

an arbitrary data arrangement The second part of this study will use such techniques to improve

parabolic hyperbolic or elliptic partial differential equations We will demonstrate that the study

of function approximations has a definite advantage to computational methods for partial

differential equations

One very important use of computers is the simulation of multidimensional spatial processes

In this paper we assumed that some finite physical quantity F is piecewise continuous in some

finite domain In many applications F is known only at a finite number of locations

xk k = 1 2 N where xk = x~ for a univariate problem and Xk = (x~yk )X for the

multivariate problem

From a finite amount of information regarding F we seek the best approximation which can

not only supply accurate estimates of F at arbitrary locations on the domain but will also provide

accurate estimates of the partial derivatives and definite integrals of F anywhere on the domain

The domain of F will consist of points xk of arbitrary ordering and sub-clustering A rectangular

grid is a very special case of a data ordering

Let us assume that an interpolation function f approximates F in the sense that

f(Xk)=F(Xk) k = l 2 N (1)

127

Example

solve linear system for

uniform amp random

choose

reconstruct interpolant

Example

linear system for

Bad news

is dense

is almost singular condition number typically

Must use Singular Value Decomposition expensive

Convergence is very rapid approx exponential

Good news

Bowen-York initial data

Bowen-York initial data

Bowen-York initial data

Pros and cons

Free to place nodes wherever we like

Exponential convergence

Must solve exceedingly ill-conditioned system

Requires care for asymptoticly flat geometries

Only two papers with results volunteers most welcome

Very accurate for very steep functions

Living Rev Relativity 12 (2009) 1httpwwwlivingreviewsorglrr-2009-1

Spectral Methods for Numerical Relativity

Philippe GrandclementLaboratoire Univers et Theories

UMR 8102 du CNRS Observatoire de ParisF-92195 Meudon Cedex France

email PhilippeGrandclementobspmfrhttpwwwluthobspmfrminisitephpnom=Grandclement

Jerome NovakLaboratoire Univers et Theories

UMR 8102 du CNRS Observatoire de ParisF-92195 Meudon Cedex Franceemail JeromeNovakobspmfr

httpwwwluthobspmfrminisitephpnom=Novak

Living Reviews in RelativityISSN 1433-8351

Accepted on 23 October 2008Published on 9 January 2009

Abstract

Equations arising in general relativity are usually too complicated to be solved analyticallyand one must rely on numerical methods to solve sets of coupled partial dierential equationsAmong the possible choices this paper focuses on a class called spectral methods in whichtypically the various functions are expanded in sets of orthogonal polynomials or functionsFirst a theoretical introduction of spectral expansion is given with a particular emphasis on thefast convergence of the spectral approximation We then present dierent approaches to solvingpartial dierential equations first limiting ourselves to the one-dimensional case with one ormore domains Generalization to more dimensions is then discussed In particular the case oftime evolutions is carefully studied and the stability of such evolutions investigated We thenpresent results obtained by various groups in the field of general relativity by means of spectralmethods Work which does not involve explicit time-evolutions is discussed going fromrapidly-rotating strange stars to the computation of black-holendashbinary initial data Finallythe evolution of various systems of astrophysical interest are presented from supernovae corecollapse to black-holendashbinary mergers

This review is licensed under a Creative CommonsAttribution-Non-Commercial-NoDerivs 30 Germany Licensehttpcreativecommonsorglicensesby-nc-nd30de

Spectral methods

A gentle introduction

Spectral interpolants

is a function

is a polynomial approximation of degree to

Convergence3

-1 -05 0 05 1x

-15

-1

-05

0

05

1

u = cos3(x2) - (x+1)

38

Pu

Iu

N=4

-1 -05 0 05 1x

-15

-1

-05

0

05

1

u = cos3(x2) - (x+1)

38

Pu

Iu

N=8

FIG 2 Same as Fig 1 with also the interpolant of u The collocation points are denoted by the circles

3 Interpolation

If one applies the Gauss quadratures to approximate the coecient of the expansion one obtains

un =1

n

N

j=0

u (xj) pn (xj)wj with n =N

j=0

p2n (xj)wj (5)

Let us precise that this is not exact in the sense that un = un However the computation of u only requires toevaluate u at the N + 1 collocation points The interpolant of u is then defined as the following polynomial

INu =N

n=0

unpn (x) (6)

The dierence between Inu and Pnu is called the aliasing error The interpolant of u is the spectral approximate ofu and one can show that it is the only polynomials of degree N that coincides with u at each collocation point

[Inu] (xi) = u (xi) i N (7)

Figure 2 shows the same function as Fig1 but the interpolant is also plotted One can see that indeed INu coincideswith u at the collocation points that are indicated by the circles Once again even with as few points as N = 8 nodierence can be seen between the various functions

Figure 3 shows the maximum dierence between INu and u on as a function of the degree of the approximation N We can observe the very general feature of spectral methods that the error decreases exponentially until one reachesthe machine accuracy (here 1014 the computation being done in double precision) This very fast convergenceexplains why spectral methods are so ecient especially compared to finite dierence ones where the error followsonly a power-law in terms of N We will later be more quantitative about the convergence properties of the spectralapproximation

Let us note that a function u can be described either by its value at each collocation point u (xi) or by the coecientsof the interpolant ui If the values at collocation points are known one is working in the configuration space and inthe coecient space if u is given in terms of its coecients There is a bijection between the two descriptions and onesimply goes from one space to another by using

bull un =1

n

N

j=0

u (xj) pn (xj) wj (configuration $ coecient)

bull u (xn) =N

j=0

ujpj (xn) (coecient $ configuration)

Depending on the operation one has to perform one choice of space is usually more suited than the other Forinstance let us assume that one wants to compute the derivative of u This is easily done if u is known in the

Exponential convergence 4

0 5 10 15 20 25 30 35

Number of coefficients

1e-15

1e-12

1e-09

1e-06

1e-03

1e+00m

ax |I N

u -

u|

FIG 3 Maximum dierence between INu and u as a function of the degree of the approximation N

-1 -05 0 05 1x

-3

-2

-1

0

1

2

ursquoIN

(ursquo)

(IN

u)rsquo

N=4

-1 -05 0 05 1x

-3

-2

-1

0

1

2

ursquoIN

(ursquo)

(IN

u)rsquo

N=8

FIG 4 First derivative of u = cos3 (x2) (x + 1)3 8 interpolant of the derivative and derivative of the interpolant forN = 4 and N = 8

coecient space Indeed one can simply approximate u by the derivative of the interpolant

u (x) [INu] =N

n=0

unpn (x) (8)

Such an approximation only requires the knowledge of the coecients of u and how the basis polynomials are derivedLet us note that the obtained polynomial even if it is a good approximation of u is not the interpolant of u Inother terms the interpolation and the derivation are two operations that do not commute (INu) = IN (u) This isclearly illustrated on Fig 4 where the derivative of u = cos3 (x2) (x + 1)3 8 is plotted along with the functions(INu) and IN (u) In particular one can note that the functions used to represent u ie (INu) does not coincidewith u at the collocation points However even with N = 8 only the three functions can not be distinguished byeye

The maximum dierence between u and (INu) as a function of N is shown on Fig 5 Once again the convergenceis exponential

Gibbs phenomena9

-1 -05 0 05 1x

-02

0

02

04

06

08

1

FIG 8 Step function and some interpolants for N = 7 N = 15 and N = 31 As N increases the maximum dierence staysconstant and the number of oscillations increases

4 5 6 7 8 9 10 20 30 40 50 60 70 8090

N

10-2

10-1

100

-1

1 |I N

u -

u| dx

FIG 9

1

1|INu u| dx as a function of N u being a step function The convergence obeys a power-law which decays slower

than N1

B A test problem

We propose to solve the equation

d2u

dx2 4

du

dx+ 4u = exp (x) + C (25)

with x [1 1] and C = 4e

1 + e2

As boundary conditions we simply ask that the solution is zero at theboundaries

u (1) = 0 and u (1) = 0 (26)

Representations

can be represented by either or

for algebraic operators use

for differential operators use

Differential equations

solve for then recover

Spectral methods in Numerical Relativity

Black hole evolution by spectral methods

Lawrence E Kidder Mark A Scheel and Saul A TeukolskyCenter for Radiophysics and Space Research Cornell University Ithaca New York 14853

Eric D Carlson and Gregory B CookDepartment of Physics Wake Forest University Winston-Salem North Carolina 27109

Received 15 May 2000 published 26 September 2000

Current methods of evolving a spacetime containing one or more black holes are plagued by instabilities that

prohibit long-term evolution Some of these instabilities may be due to the numerical method used tradition-

ally finite differencing In this paper we explore the use of a pseudospectral collocation PSC method for theevolution of a spherically symmetric black hole spacetime in one dimension using a hyperbolic formulation of

Einsteinrsquos equations We demonstrate that our PSC method is able to evolve a spherically symmetric black

hole spacetime forever without enforcing constraints even if we add dynamics via a Klein-Gordon scalar field

We find that in contrast with finite-differencing methods black hole excision is a trivial operation using PSC

applied to a hyperbolic formulation of Einsteinrsquos equations We discuss the extension of this method to three

spatial dimensions

PACS numbers 0425Dm 0270Hm

I INTRODUCTION AND SUMMARY

A major thrust of research in classical general relativity in

the past decade has been to devise algorithms to solve Ein-

steinrsquos equations numerically Despite advances in our ana-

lytic understanding of general relativity we still do not knowwhat all the features of the theory really are Numerical so-lutions will continue to provide fresh insights into the theoryas they have in the past for example the critical behavior inblack hole formation 1$ and the formation of toroidal blackholes 2$New urgency has been injected into numerical relativity

by the imminent deployment of the Laser InterferometricGravitational Wave Observatory LIGO The prime targetfor LIGO is coalescence of binary neutron star and blackhole systems The waveform is reasonably well predicted bythe post-Newtonian approximation when the binary compo-nents are at large separation However extracting the mostimportant physics requires us to be able to deal with fullynon-linear general relativity as the system spirals togetherand coalesces Moreover a number of people believe thatthere is a significant event rate for the coalescence of mas-sive black hole systems (20M) 3$ In this case LIGO ismost sensitive to waves emitted from the strong field regimeIndeed without some theoretical guidance as to what to ex-pect from this regime it is possible we may miss theseevents entirely 4$However the goal of developing a general algorithm that

can solve Einsteinrsquos equations for two black holes has re-mained elusive All attempts to date have been plagued byinstabilities These instabilities are caused by an interplay ofthree factors 1 Einsteinrsquos equations are an overdeterminedsystem with the evolution equations subject to constraintsSo if for example you choose to solve only the evolutionequations then there can be unstable solutions that are in factsolutions of the evolution equations but do not satisfy theconstraints Small numerical errors may cause these solu-tions to appear and swamp the true solution lsquolsquoconstraint-

violating modesrsquorsquo 2 The coordinate freedom inherent in

the theory means that it is very easy to impose coordinate

conditions that lead to numerical instabilities lsquolsquogaugemodesrsquorsquo 3 Experience has shown that the kind of bound-ary conditions we choose and how we implement them can

affect the stability of an algorithm enormously

Similar instabilities have hampered efforts to solve therelated problem of binary neutron stars only very recently56$ has there been some success in finding stable algo-rithms However black hole evolutions face an additionalobstacle that is absent in the case of neutron stars for neu-tron stars the gravitational field is everywhere regular but forblack holes one must somehow deal with the physical singu-larity that lurks inside each holeThere are two main approaches for handling these singu-

larities The first is to use gauge conditions eg maximalslicing that avoid the singularities altogether Such condi-tions however lead to large gradients in the gravitationalfield variables near the horizon These grow exponentially intime and ultimately cannot be resolved by the numerical evo-lution causing the code to crash The alternative approach isto excise the region containing the singularity from the com-putational domain and evolve only the exterior region If theexcision boundary is placed inside the horizon of the blackhole causality assures us that we do not need to impose aphysical boundary condition thereHowever black hole excision is only known to be math-

ematically well posed if the evolution equations are hyper-bolic with characteristic speeds less than or equal to c In thiscase the structure of the equations guarantees that even un-physical modes present in the solution gauge modesconstraint-violating modes behave causally and cannotpropagate out of the horizon For many representations ofgeneral relativity such as the usual Arnowitt-Deser-MisnerADM 7$ formulation the evolution equations are of nomathematical type for which well-posedness has beenproved so the suitability of these formulations for black holeexcision must be determined empirically on a case-by-case

PHYSICAL REVIEW D VOLUME 62 084032

0556-2821200062808403220$1500 copy2000 The American Physical Society62 084032-1

Stable evolutions in 1+1 d

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

Pros and cons

No dissipation

Exponential convergence

Superb results for little effort

Gibbs phenomena

Discrete equations are fully coupled

Lack of dissipation allows high frequency errors to remain

Can not freely choose location of nodes

Regge calculus

Regge calculus

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is piecewise flat

Curvature is a distribution

Defect angle at each vertex

2d example

2d example

2d example

2d example

2d example

3d example

3d example

3d example

3d example

3d example

4d example

Field equations

Some questions

Can smooth metrics be accurately approximated by Regge lattices

Do the Regge equations reduce to the Einstein equations in some suitable limit

Do solutions of the Regge equations converge to solutions of the Einstein equations

Some answers

Regge

Reggesolutions

Yes

Yes as an average

Allendorfer Weyl

Cheeger Muller Schrader

Continuum

Einsteinsolutions

Applications

FRW cosmologies

Brill wave initial data

Quantum gravityKasner cosmologies

Misner initial data

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 6: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Non-standard methods

Numerical Relativity

Regge calculus

Multiquadrics

Spectral methods

Smooth lattices

Tetrad methods Buchman amp BardeenPhysRevD67(2003)084017 van Putten PhysRevD55(1997)4705

Discrete differential forms Frauendiener CQG23(2006)S369 Richter amp Frauendiener CQG24(2007)433

Finite volumes Alic etalPhysRevD76(2007)104007

Finite elements Korobkin etalCQG26(2009)145007 ZambuschCQG26(2009)175011

Multiquadrics

VOL 76 NO 8 7OURNAL OF GEOPHYSICAL RESEARCH MARCH 10 1bull71

Multiquadric Equations of Topography and Other Irregular Surfaces

ROLLAbullD L

Department o] Civil Engineering and Engineering Research Institute Iowa State University Ames 50010

A new analytical method of representing irregular surfaces that involves the summation of equations of quadric surfaces having unknown coefficients is described The quadric surfaces are located at significant points throughout the region to be mapped Procedures are given for solving multiquadric equations of topography that are based on coordinate data Con- toured multiquadric surfaces are compared with topography and other irregular surfaces from which the multiquadric equation was derived

Topography can be represented by various analytical numerical and digital methods in

addition to the classical contour map The

extremes in generalization or detail that result

from use of these methods are perhaps demon-

strated best by Lee and Kaula [1967] and by

Gilbert [1968] Lee and Kaula described the

topography of the whole earth in the form of

thirty-sixth-degree spherical harmonics Gilbert

reported the magnetic tape storage of more

than six million increments of height informa-

tion in digital form measured or interpolated

from one ordinary map sheet In Lee and Kaulas work we have an ex-

treme generalization of existing topographic

information over a wide area by highly analyti-

cal methods whereas Gilberts work is extremely

detailed but scarcely analytical As valuable

as these techniques are in certain cases they

are related more to map utilization than to map

making Basically the problem they solve is

given continuous topographic information in a

certain region reduce it to an equivalent set of

discrete data eg spherical harmonic coeffi-

cients or digital terrain increments

Other investigators including myself are

concerned with a procedural inverse of the

above problem namely given a set of discrete

data on a topographic surface reduce it to a

satisfactory continuous function representing

the topographic surface Practical solutions to

this problem will tend to eliminate the classical

Copyright 1971 by the American Geophysical Union

contour map as the first step in representing terrain information

An equation of topography can be evaluated

digitally or analytically without its having

been reduced to graphical form The same equa-

tion can be treated analytically for the auto-

matic production of contoured maps Automatic

contouring can become a computer-plotter

problem in analytical geometry ie to deter-

mine and plot the intercept equations of hori- zontal planes passed through a three-dimen-

sional equation of topography This approach could also lead to reconsideration of the need

for digitized map data Problems involving map use such as determining unobstructed

lines of sight areas of deftlade volumes of

earth and minimum length of surface curves

may involve the more direct application of

analytical geometry and calculus to the inter-

relationship of these parameters with a mathe-

matical surface of topography For these rea-

sons the question of representing a topographic surface in detail by unique equations deserves increased consideration

bullNUMERICAL SURFACE TECbullNmUES

Fourier and polynomial series approximations have been applied in recent years to both sur- face and subsurface trend analysis in geological

mapping Investigators in this field have been

Krumbein [1966] Mandelbaum [1963] James

[1966] and Merriam and Sheath [1966] There

has been a natural tendency to apply these

trend surface techniques to the problem of

1905

Multiquadrics

Originally used for interpolation on scattered data

Adapted by Kansa to solve ODEs

Given compute such that

Multiquadrics

Computers Math Appfic Vol 19 No 89 pp 127-145 1990 0097-494390 $300 + 000 Printed in Great Britain Pergamon Press plc

M U L T I Q U A D R I C S - - A S C A T T E R E D D A T A

A P P R O X I M A T I O N S C H E M E W I T H A P P L I C A T I O N S T O

C O M P U T A T I O N A L F L U I D - D Y N A M I C S - - I

SURFACE APPROXIMATIONS AND PARTIAL DERIVATIVE ESTIMATES

E J KANSA

Lawrence Livermore National Laboratory L-200 PO Box 808 Livermore CA 94550 USA

A~traet - -We present a powerful enhanced multiquadrics (MQ) scheme developed for spatial approxi- mations MQ is a true scattered data grid free scheme for representing surfaces and bodies in an arbitrary number of dimensions It is continuously differentiable and integrable and is capable of representing functions with steep gradients with very high accuracy Monotonicity and convexity are observed properties as a result of such high accuracy

Numerical results show that our modified MQ scheme is an excellent method not only for very accurate interpolation but also for partial derivative estimates MQ is applied to a higher order arbitrary Lagrangian-Eulerian (ALE) rezoning In the second paper of this series MQ is applied to parabolic hyperbolic and elliptic partial differential equations The parabolic problem uses an implicit time-marching scheme whereas the hyperbolic problem uses an explicit time-marching scheme We show that MQ is also exceptionally accurate and efficient The theory of Madych and Nelson shows that the MQ interpolant belongs to a space of functions which minimizes a semi-norm and gives credence to our results

1 BACKGROUND

The study of arbitrarily shaped curves surfaces and bodies having arbitrary data orderings has

immediate application to computational fluid-dynamics The governing equations not only include

source terms but gradients divergences and Laplacians In addition many physical processes occur

over a wide range of length scales To obtain quantitatively accurate approximations of the physics

quantitatively accurate estimates of the spatial variations of such variables are required In two

and three dimensions the range of such quantitatively accurate problems possible on current

multiprocessing super computers using standard finite difference or finite element codes is limited

The question is whether there exist alternative techniques or combinations of techniques which can

broaden the scope of problems to be solved by permitting steep gradients to be modelled using

fewer data points Toward that goal our study consists of two parts The first part will investigate

a new numerical technique of curve surface and body approximations of exceptional accuracy over

an arbitrary data arrangement The second part of this study will use such techniques to improve

parabolic hyperbolic or elliptic partial differential equations We will demonstrate that the study

of function approximations has a definite advantage to computational methods for partial

differential equations

One very important use of computers is the simulation of multidimensional spatial processes

In this paper we assumed that some finite physical quantity F is piecewise continuous in some

finite domain In many applications F is known only at a finite number of locations

xk k = 1 2 N where xk = x~ for a univariate problem and Xk = (x~yk )X for the

multivariate problem

From a finite amount of information regarding F we seek the best approximation which can

not only supply accurate estimates of F at arbitrary locations on the domain but will also provide

accurate estimates of the partial derivatives and definite integrals of F anywhere on the domain

The domain of F will consist of points xk of arbitrary ordering and sub-clustering A rectangular

grid is a very special case of a data ordering

Let us assume that an interpolation function f approximates F in the sense that

f(Xk)=F(Xk) k = l 2 N (1)

127

Example

solve linear system for

uniform amp random

choose

reconstruct interpolant

Example

linear system for

Bad news

is dense

is almost singular condition number typically

Must use Singular Value Decomposition expensive

Convergence is very rapid approx exponential

Good news

Bowen-York initial data

Bowen-York initial data

Bowen-York initial data

Pros and cons

Free to place nodes wherever we like

Exponential convergence

Must solve exceedingly ill-conditioned system

Requires care for asymptoticly flat geometries

Only two papers with results volunteers most welcome

Very accurate for very steep functions

Living Rev Relativity 12 (2009) 1httpwwwlivingreviewsorglrr-2009-1

Spectral Methods for Numerical Relativity

Philippe GrandclementLaboratoire Univers et Theories

UMR 8102 du CNRS Observatoire de ParisF-92195 Meudon Cedex France

email PhilippeGrandclementobspmfrhttpwwwluthobspmfrminisitephpnom=Grandclement

Jerome NovakLaboratoire Univers et Theories

UMR 8102 du CNRS Observatoire de ParisF-92195 Meudon Cedex Franceemail JeromeNovakobspmfr

httpwwwluthobspmfrminisitephpnom=Novak

Living Reviews in RelativityISSN 1433-8351

Accepted on 23 October 2008Published on 9 January 2009

Abstract

Equations arising in general relativity are usually too complicated to be solved analyticallyand one must rely on numerical methods to solve sets of coupled partial dierential equationsAmong the possible choices this paper focuses on a class called spectral methods in whichtypically the various functions are expanded in sets of orthogonal polynomials or functionsFirst a theoretical introduction of spectral expansion is given with a particular emphasis on thefast convergence of the spectral approximation We then present dierent approaches to solvingpartial dierential equations first limiting ourselves to the one-dimensional case with one ormore domains Generalization to more dimensions is then discussed In particular the case oftime evolutions is carefully studied and the stability of such evolutions investigated We thenpresent results obtained by various groups in the field of general relativity by means of spectralmethods Work which does not involve explicit time-evolutions is discussed going fromrapidly-rotating strange stars to the computation of black-holendashbinary initial data Finallythe evolution of various systems of astrophysical interest are presented from supernovae corecollapse to black-holendashbinary mergers

This review is licensed under a Creative CommonsAttribution-Non-Commercial-NoDerivs 30 Germany Licensehttpcreativecommonsorglicensesby-nc-nd30de

Spectral methods

A gentle introduction

Spectral interpolants

is a function

is a polynomial approximation of degree to

Convergence3

-1 -05 0 05 1x

-15

-1

-05

0

05

1

u = cos3(x2) - (x+1)

38

Pu

Iu

N=4

-1 -05 0 05 1x

-15

-1

-05

0

05

1

u = cos3(x2) - (x+1)

38

Pu

Iu

N=8

FIG 2 Same as Fig 1 with also the interpolant of u The collocation points are denoted by the circles

3 Interpolation

If one applies the Gauss quadratures to approximate the coecient of the expansion one obtains

un =1

n

N

j=0

u (xj) pn (xj)wj with n =N

j=0

p2n (xj)wj (5)

Let us precise that this is not exact in the sense that un = un However the computation of u only requires toevaluate u at the N + 1 collocation points The interpolant of u is then defined as the following polynomial

INu =N

n=0

unpn (x) (6)

The dierence between Inu and Pnu is called the aliasing error The interpolant of u is the spectral approximate ofu and one can show that it is the only polynomials of degree N that coincides with u at each collocation point

[Inu] (xi) = u (xi) i N (7)

Figure 2 shows the same function as Fig1 but the interpolant is also plotted One can see that indeed INu coincideswith u at the collocation points that are indicated by the circles Once again even with as few points as N = 8 nodierence can be seen between the various functions

Figure 3 shows the maximum dierence between INu and u on as a function of the degree of the approximation N We can observe the very general feature of spectral methods that the error decreases exponentially until one reachesthe machine accuracy (here 1014 the computation being done in double precision) This very fast convergenceexplains why spectral methods are so ecient especially compared to finite dierence ones where the error followsonly a power-law in terms of N We will later be more quantitative about the convergence properties of the spectralapproximation

Let us note that a function u can be described either by its value at each collocation point u (xi) or by the coecientsof the interpolant ui If the values at collocation points are known one is working in the configuration space and inthe coecient space if u is given in terms of its coecients There is a bijection between the two descriptions and onesimply goes from one space to another by using

bull un =1

n

N

j=0

u (xj) pn (xj) wj (configuration $ coecient)

bull u (xn) =N

j=0

ujpj (xn) (coecient $ configuration)

Depending on the operation one has to perform one choice of space is usually more suited than the other Forinstance let us assume that one wants to compute the derivative of u This is easily done if u is known in the

Exponential convergence 4

0 5 10 15 20 25 30 35

Number of coefficients

1e-15

1e-12

1e-09

1e-06

1e-03

1e+00m

ax |I N

u -

u|

FIG 3 Maximum dierence between INu and u as a function of the degree of the approximation N

-1 -05 0 05 1x

-3

-2

-1

0

1

2

ursquoIN

(ursquo)

(IN

u)rsquo

N=4

-1 -05 0 05 1x

-3

-2

-1

0

1

2

ursquoIN

(ursquo)

(IN

u)rsquo

N=8

FIG 4 First derivative of u = cos3 (x2) (x + 1)3 8 interpolant of the derivative and derivative of the interpolant forN = 4 and N = 8

coecient space Indeed one can simply approximate u by the derivative of the interpolant

u (x) [INu] =N

n=0

unpn (x) (8)

Such an approximation only requires the knowledge of the coecients of u and how the basis polynomials are derivedLet us note that the obtained polynomial even if it is a good approximation of u is not the interpolant of u Inother terms the interpolation and the derivation are two operations that do not commute (INu) = IN (u) This isclearly illustrated on Fig 4 where the derivative of u = cos3 (x2) (x + 1)3 8 is plotted along with the functions(INu) and IN (u) In particular one can note that the functions used to represent u ie (INu) does not coincidewith u at the collocation points However even with N = 8 only the three functions can not be distinguished byeye

The maximum dierence between u and (INu) as a function of N is shown on Fig 5 Once again the convergenceis exponential

Gibbs phenomena9

-1 -05 0 05 1x

-02

0

02

04

06

08

1

FIG 8 Step function and some interpolants for N = 7 N = 15 and N = 31 As N increases the maximum dierence staysconstant and the number of oscillations increases

4 5 6 7 8 9 10 20 30 40 50 60 70 8090

N

10-2

10-1

100

-1

1 |I N

u -

u| dx

FIG 9

1

1|INu u| dx as a function of N u being a step function The convergence obeys a power-law which decays slower

than N1

B A test problem

We propose to solve the equation

d2u

dx2 4

du

dx+ 4u = exp (x) + C (25)

with x [1 1] and C = 4e

1 + e2

As boundary conditions we simply ask that the solution is zero at theboundaries

u (1) = 0 and u (1) = 0 (26)

Representations

can be represented by either or

for algebraic operators use

for differential operators use

Differential equations

solve for then recover

Spectral methods in Numerical Relativity

Black hole evolution by spectral methods

Lawrence E Kidder Mark A Scheel and Saul A TeukolskyCenter for Radiophysics and Space Research Cornell University Ithaca New York 14853

Eric D Carlson and Gregory B CookDepartment of Physics Wake Forest University Winston-Salem North Carolina 27109

Received 15 May 2000 published 26 September 2000

Current methods of evolving a spacetime containing one or more black holes are plagued by instabilities that

prohibit long-term evolution Some of these instabilities may be due to the numerical method used tradition-

ally finite differencing In this paper we explore the use of a pseudospectral collocation PSC method for theevolution of a spherically symmetric black hole spacetime in one dimension using a hyperbolic formulation of

Einsteinrsquos equations We demonstrate that our PSC method is able to evolve a spherically symmetric black

hole spacetime forever without enforcing constraints even if we add dynamics via a Klein-Gordon scalar field

We find that in contrast with finite-differencing methods black hole excision is a trivial operation using PSC

applied to a hyperbolic formulation of Einsteinrsquos equations We discuss the extension of this method to three

spatial dimensions

PACS numbers 0425Dm 0270Hm

I INTRODUCTION AND SUMMARY

A major thrust of research in classical general relativity in

the past decade has been to devise algorithms to solve Ein-

steinrsquos equations numerically Despite advances in our ana-

lytic understanding of general relativity we still do not knowwhat all the features of the theory really are Numerical so-lutions will continue to provide fresh insights into the theoryas they have in the past for example the critical behavior inblack hole formation 1$ and the formation of toroidal blackholes 2$New urgency has been injected into numerical relativity

by the imminent deployment of the Laser InterferometricGravitational Wave Observatory LIGO The prime targetfor LIGO is coalescence of binary neutron star and blackhole systems The waveform is reasonably well predicted bythe post-Newtonian approximation when the binary compo-nents are at large separation However extracting the mostimportant physics requires us to be able to deal with fullynon-linear general relativity as the system spirals togetherand coalesces Moreover a number of people believe thatthere is a significant event rate for the coalescence of mas-sive black hole systems (20M) 3$ In this case LIGO ismost sensitive to waves emitted from the strong field regimeIndeed without some theoretical guidance as to what to ex-pect from this regime it is possible we may miss theseevents entirely 4$However the goal of developing a general algorithm that

can solve Einsteinrsquos equations for two black holes has re-mained elusive All attempts to date have been plagued byinstabilities These instabilities are caused by an interplay ofthree factors 1 Einsteinrsquos equations are an overdeterminedsystem with the evolution equations subject to constraintsSo if for example you choose to solve only the evolutionequations then there can be unstable solutions that are in factsolutions of the evolution equations but do not satisfy theconstraints Small numerical errors may cause these solu-tions to appear and swamp the true solution lsquolsquoconstraint-

violating modesrsquorsquo 2 The coordinate freedom inherent in

the theory means that it is very easy to impose coordinate

conditions that lead to numerical instabilities lsquolsquogaugemodesrsquorsquo 3 Experience has shown that the kind of bound-ary conditions we choose and how we implement them can

affect the stability of an algorithm enormously

Similar instabilities have hampered efforts to solve therelated problem of binary neutron stars only very recently56$ has there been some success in finding stable algo-rithms However black hole evolutions face an additionalobstacle that is absent in the case of neutron stars for neu-tron stars the gravitational field is everywhere regular but forblack holes one must somehow deal with the physical singu-larity that lurks inside each holeThere are two main approaches for handling these singu-

larities The first is to use gauge conditions eg maximalslicing that avoid the singularities altogether Such condi-tions however lead to large gradients in the gravitationalfield variables near the horizon These grow exponentially intime and ultimately cannot be resolved by the numerical evo-lution causing the code to crash The alternative approach isto excise the region containing the singularity from the com-putational domain and evolve only the exterior region If theexcision boundary is placed inside the horizon of the blackhole causality assures us that we do not need to impose aphysical boundary condition thereHowever black hole excision is only known to be math-

ematically well posed if the evolution equations are hyper-bolic with characteristic speeds less than or equal to c In thiscase the structure of the equations guarantees that even un-physical modes present in the solution gauge modesconstraint-violating modes behave causally and cannotpropagate out of the horizon For many representations ofgeneral relativity such as the usual Arnowitt-Deser-MisnerADM 7$ formulation the evolution equations are of nomathematical type for which well-posedness has beenproved so the suitability of these formulations for black holeexcision must be determined empirically on a case-by-case

PHYSICAL REVIEW D VOLUME 62 084032

0556-2821200062808403220$1500 copy2000 The American Physical Society62 084032-1

Stable evolutions in 1+1 d

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

Pros and cons

No dissipation

Exponential convergence

Superb results for little effort

Gibbs phenomena

Discrete equations are fully coupled

Lack of dissipation allows high frequency errors to remain

Can not freely choose location of nodes

Regge calculus

Regge calculus

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is piecewise flat

Curvature is a distribution

Defect angle at each vertex

2d example

2d example

2d example

2d example

2d example

3d example

3d example

3d example

3d example

3d example

4d example

Field equations

Some questions

Can smooth metrics be accurately approximated by Regge lattices

Do the Regge equations reduce to the Einstein equations in some suitable limit

Do solutions of the Regge equations converge to solutions of the Einstein equations

Some answers

Regge

Reggesolutions

Yes

Yes as an average

Allendorfer Weyl

Cheeger Muller Schrader

Continuum

Einsteinsolutions

Applications

FRW cosmologies

Brill wave initial data

Quantum gravityKasner cosmologies

Misner initial data

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 7: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Multiquadrics

VOL 76 NO 8 7OURNAL OF GEOPHYSICAL RESEARCH MARCH 10 1bull71

Multiquadric Equations of Topography and Other Irregular Surfaces

ROLLAbullD L

Department o] Civil Engineering and Engineering Research Institute Iowa State University Ames 50010

A new analytical method of representing irregular surfaces that involves the summation of equations of quadric surfaces having unknown coefficients is described The quadric surfaces are located at significant points throughout the region to be mapped Procedures are given for solving multiquadric equations of topography that are based on coordinate data Con- toured multiquadric surfaces are compared with topography and other irregular surfaces from which the multiquadric equation was derived

Topography can be represented by various analytical numerical and digital methods in

addition to the classical contour map The

extremes in generalization or detail that result

from use of these methods are perhaps demon-

strated best by Lee and Kaula [1967] and by

Gilbert [1968] Lee and Kaula described the

topography of the whole earth in the form of

thirty-sixth-degree spherical harmonics Gilbert

reported the magnetic tape storage of more

than six million increments of height informa-

tion in digital form measured or interpolated

from one ordinary map sheet In Lee and Kaulas work we have an ex-

treme generalization of existing topographic

information over a wide area by highly analyti-

cal methods whereas Gilberts work is extremely

detailed but scarcely analytical As valuable

as these techniques are in certain cases they

are related more to map utilization than to map

making Basically the problem they solve is

given continuous topographic information in a

certain region reduce it to an equivalent set of

discrete data eg spherical harmonic coeffi-

cients or digital terrain increments

Other investigators including myself are

concerned with a procedural inverse of the

above problem namely given a set of discrete

data on a topographic surface reduce it to a

satisfactory continuous function representing

the topographic surface Practical solutions to

this problem will tend to eliminate the classical

Copyright 1971 by the American Geophysical Union

contour map as the first step in representing terrain information

An equation of topography can be evaluated

digitally or analytically without its having

been reduced to graphical form The same equa-

tion can be treated analytically for the auto-

matic production of contoured maps Automatic

contouring can become a computer-plotter

problem in analytical geometry ie to deter-

mine and plot the intercept equations of hori- zontal planes passed through a three-dimen-

sional equation of topography This approach could also lead to reconsideration of the need

for digitized map data Problems involving map use such as determining unobstructed

lines of sight areas of deftlade volumes of

earth and minimum length of surface curves

may involve the more direct application of

analytical geometry and calculus to the inter-

relationship of these parameters with a mathe-

matical surface of topography For these rea-

sons the question of representing a topographic surface in detail by unique equations deserves increased consideration

bullNUMERICAL SURFACE TECbullNmUES

Fourier and polynomial series approximations have been applied in recent years to both sur- face and subsurface trend analysis in geological

mapping Investigators in this field have been

Krumbein [1966] Mandelbaum [1963] James

[1966] and Merriam and Sheath [1966] There

has been a natural tendency to apply these

trend surface techniques to the problem of

1905

Multiquadrics

Originally used for interpolation on scattered data

Adapted by Kansa to solve ODEs

Given compute such that

Multiquadrics

Computers Math Appfic Vol 19 No 89 pp 127-145 1990 0097-494390 $300 + 000 Printed in Great Britain Pergamon Press plc

M U L T I Q U A D R I C S - - A S C A T T E R E D D A T A

A P P R O X I M A T I O N S C H E M E W I T H A P P L I C A T I O N S T O

C O M P U T A T I O N A L F L U I D - D Y N A M I C S - - I

SURFACE APPROXIMATIONS AND PARTIAL DERIVATIVE ESTIMATES

E J KANSA

Lawrence Livermore National Laboratory L-200 PO Box 808 Livermore CA 94550 USA

A~traet - -We present a powerful enhanced multiquadrics (MQ) scheme developed for spatial approxi- mations MQ is a true scattered data grid free scheme for representing surfaces and bodies in an arbitrary number of dimensions It is continuously differentiable and integrable and is capable of representing functions with steep gradients with very high accuracy Monotonicity and convexity are observed properties as a result of such high accuracy

Numerical results show that our modified MQ scheme is an excellent method not only for very accurate interpolation but also for partial derivative estimates MQ is applied to a higher order arbitrary Lagrangian-Eulerian (ALE) rezoning In the second paper of this series MQ is applied to parabolic hyperbolic and elliptic partial differential equations The parabolic problem uses an implicit time-marching scheme whereas the hyperbolic problem uses an explicit time-marching scheme We show that MQ is also exceptionally accurate and efficient The theory of Madych and Nelson shows that the MQ interpolant belongs to a space of functions which minimizes a semi-norm and gives credence to our results

1 BACKGROUND

The study of arbitrarily shaped curves surfaces and bodies having arbitrary data orderings has

immediate application to computational fluid-dynamics The governing equations not only include

source terms but gradients divergences and Laplacians In addition many physical processes occur

over a wide range of length scales To obtain quantitatively accurate approximations of the physics

quantitatively accurate estimates of the spatial variations of such variables are required In two

and three dimensions the range of such quantitatively accurate problems possible on current

multiprocessing super computers using standard finite difference or finite element codes is limited

The question is whether there exist alternative techniques or combinations of techniques which can

broaden the scope of problems to be solved by permitting steep gradients to be modelled using

fewer data points Toward that goal our study consists of two parts The first part will investigate

a new numerical technique of curve surface and body approximations of exceptional accuracy over

an arbitrary data arrangement The second part of this study will use such techniques to improve

parabolic hyperbolic or elliptic partial differential equations We will demonstrate that the study

of function approximations has a definite advantage to computational methods for partial

differential equations

One very important use of computers is the simulation of multidimensional spatial processes

In this paper we assumed that some finite physical quantity F is piecewise continuous in some

finite domain In many applications F is known only at a finite number of locations

xk k = 1 2 N where xk = x~ for a univariate problem and Xk = (x~yk )X for the

multivariate problem

From a finite amount of information regarding F we seek the best approximation which can

not only supply accurate estimates of F at arbitrary locations on the domain but will also provide

accurate estimates of the partial derivatives and definite integrals of F anywhere on the domain

The domain of F will consist of points xk of arbitrary ordering and sub-clustering A rectangular

grid is a very special case of a data ordering

Let us assume that an interpolation function f approximates F in the sense that

f(Xk)=F(Xk) k = l 2 N (1)

127

Example

solve linear system for

uniform amp random

choose

reconstruct interpolant

Example

linear system for

Bad news

is dense

is almost singular condition number typically

Must use Singular Value Decomposition expensive

Convergence is very rapid approx exponential

Good news

Bowen-York initial data

Bowen-York initial data

Bowen-York initial data

Pros and cons

Free to place nodes wherever we like

Exponential convergence

Must solve exceedingly ill-conditioned system

Requires care for asymptoticly flat geometries

Only two papers with results volunteers most welcome

Very accurate for very steep functions

Living Rev Relativity 12 (2009) 1httpwwwlivingreviewsorglrr-2009-1

Spectral Methods for Numerical Relativity

Philippe GrandclementLaboratoire Univers et Theories

UMR 8102 du CNRS Observatoire de ParisF-92195 Meudon Cedex France

email PhilippeGrandclementobspmfrhttpwwwluthobspmfrminisitephpnom=Grandclement

Jerome NovakLaboratoire Univers et Theories

UMR 8102 du CNRS Observatoire de ParisF-92195 Meudon Cedex Franceemail JeromeNovakobspmfr

httpwwwluthobspmfrminisitephpnom=Novak

Living Reviews in RelativityISSN 1433-8351

Accepted on 23 October 2008Published on 9 January 2009

Abstract

Equations arising in general relativity are usually too complicated to be solved analyticallyand one must rely on numerical methods to solve sets of coupled partial dierential equationsAmong the possible choices this paper focuses on a class called spectral methods in whichtypically the various functions are expanded in sets of orthogonal polynomials or functionsFirst a theoretical introduction of spectral expansion is given with a particular emphasis on thefast convergence of the spectral approximation We then present dierent approaches to solvingpartial dierential equations first limiting ourselves to the one-dimensional case with one ormore domains Generalization to more dimensions is then discussed In particular the case oftime evolutions is carefully studied and the stability of such evolutions investigated We thenpresent results obtained by various groups in the field of general relativity by means of spectralmethods Work which does not involve explicit time-evolutions is discussed going fromrapidly-rotating strange stars to the computation of black-holendashbinary initial data Finallythe evolution of various systems of astrophysical interest are presented from supernovae corecollapse to black-holendashbinary mergers

This review is licensed under a Creative CommonsAttribution-Non-Commercial-NoDerivs 30 Germany Licensehttpcreativecommonsorglicensesby-nc-nd30de

Spectral methods

A gentle introduction

Spectral interpolants

is a function

is a polynomial approximation of degree to

Convergence3

-1 -05 0 05 1x

-15

-1

-05

0

05

1

u = cos3(x2) - (x+1)

38

Pu

Iu

N=4

-1 -05 0 05 1x

-15

-1

-05

0

05

1

u = cos3(x2) - (x+1)

38

Pu

Iu

N=8

FIG 2 Same as Fig 1 with also the interpolant of u The collocation points are denoted by the circles

3 Interpolation

If one applies the Gauss quadratures to approximate the coecient of the expansion one obtains

un =1

n

N

j=0

u (xj) pn (xj)wj with n =N

j=0

p2n (xj)wj (5)

Let us precise that this is not exact in the sense that un = un However the computation of u only requires toevaluate u at the N + 1 collocation points The interpolant of u is then defined as the following polynomial

INu =N

n=0

unpn (x) (6)

The dierence between Inu and Pnu is called the aliasing error The interpolant of u is the spectral approximate ofu and one can show that it is the only polynomials of degree N that coincides with u at each collocation point

[Inu] (xi) = u (xi) i N (7)

Figure 2 shows the same function as Fig1 but the interpolant is also plotted One can see that indeed INu coincideswith u at the collocation points that are indicated by the circles Once again even with as few points as N = 8 nodierence can be seen between the various functions

Figure 3 shows the maximum dierence between INu and u on as a function of the degree of the approximation N We can observe the very general feature of spectral methods that the error decreases exponentially until one reachesthe machine accuracy (here 1014 the computation being done in double precision) This very fast convergenceexplains why spectral methods are so ecient especially compared to finite dierence ones where the error followsonly a power-law in terms of N We will later be more quantitative about the convergence properties of the spectralapproximation

Let us note that a function u can be described either by its value at each collocation point u (xi) or by the coecientsof the interpolant ui If the values at collocation points are known one is working in the configuration space and inthe coecient space if u is given in terms of its coecients There is a bijection between the two descriptions and onesimply goes from one space to another by using

bull un =1

n

N

j=0

u (xj) pn (xj) wj (configuration $ coecient)

bull u (xn) =N

j=0

ujpj (xn) (coecient $ configuration)

Depending on the operation one has to perform one choice of space is usually more suited than the other Forinstance let us assume that one wants to compute the derivative of u This is easily done if u is known in the

Exponential convergence 4

0 5 10 15 20 25 30 35

Number of coefficients

1e-15

1e-12

1e-09

1e-06

1e-03

1e+00m

ax |I N

u -

u|

FIG 3 Maximum dierence between INu and u as a function of the degree of the approximation N

-1 -05 0 05 1x

-3

-2

-1

0

1

2

ursquoIN

(ursquo)

(IN

u)rsquo

N=4

-1 -05 0 05 1x

-3

-2

-1

0

1

2

ursquoIN

(ursquo)

(IN

u)rsquo

N=8

FIG 4 First derivative of u = cos3 (x2) (x + 1)3 8 interpolant of the derivative and derivative of the interpolant forN = 4 and N = 8

coecient space Indeed one can simply approximate u by the derivative of the interpolant

u (x) [INu] =N

n=0

unpn (x) (8)

Such an approximation only requires the knowledge of the coecients of u and how the basis polynomials are derivedLet us note that the obtained polynomial even if it is a good approximation of u is not the interpolant of u Inother terms the interpolation and the derivation are two operations that do not commute (INu) = IN (u) This isclearly illustrated on Fig 4 where the derivative of u = cos3 (x2) (x + 1)3 8 is plotted along with the functions(INu) and IN (u) In particular one can note that the functions used to represent u ie (INu) does not coincidewith u at the collocation points However even with N = 8 only the three functions can not be distinguished byeye

The maximum dierence between u and (INu) as a function of N is shown on Fig 5 Once again the convergenceis exponential

Gibbs phenomena9

-1 -05 0 05 1x

-02

0

02

04

06

08

1

FIG 8 Step function and some interpolants for N = 7 N = 15 and N = 31 As N increases the maximum dierence staysconstant and the number of oscillations increases

4 5 6 7 8 9 10 20 30 40 50 60 70 8090

N

10-2

10-1

100

-1

1 |I N

u -

u| dx

FIG 9

1

1|INu u| dx as a function of N u being a step function The convergence obeys a power-law which decays slower

than N1

B A test problem

We propose to solve the equation

d2u

dx2 4

du

dx+ 4u = exp (x) + C (25)

with x [1 1] and C = 4e

1 + e2

As boundary conditions we simply ask that the solution is zero at theboundaries

u (1) = 0 and u (1) = 0 (26)

Representations

can be represented by either or

for algebraic operators use

for differential operators use

Differential equations

solve for then recover

Spectral methods in Numerical Relativity

Black hole evolution by spectral methods

Lawrence E Kidder Mark A Scheel and Saul A TeukolskyCenter for Radiophysics and Space Research Cornell University Ithaca New York 14853

Eric D Carlson and Gregory B CookDepartment of Physics Wake Forest University Winston-Salem North Carolina 27109

Received 15 May 2000 published 26 September 2000

Current methods of evolving a spacetime containing one or more black holes are plagued by instabilities that

prohibit long-term evolution Some of these instabilities may be due to the numerical method used tradition-

ally finite differencing In this paper we explore the use of a pseudospectral collocation PSC method for theevolution of a spherically symmetric black hole spacetime in one dimension using a hyperbolic formulation of

Einsteinrsquos equations We demonstrate that our PSC method is able to evolve a spherically symmetric black

hole spacetime forever without enforcing constraints even if we add dynamics via a Klein-Gordon scalar field

We find that in contrast with finite-differencing methods black hole excision is a trivial operation using PSC

applied to a hyperbolic formulation of Einsteinrsquos equations We discuss the extension of this method to three

spatial dimensions

PACS numbers 0425Dm 0270Hm

I INTRODUCTION AND SUMMARY

A major thrust of research in classical general relativity in

the past decade has been to devise algorithms to solve Ein-

steinrsquos equations numerically Despite advances in our ana-

lytic understanding of general relativity we still do not knowwhat all the features of the theory really are Numerical so-lutions will continue to provide fresh insights into the theoryas they have in the past for example the critical behavior inblack hole formation 1$ and the formation of toroidal blackholes 2$New urgency has been injected into numerical relativity

by the imminent deployment of the Laser InterferometricGravitational Wave Observatory LIGO The prime targetfor LIGO is coalescence of binary neutron star and blackhole systems The waveform is reasonably well predicted bythe post-Newtonian approximation when the binary compo-nents are at large separation However extracting the mostimportant physics requires us to be able to deal with fullynon-linear general relativity as the system spirals togetherand coalesces Moreover a number of people believe thatthere is a significant event rate for the coalescence of mas-sive black hole systems (20M) 3$ In this case LIGO ismost sensitive to waves emitted from the strong field regimeIndeed without some theoretical guidance as to what to ex-pect from this regime it is possible we may miss theseevents entirely 4$However the goal of developing a general algorithm that

can solve Einsteinrsquos equations for two black holes has re-mained elusive All attempts to date have been plagued byinstabilities These instabilities are caused by an interplay ofthree factors 1 Einsteinrsquos equations are an overdeterminedsystem with the evolution equations subject to constraintsSo if for example you choose to solve only the evolutionequations then there can be unstable solutions that are in factsolutions of the evolution equations but do not satisfy theconstraints Small numerical errors may cause these solu-tions to appear and swamp the true solution lsquolsquoconstraint-

violating modesrsquorsquo 2 The coordinate freedom inherent in

the theory means that it is very easy to impose coordinate

conditions that lead to numerical instabilities lsquolsquogaugemodesrsquorsquo 3 Experience has shown that the kind of bound-ary conditions we choose and how we implement them can

affect the stability of an algorithm enormously

Similar instabilities have hampered efforts to solve therelated problem of binary neutron stars only very recently56$ has there been some success in finding stable algo-rithms However black hole evolutions face an additionalobstacle that is absent in the case of neutron stars for neu-tron stars the gravitational field is everywhere regular but forblack holes one must somehow deal with the physical singu-larity that lurks inside each holeThere are two main approaches for handling these singu-

larities The first is to use gauge conditions eg maximalslicing that avoid the singularities altogether Such condi-tions however lead to large gradients in the gravitationalfield variables near the horizon These grow exponentially intime and ultimately cannot be resolved by the numerical evo-lution causing the code to crash The alternative approach isto excise the region containing the singularity from the com-putational domain and evolve only the exterior region If theexcision boundary is placed inside the horizon of the blackhole causality assures us that we do not need to impose aphysical boundary condition thereHowever black hole excision is only known to be math-

ematically well posed if the evolution equations are hyper-bolic with characteristic speeds less than or equal to c In thiscase the structure of the equations guarantees that even un-physical modes present in the solution gauge modesconstraint-violating modes behave causally and cannotpropagate out of the horizon For many representations ofgeneral relativity such as the usual Arnowitt-Deser-MisnerADM 7$ formulation the evolution equations are of nomathematical type for which well-posedness has beenproved so the suitability of these formulations for black holeexcision must be determined empirically on a case-by-case

PHYSICAL REVIEW D VOLUME 62 084032

0556-2821200062808403220$1500 copy2000 The American Physical Society62 084032-1

Stable evolutions in 1+1 d

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

Pros and cons

No dissipation

Exponential convergence

Superb results for little effort

Gibbs phenomena

Discrete equations are fully coupled

Lack of dissipation allows high frequency errors to remain

Can not freely choose location of nodes

Regge calculus

Regge calculus

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is piecewise flat

Curvature is a distribution

Defect angle at each vertex

2d example

2d example

2d example

2d example

2d example

3d example

3d example

3d example

3d example

3d example

4d example

Field equations

Some questions

Can smooth metrics be accurately approximated by Regge lattices

Do the Regge equations reduce to the Einstein equations in some suitable limit

Do solutions of the Regge equations converge to solutions of the Einstein equations

Some answers

Regge

Reggesolutions

Yes

Yes as an average

Allendorfer Weyl

Cheeger Muller Schrader

Continuum

Einsteinsolutions

Applications

FRW cosmologies

Brill wave initial data

Quantum gravityKasner cosmologies

Misner initial data

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 8: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Multiquadrics

Originally used for interpolation on scattered data

Adapted by Kansa to solve ODEs

Given compute such that

Multiquadrics

Computers Math Appfic Vol 19 No 89 pp 127-145 1990 0097-494390 $300 + 000 Printed in Great Britain Pergamon Press plc

M U L T I Q U A D R I C S - - A S C A T T E R E D D A T A

A P P R O X I M A T I O N S C H E M E W I T H A P P L I C A T I O N S T O

C O M P U T A T I O N A L F L U I D - D Y N A M I C S - - I

SURFACE APPROXIMATIONS AND PARTIAL DERIVATIVE ESTIMATES

E J KANSA

Lawrence Livermore National Laboratory L-200 PO Box 808 Livermore CA 94550 USA

A~traet - -We present a powerful enhanced multiquadrics (MQ) scheme developed for spatial approxi- mations MQ is a true scattered data grid free scheme for representing surfaces and bodies in an arbitrary number of dimensions It is continuously differentiable and integrable and is capable of representing functions with steep gradients with very high accuracy Monotonicity and convexity are observed properties as a result of such high accuracy

Numerical results show that our modified MQ scheme is an excellent method not only for very accurate interpolation but also for partial derivative estimates MQ is applied to a higher order arbitrary Lagrangian-Eulerian (ALE) rezoning In the second paper of this series MQ is applied to parabolic hyperbolic and elliptic partial differential equations The parabolic problem uses an implicit time-marching scheme whereas the hyperbolic problem uses an explicit time-marching scheme We show that MQ is also exceptionally accurate and efficient The theory of Madych and Nelson shows that the MQ interpolant belongs to a space of functions which minimizes a semi-norm and gives credence to our results

1 BACKGROUND

The study of arbitrarily shaped curves surfaces and bodies having arbitrary data orderings has

immediate application to computational fluid-dynamics The governing equations not only include

source terms but gradients divergences and Laplacians In addition many physical processes occur

over a wide range of length scales To obtain quantitatively accurate approximations of the physics

quantitatively accurate estimates of the spatial variations of such variables are required In two

and three dimensions the range of such quantitatively accurate problems possible on current

multiprocessing super computers using standard finite difference or finite element codes is limited

The question is whether there exist alternative techniques or combinations of techniques which can

broaden the scope of problems to be solved by permitting steep gradients to be modelled using

fewer data points Toward that goal our study consists of two parts The first part will investigate

a new numerical technique of curve surface and body approximations of exceptional accuracy over

an arbitrary data arrangement The second part of this study will use such techniques to improve

parabolic hyperbolic or elliptic partial differential equations We will demonstrate that the study

of function approximations has a definite advantage to computational methods for partial

differential equations

One very important use of computers is the simulation of multidimensional spatial processes

In this paper we assumed that some finite physical quantity F is piecewise continuous in some

finite domain In many applications F is known only at a finite number of locations

xk k = 1 2 N where xk = x~ for a univariate problem and Xk = (x~yk )X for the

multivariate problem

From a finite amount of information regarding F we seek the best approximation which can

not only supply accurate estimates of F at arbitrary locations on the domain but will also provide

accurate estimates of the partial derivatives and definite integrals of F anywhere on the domain

The domain of F will consist of points xk of arbitrary ordering and sub-clustering A rectangular

grid is a very special case of a data ordering

Let us assume that an interpolation function f approximates F in the sense that

f(Xk)=F(Xk) k = l 2 N (1)

127

Example

solve linear system for

uniform amp random

choose

reconstruct interpolant

Example

linear system for

Bad news

is dense

is almost singular condition number typically

Must use Singular Value Decomposition expensive

Convergence is very rapid approx exponential

Good news

Bowen-York initial data

Bowen-York initial data

Bowen-York initial data

Pros and cons

Free to place nodes wherever we like

Exponential convergence

Must solve exceedingly ill-conditioned system

Requires care for asymptoticly flat geometries

Only two papers with results volunteers most welcome

Very accurate for very steep functions

Living Rev Relativity 12 (2009) 1httpwwwlivingreviewsorglrr-2009-1

Spectral Methods for Numerical Relativity

Philippe GrandclementLaboratoire Univers et Theories

UMR 8102 du CNRS Observatoire de ParisF-92195 Meudon Cedex France

email PhilippeGrandclementobspmfrhttpwwwluthobspmfrminisitephpnom=Grandclement

Jerome NovakLaboratoire Univers et Theories

UMR 8102 du CNRS Observatoire de ParisF-92195 Meudon Cedex Franceemail JeromeNovakobspmfr

httpwwwluthobspmfrminisitephpnom=Novak

Living Reviews in RelativityISSN 1433-8351

Accepted on 23 October 2008Published on 9 January 2009

Abstract

Equations arising in general relativity are usually too complicated to be solved analyticallyand one must rely on numerical methods to solve sets of coupled partial dierential equationsAmong the possible choices this paper focuses on a class called spectral methods in whichtypically the various functions are expanded in sets of orthogonal polynomials or functionsFirst a theoretical introduction of spectral expansion is given with a particular emphasis on thefast convergence of the spectral approximation We then present dierent approaches to solvingpartial dierential equations first limiting ourselves to the one-dimensional case with one ormore domains Generalization to more dimensions is then discussed In particular the case oftime evolutions is carefully studied and the stability of such evolutions investigated We thenpresent results obtained by various groups in the field of general relativity by means of spectralmethods Work which does not involve explicit time-evolutions is discussed going fromrapidly-rotating strange stars to the computation of black-holendashbinary initial data Finallythe evolution of various systems of astrophysical interest are presented from supernovae corecollapse to black-holendashbinary mergers

This review is licensed under a Creative CommonsAttribution-Non-Commercial-NoDerivs 30 Germany Licensehttpcreativecommonsorglicensesby-nc-nd30de

Spectral methods

A gentle introduction

Spectral interpolants

is a function

is a polynomial approximation of degree to

Convergence3

-1 -05 0 05 1x

-15

-1

-05

0

05

1

u = cos3(x2) - (x+1)

38

Pu

Iu

N=4

-1 -05 0 05 1x

-15

-1

-05

0

05

1

u = cos3(x2) - (x+1)

38

Pu

Iu

N=8

FIG 2 Same as Fig 1 with also the interpolant of u The collocation points are denoted by the circles

3 Interpolation

If one applies the Gauss quadratures to approximate the coecient of the expansion one obtains

un =1

n

N

j=0

u (xj) pn (xj)wj with n =N

j=0

p2n (xj)wj (5)

Let us precise that this is not exact in the sense that un = un However the computation of u only requires toevaluate u at the N + 1 collocation points The interpolant of u is then defined as the following polynomial

INu =N

n=0

unpn (x) (6)

The dierence between Inu and Pnu is called the aliasing error The interpolant of u is the spectral approximate ofu and one can show that it is the only polynomials of degree N that coincides with u at each collocation point

[Inu] (xi) = u (xi) i N (7)

Figure 2 shows the same function as Fig1 but the interpolant is also plotted One can see that indeed INu coincideswith u at the collocation points that are indicated by the circles Once again even with as few points as N = 8 nodierence can be seen between the various functions

Figure 3 shows the maximum dierence between INu and u on as a function of the degree of the approximation N We can observe the very general feature of spectral methods that the error decreases exponentially until one reachesthe machine accuracy (here 1014 the computation being done in double precision) This very fast convergenceexplains why spectral methods are so ecient especially compared to finite dierence ones where the error followsonly a power-law in terms of N We will later be more quantitative about the convergence properties of the spectralapproximation

Let us note that a function u can be described either by its value at each collocation point u (xi) or by the coecientsof the interpolant ui If the values at collocation points are known one is working in the configuration space and inthe coecient space if u is given in terms of its coecients There is a bijection between the two descriptions and onesimply goes from one space to another by using

bull un =1

n

N

j=0

u (xj) pn (xj) wj (configuration $ coecient)

bull u (xn) =N

j=0

ujpj (xn) (coecient $ configuration)

Depending on the operation one has to perform one choice of space is usually more suited than the other Forinstance let us assume that one wants to compute the derivative of u This is easily done if u is known in the

Exponential convergence 4

0 5 10 15 20 25 30 35

Number of coefficients

1e-15

1e-12

1e-09

1e-06

1e-03

1e+00m

ax |I N

u -

u|

FIG 3 Maximum dierence between INu and u as a function of the degree of the approximation N

-1 -05 0 05 1x

-3

-2

-1

0

1

2

ursquoIN

(ursquo)

(IN

u)rsquo

N=4

-1 -05 0 05 1x

-3

-2

-1

0

1

2

ursquoIN

(ursquo)

(IN

u)rsquo

N=8

FIG 4 First derivative of u = cos3 (x2) (x + 1)3 8 interpolant of the derivative and derivative of the interpolant forN = 4 and N = 8

coecient space Indeed one can simply approximate u by the derivative of the interpolant

u (x) [INu] =N

n=0

unpn (x) (8)

Such an approximation only requires the knowledge of the coecients of u and how the basis polynomials are derivedLet us note that the obtained polynomial even if it is a good approximation of u is not the interpolant of u Inother terms the interpolation and the derivation are two operations that do not commute (INu) = IN (u) This isclearly illustrated on Fig 4 where the derivative of u = cos3 (x2) (x + 1)3 8 is plotted along with the functions(INu) and IN (u) In particular one can note that the functions used to represent u ie (INu) does not coincidewith u at the collocation points However even with N = 8 only the three functions can not be distinguished byeye

The maximum dierence between u and (INu) as a function of N is shown on Fig 5 Once again the convergenceis exponential

Gibbs phenomena9

-1 -05 0 05 1x

-02

0

02

04

06

08

1

FIG 8 Step function and some interpolants for N = 7 N = 15 and N = 31 As N increases the maximum dierence staysconstant and the number of oscillations increases

4 5 6 7 8 9 10 20 30 40 50 60 70 8090

N

10-2

10-1

100

-1

1 |I N

u -

u| dx

FIG 9

1

1|INu u| dx as a function of N u being a step function The convergence obeys a power-law which decays slower

than N1

B A test problem

We propose to solve the equation

d2u

dx2 4

du

dx+ 4u = exp (x) + C (25)

with x [1 1] and C = 4e

1 + e2

As boundary conditions we simply ask that the solution is zero at theboundaries

u (1) = 0 and u (1) = 0 (26)

Representations

can be represented by either or

for algebraic operators use

for differential operators use

Differential equations

solve for then recover

Spectral methods in Numerical Relativity

Black hole evolution by spectral methods

Lawrence E Kidder Mark A Scheel and Saul A TeukolskyCenter for Radiophysics and Space Research Cornell University Ithaca New York 14853

Eric D Carlson and Gregory B CookDepartment of Physics Wake Forest University Winston-Salem North Carolina 27109

Received 15 May 2000 published 26 September 2000

Current methods of evolving a spacetime containing one or more black holes are plagued by instabilities that

prohibit long-term evolution Some of these instabilities may be due to the numerical method used tradition-

ally finite differencing In this paper we explore the use of a pseudospectral collocation PSC method for theevolution of a spherically symmetric black hole spacetime in one dimension using a hyperbolic formulation of

Einsteinrsquos equations We demonstrate that our PSC method is able to evolve a spherically symmetric black

hole spacetime forever without enforcing constraints even if we add dynamics via a Klein-Gordon scalar field

We find that in contrast with finite-differencing methods black hole excision is a trivial operation using PSC

applied to a hyperbolic formulation of Einsteinrsquos equations We discuss the extension of this method to three

spatial dimensions

PACS numbers 0425Dm 0270Hm

I INTRODUCTION AND SUMMARY

A major thrust of research in classical general relativity in

the past decade has been to devise algorithms to solve Ein-

steinrsquos equations numerically Despite advances in our ana-

lytic understanding of general relativity we still do not knowwhat all the features of the theory really are Numerical so-lutions will continue to provide fresh insights into the theoryas they have in the past for example the critical behavior inblack hole formation 1$ and the formation of toroidal blackholes 2$New urgency has been injected into numerical relativity

by the imminent deployment of the Laser InterferometricGravitational Wave Observatory LIGO The prime targetfor LIGO is coalescence of binary neutron star and blackhole systems The waveform is reasonably well predicted bythe post-Newtonian approximation when the binary compo-nents are at large separation However extracting the mostimportant physics requires us to be able to deal with fullynon-linear general relativity as the system spirals togetherand coalesces Moreover a number of people believe thatthere is a significant event rate for the coalescence of mas-sive black hole systems (20M) 3$ In this case LIGO ismost sensitive to waves emitted from the strong field regimeIndeed without some theoretical guidance as to what to ex-pect from this regime it is possible we may miss theseevents entirely 4$However the goal of developing a general algorithm that

can solve Einsteinrsquos equations for two black holes has re-mained elusive All attempts to date have been plagued byinstabilities These instabilities are caused by an interplay ofthree factors 1 Einsteinrsquos equations are an overdeterminedsystem with the evolution equations subject to constraintsSo if for example you choose to solve only the evolutionequations then there can be unstable solutions that are in factsolutions of the evolution equations but do not satisfy theconstraints Small numerical errors may cause these solu-tions to appear and swamp the true solution lsquolsquoconstraint-

violating modesrsquorsquo 2 The coordinate freedom inherent in

the theory means that it is very easy to impose coordinate

conditions that lead to numerical instabilities lsquolsquogaugemodesrsquorsquo 3 Experience has shown that the kind of bound-ary conditions we choose and how we implement them can

affect the stability of an algorithm enormously

Similar instabilities have hampered efforts to solve therelated problem of binary neutron stars only very recently56$ has there been some success in finding stable algo-rithms However black hole evolutions face an additionalobstacle that is absent in the case of neutron stars for neu-tron stars the gravitational field is everywhere regular but forblack holes one must somehow deal with the physical singu-larity that lurks inside each holeThere are two main approaches for handling these singu-

larities The first is to use gauge conditions eg maximalslicing that avoid the singularities altogether Such condi-tions however lead to large gradients in the gravitationalfield variables near the horizon These grow exponentially intime and ultimately cannot be resolved by the numerical evo-lution causing the code to crash The alternative approach isto excise the region containing the singularity from the com-putational domain and evolve only the exterior region If theexcision boundary is placed inside the horizon of the blackhole causality assures us that we do not need to impose aphysical boundary condition thereHowever black hole excision is only known to be math-

ematically well posed if the evolution equations are hyper-bolic with characteristic speeds less than or equal to c In thiscase the structure of the equations guarantees that even un-physical modes present in the solution gauge modesconstraint-violating modes behave causally and cannotpropagate out of the horizon For many representations ofgeneral relativity such as the usual Arnowitt-Deser-MisnerADM 7$ formulation the evolution equations are of nomathematical type for which well-posedness has beenproved so the suitability of these formulations for black holeexcision must be determined empirically on a case-by-case

PHYSICAL REVIEW D VOLUME 62 084032

0556-2821200062808403220$1500 copy2000 The American Physical Society62 084032-1

Stable evolutions in 1+1 d

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

Pros and cons

No dissipation

Exponential convergence

Superb results for little effort

Gibbs phenomena

Discrete equations are fully coupled

Lack of dissipation allows high frequency errors to remain

Can not freely choose location of nodes

Regge calculus

Regge calculus

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is piecewise flat

Curvature is a distribution

Defect angle at each vertex

2d example

2d example

2d example

2d example

2d example

3d example

3d example

3d example

3d example

3d example

4d example

Field equations

Some questions

Can smooth metrics be accurately approximated by Regge lattices

Do the Regge equations reduce to the Einstein equations in some suitable limit

Do solutions of the Regge equations converge to solutions of the Einstein equations

Some answers

Regge

Reggesolutions

Yes

Yes as an average

Allendorfer Weyl

Cheeger Muller Schrader

Continuum

Einsteinsolutions

Applications

FRW cosmologies

Brill wave initial data

Quantum gravityKasner cosmologies

Misner initial data

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 9: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Multiquadrics

Computers Math Appfic Vol 19 No 89 pp 127-145 1990 0097-494390 $300 + 000 Printed in Great Britain Pergamon Press plc

M U L T I Q U A D R I C S - - A S C A T T E R E D D A T A

A P P R O X I M A T I O N S C H E M E W I T H A P P L I C A T I O N S T O

C O M P U T A T I O N A L F L U I D - D Y N A M I C S - - I

SURFACE APPROXIMATIONS AND PARTIAL DERIVATIVE ESTIMATES

E J KANSA

Lawrence Livermore National Laboratory L-200 PO Box 808 Livermore CA 94550 USA

A~traet - -We present a powerful enhanced multiquadrics (MQ) scheme developed for spatial approxi- mations MQ is a true scattered data grid free scheme for representing surfaces and bodies in an arbitrary number of dimensions It is continuously differentiable and integrable and is capable of representing functions with steep gradients with very high accuracy Monotonicity and convexity are observed properties as a result of such high accuracy

Numerical results show that our modified MQ scheme is an excellent method not only for very accurate interpolation but also for partial derivative estimates MQ is applied to a higher order arbitrary Lagrangian-Eulerian (ALE) rezoning In the second paper of this series MQ is applied to parabolic hyperbolic and elliptic partial differential equations The parabolic problem uses an implicit time-marching scheme whereas the hyperbolic problem uses an explicit time-marching scheme We show that MQ is also exceptionally accurate and efficient The theory of Madych and Nelson shows that the MQ interpolant belongs to a space of functions which minimizes a semi-norm and gives credence to our results

1 BACKGROUND

The study of arbitrarily shaped curves surfaces and bodies having arbitrary data orderings has

immediate application to computational fluid-dynamics The governing equations not only include

source terms but gradients divergences and Laplacians In addition many physical processes occur

over a wide range of length scales To obtain quantitatively accurate approximations of the physics

quantitatively accurate estimates of the spatial variations of such variables are required In two

and three dimensions the range of such quantitatively accurate problems possible on current

multiprocessing super computers using standard finite difference or finite element codes is limited

The question is whether there exist alternative techniques or combinations of techniques which can

broaden the scope of problems to be solved by permitting steep gradients to be modelled using

fewer data points Toward that goal our study consists of two parts The first part will investigate

a new numerical technique of curve surface and body approximations of exceptional accuracy over

an arbitrary data arrangement The second part of this study will use such techniques to improve

parabolic hyperbolic or elliptic partial differential equations We will demonstrate that the study

of function approximations has a definite advantage to computational methods for partial

differential equations

One very important use of computers is the simulation of multidimensional spatial processes

In this paper we assumed that some finite physical quantity F is piecewise continuous in some

finite domain In many applications F is known only at a finite number of locations

xk k = 1 2 N where xk = x~ for a univariate problem and Xk = (x~yk )X for the

multivariate problem

From a finite amount of information regarding F we seek the best approximation which can

not only supply accurate estimates of F at arbitrary locations on the domain but will also provide

accurate estimates of the partial derivatives and definite integrals of F anywhere on the domain

The domain of F will consist of points xk of arbitrary ordering and sub-clustering A rectangular

grid is a very special case of a data ordering

Let us assume that an interpolation function f approximates F in the sense that

f(Xk)=F(Xk) k = l 2 N (1)

127

Example

solve linear system for

uniform amp random

choose

reconstruct interpolant

Example

linear system for

Bad news

is dense

is almost singular condition number typically

Must use Singular Value Decomposition expensive

Convergence is very rapid approx exponential

Good news

Bowen-York initial data

Bowen-York initial data

Bowen-York initial data

Pros and cons

Free to place nodes wherever we like

Exponential convergence

Must solve exceedingly ill-conditioned system

Requires care for asymptoticly flat geometries

Only two papers with results volunteers most welcome

Very accurate for very steep functions

Living Rev Relativity 12 (2009) 1httpwwwlivingreviewsorglrr-2009-1

Spectral Methods for Numerical Relativity

Philippe GrandclementLaboratoire Univers et Theories

UMR 8102 du CNRS Observatoire de ParisF-92195 Meudon Cedex France

email PhilippeGrandclementobspmfrhttpwwwluthobspmfrminisitephpnom=Grandclement

Jerome NovakLaboratoire Univers et Theories

UMR 8102 du CNRS Observatoire de ParisF-92195 Meudon Cedex Franceemail JeromeNovakobspmfr

httpwwwluthobspmfrminisitephpnom=Novak

Living Reviews in RelativityISSN 1433-8351

Accepted on 23 October 2008Published on 9 January 2009

Abstract

Equations arising in general relativity are usually too complicated to be solved analyticallyand one must rely on numerical methods to solve sets of coupled partial dierential equationsAmong the possible choices this paper focuses on a class called spectral methods in whichtypically the various functions are expanded in sets of orthogonal polynomials or functionsFirst a theoretical introduction of spectral expansion is given with a particular emphasis on thefast convergence of the spectral approximation We then present dierent approaches to solvingpartial dierential equations first limiting ourselves to the one-dimensional case with one ormore domains Generalization to more dimensions is then discussed In particular the case oftime evolutions is carefully studied and the stability of such evolutions investigated We thenpresent results obtained by various groups in the field of general relativity by means of spectralmethods Work which does not involve explicit time-evolutions is discussed going fromrapidly-rotating strange stars to the computation of black-holendashbinary initial data Finallythe evolution of various systems of astrophysical interest are presented from supernovae corecollapse to black-holendashbinary mergers

This review is licensed under a Creative CommonsAttribution-Non-Commercial-NoDerivs 30 Germany Licensehttpcreativecommonsorglicensesby-nc-nd30de

Spectral methods

A gentle introduction

Spectral interpolants

is a function

is a polynomial approximation of degree to

Convergence3

-1 -05 0 05 1x

-15

-1

-05

0

05

1

u = cos3(x2) - (x+1)

38

Pu

Iu

N=4

-1 -05 0 05 1x

-15

-1

-05

0

05

1

u = cos3(x2) - (x+1)

38

Pu

Iu

N=8

FIG 2 Same as Fig 1 with also the interpolant of u The collocation points are denoted by the circles

3 Interpolation

If one applies the Gauss quadratures to approximate the coecient of the expansion one obtains

un =1

n

N

j=0

u (xj) pn (xj)wj with n =N

j=0

p2n (xj)wj (5)

Let us precise that this is not exact in the sense that un = un However the computation of u only requires toevaluate u at the N + 1 collocation points The interpolant of u is then defined as the following polynomial

INu =N

n=0

unpn (x) (6)

The dierence between Inu and Pnu is called the aliasing error The interpolant of u is the spectral approximate ofu and one can show that it is the only polynomials of degree N that coincides with u at each collocation point

[Inu] (xi) = u (xi) i N (7)

Figure 2 shows the same function as Fig1 but the interpolant is also plotted One can see that indeed INu coincideswith u at the collocation points that are indicated by the circles Once again even with as few points as N = 8 nodierence can be seen between the various functions

Figure 3 shows the maximum dierence between INu and u on as a function of the degree of the approximation N We can observe the very general feature of spectral methods that the error decreases exponentially until one reachesthe machine accuracy (here 1014 the computation being done in double precision) This very fast convergenceexplains why spectral methods are so ecient especially compared to finite dierence ones where the error followsonly a power-law in terms of N We will later be more quantitative about the convergence properties of the spectralapproximation

Let us note that a function u can be described either by its value at each collocation point u (xi) or by the coecientsof the interpolant ui If the values at collocation points are known one is working in the configuration space and inthe coecient space if u is given in terms of its coecients There is a bijection between the two descriptions and onesimply goes from one space to another by using

bull un =1

n

N

j=0

u (xj) pn (xj) wj (configuration $ coecient)

bull u (xn) =N

j=0

ujpj (xn) (coecient $ configuration)

Depending on the operation one has to perform one choice of space is usually more suited than the other Forinstance let us assume that one wants to compute the derivative of u This is easily done if u is known in the

Exponential convergence 4

0 5 10 15 20 25 30 35

Number of coefficients

1e-15

1e-12

1e-09

1e-06

1e-03

1e+00m

ax |I N

u -

u|

FIG 3 Maximum dierence between INu and u as a function of the degree of the approximation N

-1 -05 0 05 1x

-3

-2

-1

0

1

2

ursquoIN

(ursquo)

(IN

u)rsquo

N=4

-1 -05 0 05 1x

-3

-2

-1

0

1

2

ursquoIN

(ursquo)

(IN

u)rsquo

N=8

FIG 4 First derivative of u = cos3 (x2) (x + 1)3 8 interpolant of the derivative and derivative of the interpolant forN = 4 and N = 8

coecient space Indeed one can simply approximate u by the derivative of the interpolant

u (x) [INu] =N

n=0

unpn (x) (8)

Such an approximation only requires the knowledge of the coecients of u and how the basis polynomials are derivedLet us note that the obtained polynomial even if it is a good approximation of u is not the interpolant of u Inother terms the interpolation and the derivation are two operations that do not commute (INu) = IN (u) This isclearly illustrated on Fig 4 where the derivative of u = cos3 (x2) (x + 1)3 8 is plotted along with the functions(INu) and IN (u) In particular one can note that the functions used to represent u ie (INu) does not coincidewith u at the collocation points However even with N = 8 only the three functions can not be distinguished byeye

The maximum dierence between u and (INu) as a function of N is shown on Fig 5 Once again the convergenceis exponential

Gibbs phenomena9

-1 -05 0 05 1x

-02

0

02

04

06

08

1

FIG 8 Step function and some interpolants for N = 7 N = 15 and N = 31 As N increases the maximum dierence staysconstant and the number of oscillations increases

4 5 6 7 8 9 10 20 30 40 50 60 70 8090

N

10-2

10-1

100

-1

1 |I N

u -

u| dx

FIG 9

1

1|INu u| dx as a function of N u being a step function The convergence obeys a power-law which decays slower

than N1

B A test problem

We propose to solve the equation

d2u

dx2 4

du

dx+ 4u = exp (x) + C (25)

with x [1 1] and C = 4e

1 + e2

As boundary conditions we simply ask that the solution is zero at theboundaries

u (1) = 0 and u (1) = 0 (26)

Representations

can be represented by either or

for algebraic operators use

for differential operators use

Differential equations

solve for then recover

Spectral methods in Numerical Relativity

Black hole evolution by spectral methods

Lawrence E Kidder Mark A Scheel and Saul A TeukolskyCenter for Radiophysics and Space Research Cornell University Ithaca New York 14853

Eric D Carlson and Gregory B CookDepartment of Physics Wake Forest University Winston-Salem North Carolina 27109

Received 15 May 2000 published 26 September 2000

Current methods of evolving a spacetime containing one or more black holes are plagued by instabilities that

prohibit long-term evolution Some of these instabilities may be due to the numerical method used tradition-

ally finite differencing In this paper we explore the use of a pseudospectral collocation PSC method for theevolution of a spherically symmetric black hole spacetime in one dimension using a hyperbolic formulation of

Einsteinrsquos equations We demonstrate that our PSC method is able to evolve a spherically symmetric black

hole spacetime forever without enforcing constraints even if we add dynamics via a Klein-Gordon scalar field

We find that in contrast with finite-differencing methods black hole excision is a trivial operation using PSC

applied to a hyperbolic formulation of Einsteinrsquos equations We discuss the extension of this method to three

spatial dimensions

PACS numbers 0425Dm 0270Hm

I INTRODUCTION AND SUMMARY

A major thrust of research in classical general relativity in

the past decade has been to devise algorithms to solve Ein-

steinrsquos equations numerically Despite advances in our ana-

lytic understanding of general relativity we still do not knowwhat all the features of the theory really are Numerical so-lutions will continue to provide fresh insights into the theoryas they have in the past for example the critical behavior inblack hole formation 1$ and the formation of toroidal blackholes 2$New urgency has been injected into numerical relativity

by the imminent deployment of the Laser InterferometricGravitational Wave Observatory LIGO The prime targetfor LIGO is coalescence of binary neutron star and blackhole systems The waveform is reasonably well predicted bythe post-Newtonian approximation when the binary compo-nents are at large separation However extracting the mostimportant physics requires us to be able to deal with fullynon-linear general relativity as the system spirals togetherand coalesces Moreover a number of people believe thatthere is a significant event rate for the coalescence of mas-sive black hole systems (20M) 3$ In this case LIGO ismost sensitive to waves emitted from the strong field regimeIndeed without some theoretical guidance as to what to ex-pect from this regime it is possible we may miss theseevents entirely 4$However the goal of developing a general algorithm that

can solve Einsteinrsquos equations for two black holes has re-mained elusive All attempts to date have been plagued byinstabilities These instabilities are caused by an interplay ofthree factors 1 Einsteinrsquos equations are an overdeterminedsystem with the evolution equations subject to constraintsSo if for example you choose to solve only the evolutionequations then there can be unstable solutions that are in factsolutions of the evolution equations but do not satisfy theconstraints Small numerical errors may cause these solu-tions to appear and swamp the true solution lsquolsquoconstraint-

violating modesrsquorsquo 2 The coordinate freedom inherent in

the theory means that it is very easy to impose coordinate

conditions that lead to numerical instabilities lsquolsquogaugemodesrsquorsquo 3 Experience has shown that the kind of bound-ary conditions we choose and how we implement them can

affect the stability of an algorithm enormously

Similar instabilities have hampered efforts to solve therelated problem of binary neutron stars only very recently56$ has there been some success in finding stable algo-rithms However black hole evolutions face an additionalobstacle that is absent in the case of neutron stars for neu-tron stars the gravitational field is everywhere regular but forblack holes one must somehow deal with the physical singu-larity that lurks inside each holeThere are two main approaches for handling these singu-

larities The first is to use gauge conditions eg maximalslicing that avoid the singularities altogether Such condi-tions however lead to large gradients in the gravitationalfield variables near the horizon These grow exponentially intime and ultimately cannot be resolved by the numerical evo-lution causing the code to crash The alternative approach isto excise the region containing the singularity from the com-putational domain and evolve only the exterior region If theexcision boundary is placed inside the horizon of the blackhole causality assures us that we do not need to impose aphysical boundary condition thereHowever black hole excision is only known to be math-

ematically well posed if the evolution equations are hyper-bolic with characteristic speeds less than or equal to c In thiscase the structure of the equations guarantees that even un-physical modes present in the solution gauge modesconstraint-violating modes behave causally and cannotpropagate out of the horizon For many representations ofgeneral relativity such as the usual Arnowitt-Deser-MisnerADM 7$ formulation the evolution equations are of nomathematical type for which well-posedness has beenproved so the suitability of these formulations for black holeexcision must be determined empirically on a case-by-case

PHYSICAL REVIEW D VOLUME 62 084032

0556-2821200062808403220$1500 copy2000 The American Physical Society62 084032-1

Stable evolutions in 1+1 d

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

Pros and cons

No dissipation

Exponential convergence

Superb results for little effort

Gibbs phenomena

Discrete equations are fully coupled

Lack of dissipation allows high frequency errors to remain

Can not freely choose location of nodes

Regge calculus

Regge calculus

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is piecewise flat

Curvature is a distribution

Defect angle at each vertex

2d example

2d example

2d example

2d example

2d example

3d example

3d example

3d example

3d example

3d example

4d example

Field equations

Some questions

Can smooth metrics be accurately approximated by Regge lattices

Do the Regge equations reduce to the Einstein equations in some suitable limit

Do solutions of the Regge equations converge to solutions of the Einstein equations

Some answers

Regge

Reggesolutions

Yes

Yes as an average

Allendorfer Weyl

Cheeger Muller Schrader

Continuum

Einsteinsolutions

Applications

FRW cosmologies

Brill wave initial data

Quantum gravityKasner cosmologies

Misner initial data

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 10: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Example

solve linear system for

uniform amp random

choose

reconstruct interpolant

Example

linear system for

Bad news

is dense

is almost singular condition number typically

Must use Singular Value Decomposition expensive

Convergence is very rapid approx exponential

Good news

Bowen-York initial data

Bowen-York initial data

Bowen-York initial data

Pros and cons

Free to place nodes wherever we like

Exponential convergence

Must solve exceedingly ill-conditioned system

Requires care for asymptoticly flat geometries

Only two papers with results volunteers most welcome

Very accurate for very steep functions

Living Rev Relativity 12 (2009) 1httpwwwlivingreviewsorglrr-2009-1

Spectral Methods for Numerical Relativity

Philippe GrandclementLaboratoire Univers et Theories

UMR 8102 du CNRS Observatoire de ParisF-92195 Meudon Cedex France

email PhilippeGrandclementobspmfrhttpwwwluthobspmfrminisitephpnom=Grandclement

Jerome NovakLaboratoire Univers et Theories

UMR 8102 du CNRS Observatoire de ParisF-92195 Meudon Cedex Franceemail JeromeNovakobspmfr

httpwwwluthobspmfrminisitephpnom=Novak

Living Reviews in RelativityISSN 1433-8351

Accepted on 23 October 2008Published on 9 January 2009

Abstract

Equations arising in general relativity are usually too complicated to be solved analyticallyand one must rely on numerical methods to solve sets of coupled partial dierential equationsAmong the possible choices this paper focuses on a class called spectral methods in whichtypically the various functions are expanded in sets of orthogonal polynomials or functionsFirst a theoretical introduction of spectral expansion is given with a particular emphasis on thefast convergence of the spectral approximation We then present dierent approaches to solvingpartial dierential equations first limiting ourselves to the one-dimensional case with one ormore domains Generalization to more dimensions is then discussed In particular the case oftime evolutions is carefully studied and the stability of such evolutions investigated We thenpresent results obtained by various groups in the field of general relativity by means of spectralmethods Work which does not involve explicit time-evolutions is discussed going fromrapidly-rotating strange stars to the computation of black-holendashbinary initial data Finallythe evolution of various systems of astrophysical interest are presented from supernovae corecollapse to black-holendashbinary mergers

This review is licensed under a Creative CommonsAttribution-Non-Commercial-NoDerivs 30 Germany Licensehttpcreativecommonsorglicensesby-nc-nd30de

Spectral methods

A gentle introduction

Spectral interpolants

is a function

is a polynomial approximation of degree to

Convergence3

-1 -05 0 05 1x

-15

-1

-05

0

05

1

u = cos3(x2) - (x+1)

38

Pu

Iu

N=4

-1 -05 0 05 1x

-15

-1

-05

0

05

1

u = cos3(x2) - (x+1)

38

Pu

Iu

N=8

FIG 2 Same as Fig 1 with also the interpolant of u The collocation points are denoted by the circles

3 Interpolation

If one applies the Gauss quadratures to approximate the coecient of the expansion one obtains

un =1

n

N

j=0

u (xj) pn (xj)wj with n =N

j=0

p2n (xj)wj (5)

Let us precise that this is not exact in the sense that un = un However the computation of u only requires toevaluate u at the N + 1 collocation points The interpolant of u is then defined as the following polynomial

INu =N

n=0

unpn (x) (6)

The dierence between Inu and Pnu is called the aliasing error The interpolant of u is the spectral approximate ofu and one can show that it is the only polynomials of degree N that coincides with u at each collocation point

[Inu] (xi) = u (xi) i N (7)

Figure 2 shows the same function as Fig1 but the interpolant is also plotted One can see that indeed INu coincideswith u at the collocation points that are indicated by the circles Once again even with as few points as N = 8 nodierence can be seen between the various functions

Figure 3 shows the maximum dierence between INu and u on as a function of the degree of the approximation N We can observe the very general feature of spectral methods that the error decreases exponentially until one reachesthe machine accuracy (here 1014 the computation being done in double precision) This very fast convergenceexplains why spectral methods are so ecient especially compared to finite dierence ones where the error followsonly a power-law in terms of N We will later be more quantitative about the convergence properties of the spectralapproximation

Let us note that a function u can be described either by its value at each collocation point u (xi) or by the coecientsof the interpolant ui If the values at collocation points are known one is working in the configuration space and inthe coecient space if u is given in terms of its coecients There is a bijection between the two descriptions and onesimply goes from one space to another by using

bull un =1

n

N

j=0

u (xj) pn (xj) wj (configuration $ coecient)

bull u (xn) =N

j=0

ujpj (xn) (coecient $ configuration)

Depending on the operation one has to perform one choice of space is usually more suited than the other Forinstance let us assume that one wants to compute the derivative of u This is easily done if u is known in the

Exponential convergence 4

0 5 10 15 20 25 30 35

Number of coefficients

1e-15

1e-12

1e-09

1e-06

1e-03

1e+00m

ax |I N

u -

u|

FIG 3 Maximum dierence between INu and u as a function of the degree of the approximation N

-1 -05 0 05 1x

-3

-2

-1

0

1

2

ursquoIN

(ursquo)

(IN

u)rsquo

N=4

-1 -05 0 05 1x

-3

-2

-1

0

1

2

ursquoIN

(ursquo)

(IN

u)rsquo

N=8

FIG 4 First derivative of u = cos3 (x2) (x + 1)3 8 interpolant of the derivative and derivative of the interpolant forN = 4 and N = 8

coecient space Indeed one can simply approximate u by the derivative of the interpolant

u (x) [INu] =N

n=0

unpn (x) (8)

Such an approximation only requires the knowledge of the coecients of u and how the basis polynomials are derivedLet us note that the obtained polynomial even if it is a good approximation of u is not the interpolant of u Inother terms the interpolation and the derivation are two operations that do not commute (INu) = IN (u) This isclearly illustrated on Fig 4 where the derivative of u = cos3 (x2) (x + 1)3 8 is plotted along with the functions(INu) and IN (u) In particular one can note that the functions used to represent u ie (INu) does not coincidewith u at the collocation points However even with N = 8 only the three functions can not be distinguished byeye

The maximum dierence between u and (INu) as a function of N is shown on Fig 5 Once again the convergenceis exponential

Gibbs phenomena9

-1 -05 0 05 1x

-02

0

02

04

06

08

1

FIG 8 Step function and some interpolants for N = 7 N = 15 and N = 31 As N increases the maximum dierence staysconstant and the number of oscillations increases

4 5 6 7 8 9 10 20 30 40 50 60 70 8090

N

10-2

10-1

100

-1

1 |I N

u -

u| dx

FIG 9

1

1|INu u| dx as a function of N u being a step function The convergence obeys a power-law which decays slower

than N1

B A test problem

We propose to solve the equation

d2u

dx2 4

du

dx+ 4u = exp (x) + C (25)

with x [1 1] and C = 4e

1 + e2

As boundary conditions we simply ask that the solution is zero at theboundaries

u (1) = 0 and u (1) = 0 (26)

Representations

can be represented by either or

for algebraic operators use

for differential operators use

Differential equations

solve for then recover

Spectral methods in Numerical Relativity

Black hole evolution by spectral methods

Lawrence E Kidder Mark A Scheel and Saul A TeukolskyCenter for Radiophysics and Space Research Cornell University Ithaca New York 14853

Eric D Carlson and Gregory B CookDepartment of Physics Wake Forest University Winston-Salem North Carolina 27109

Received 15 May 2000 published 26 September 2000

Current methods of evolving a spacetime containing one or more black holes are plagued by instabilities that

prohibit long-term evolution Some of these instabilities may be due to the numerical method used tradition-

ally finite differencing In this paper we explore the use of a pseudospectral collocation PSC method for theevolution of a spherically symmetric black hole spacetime in one dimension using a hyperbolic formulation of

Einsteinrsquos equations We demonstrate that our PSC method is able to evolve a spherically symmetric black

hole spacetime forever without enforcing constraints even if we add dynamics via a Klein-Gordon scalar field

We find that in contrast with finite-differencing methods black hole excision is a trivial operation using PSC

applied to a hyperbolic formulation of Einsteinrsquos equations We discuss the extension of this method to three

spatial dimensions

PACS numbers 0425Dm 0270Hm

I INTRODUCTION AND SUMMARY

A major thrust of research in classical general relativity in

the past decade has been to devise algorithms to solve Ein-

steinrsquos equations numerically Despite advances in our ana-

lytic understanding of general relativity we still do not knowwhat all the features of the theory really are Numerical so-lutions will continue to provide fresh insights into the theoryas they have in the past for example the critical behavior inblack hole formation 1$ and the formation of toroidal blackholes 2$New urgency has been injected into numerical relativity

by the imminent deployment of the Laser InterferometricGravitational Wave Observatory LIGO The prime targetfor LIGO is coalescence of binary neutron star and blackhole systems The waveform is reasonably well predicted bythe post-Newtonian approximation when the binary compo-nents are at large separation However extracting the mostimportant physics requires us to be able to deal with fullynon-linear general relativity as the system spirals togetherand coalesces Moreover a number of people believe thatthere is a significant event rate for the coalescence of mas-sive black hole systems (20M) 3$ In this case LIGO ismost sensitive to waves emitted from the strong field regimeIndeed without some theoretical guidance as to what to ex-pect from this regime it is possible we may miss theseevents entirely 4$However the goal of developing a general algorithm that

can solve Einsteinrsquos equations for two black holes has re-mained elusive All attempts to date have been plagued byinstabilities These instabilities are caused by an interplay ofthree factors 1 Einsteinrsquos equations are an overdeterminedsystem with the evolution equations subject to constraintsSo if for example you choose to solve only the evolutionequations then there can be unstable solutions that are in factsolutions of the evolution equations but do not satisfy theconstraints Small numerical errors may cause these solu-tions to appear and swamp the true solution lsquolsquoconstraint-

violating modesrsquorsquo 2 The coordinate freedom inherent in

the theory means that it is very easy to impose coordinate

conditions that lead to numerical instabilities lsquolsquogaugemodesrsquorsquo 3 Experience has shown that the kind of bound-ary conditions we choose and how we implement them can

affect the stability of an algorithm enormously

Similar instabilities have hampered efforts to solve therelated problem of binary neutron stars only very recently56$ has there been some success in finding stable algo-rithms However black hole evolutions face an additionalobstacle that is absent in the case of neutron stars for neu-tron stars the gravitational field is everywhere regular but forblack holes one must somehow deal with the physical singu-larity that lurks inside each holeThere are two main approaches for handling these singu-

larities The first is to use gauge conditions eg maximalslicing that avoid the singularities altogether Such condi-tions however lead to large gradients in the gravitationalfield variables near the horizon These grow exponentially intime and ultimately cannot be resolved by the numerical evo-lution causing the code to crash The alternative approach isto excise the region containing the singularity from the com-putational domain and evolve only the exterior region If theexcision boundary is placed inside the horizon of the blackhole causality assures us that we do not need to impose aphysical boundary condition thereHowever black hole excision is only known to be math-

ematically well posed if the evolution equations are hyper-bolic with characteristic speeds less than or equal to c In thiscase the structure of the equations guarantees that even un-physical modes present in the solution gauge modesconstraint-violating modes behave causally and cannotpropagate out of the horizon For many representations ofgeneral relativity such as the usual Arnowitt-Deser-MisnerADM 7$ formulation the evolution equations are of nomathematical type for which well-posedness has beenproved so the suitability of these formulations for black holeexcision must be determined empirically on a case-by-case

PHYSICAL REVIEW D VOLUME 62 084032

0556-2821200062808403220$1500 copy2000 The American Physical Society62 084032-1

Stable evolutions in 1+1 d

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

Pros and cons

No dissipation

Exponential convergence

Superb results for little effort

Gibbs phenomena

Discrete equations are fully coupled

Lack of dissipation allows high frequency errors to remain

Can not freely choose location of nodes

Regge calculus

Regge calculus

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is piecewise flat

Curvature is a distribution

Defect angle at each vertex

2d example

2d example

2d example

2d example

2d example

3d example

3d example

3d example

3d example

3d example

4d example

Field equations

Some questions

Can smooth metrics be accurately approximated by Regge lattices

Do the Regge equations reduce to the Einstein equations in some suitable limit

Do solutions of the Regge equations converge to solutions of the Einstein equations

Some answers

Regge

Reggesolutions

Yes

Yes as an average

Allendorfer Weyl

Cheeger Muller Schrader

Continuum

Einsteinsolutions

Applications

FRW cosmologies

Brill wave initial data

Quantum gravityKasner cosmologies

Misner initial data

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 11: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Example

linear system for

Bad news

is dense

is almost singular condition number typically

Must use Singular Value Decomposition expensive

Convergence is very rapid approx exponential

Good news

Bowen-York initial data

Bowen-York initial data

Bowen-York initial data

Pros and cons

Free to place nodes wherever we like

Exponential convergence

Must solve exceedingly ill-conditioned system

Requires care for asymptoticly flat geometries

Only two papers with results volunteers most welcome

Very accurate for very steep functions

Living Rev Relativity 12 (2009) 1httpwwwlivingreviewsorglrr-2009-1

Spectral Methods for Numerical Relativity

Philippe GrandclementLaboratoire Univers et Theories

UMR 8102 du CNRS Observatoire de ParisF-92195 Meudon Cedex France

email PhilippeGrandclementobspmfrhttpwwwluthobspmfrminisitephpnom=Grandclement

Jerome NovakLaboratoire Univers et Theories

UMR 8102 du CNRS Observatoire de ParisF-92195 Meudon Cedex Franceemail JeromeNovakobspmfr

httpwwwluthobspmfrminisitephpnom=Novak

Living Reviews in RelativityISSN 1433-8351

Accepted on 23 October 2008Published on 9 January 2009

Abstract

Equations arising in general relativity are usually too complicated to be solved analyticallyand one must rely on numerical methods to solve sets of coupled partial dierential equationsAmong the possible choices this paper focuses on a class called spectral methods in whichtypically the various functions are expanded in sets of orthogonal polynomials or functionsFirst a theoretical introduction of spectral expansion is given with a particular emphasis on thefast convergence of the spectral approximation We then present dierent approaches to solvingpartial dierential equations first limiting ourselves to the one-dimensional case with one ormore domains Generalization to more dimensions is then discussed In particular the case oftime evolutions is carefully studied and the stability of such evolutions investigated We thenpresent results obtained by various groups in the field of general relativity by means of spectralmethods Work which does not involve explicit time-evolutions is discussed going fromrapidly-rotating strange stars to the computation of black-holendashbinary initial data Finallythe evolution of various systems of astrophysical interest are presented from supernovae corecollapse to black-holendashbinary mergers

This review is licensed under a Creative CommonsAttribution-Non-Commercial-NoDerivs 30 Germany Licensehttpcreativecommonsorglicensesby-nc-nd30de

Spectral methods

A gentle introduction

Spectral interpolants

is a function

is a polynomial approximation of degree to

Convergence3

-1 -05 0 05 1x

-15

-1

-05

0

05

1

u = cos3(x2) - (x+1)

38

Pu

Iu

N=4

-1 -05 0 05 1x

-15

-1

-05

0

05

1

u = cos3(x2) - (x+1)

38

Pu

Iu

N=8

FIG 2 Same as Fig 1 with also the interpolant of u The collocation points are denoted by the circles

3 Interpolation

If one applies the Gauss quadratures to approximate the coecient of the expansion one obtains

un =1

n

N

j=0

u (xj) pn (xj)wj with n =N

j=0

p2n (xj)wj (5)

Let us precise that this is not exact in the sense that un = un However the computation of u only requires toevaluate u at the N + 1 collocation points The interpolant of u is then defined as the following polynomial

INu =N

n=0

unpn (x) (6)

The dierence between Inu and Pnu is called the aliasing error The interpolant of u is the spectral approximate ofu and one can show that it is the only polynomials of degree N that coincides with u at each collocation point

[Inu] (xi) = u (xi) i N (7)

Figure 2 shows the same function as Fig1 but the interpolant is also plotted One can see that indeed INu coincideswith u at the collocation points that are indicated by the circles Once again even with as few points as N = 8 nodierence can be seen between the various functions

Figure 3 shows the maximum dierence between INu and u on as a function of the degree of the approximation N We can observe the very general feature of spectral methods that the error decreases exponentially until one reachesthe machine accuracy (here 1014 the computation being done in double precision) This very fast convergenceexplains why spectral methods are so ecient especially compared to finite dierence ones where the error followsonly a power-law in terms of N We will later be more quantitative about the convergence properties of the spectralapproximation

Let us note that a function u can be described either by its value at each collocation point u (xi) or by the coecientsof the interpolant ui If the values at collocation points are known one is working in the configuration space and inthe coecient space if u is given in terms of its coecients There is a bijection between the two descriptions and onesimply goes from one space to another by using

bull un =1

n

N

j=0

u (xj) pn (xj) wj (configuration $ coecient)

bull u (xn) =N

j=0

ujpj (xn) (coecient $ configuration)

Depending on the operation one has to perform one choice of space is usually more suited than the other Forinstance let us assume that one wants to compute the derivative of u This is easily done if u is known in the

Exponential convergence 4

0 5 10 15 20 25 30 35

Number of coefficients

1e-15

1e-12

1e-09

1e-06

1e-03

1e+00m

ax |I N

u -

u|

FIG 3 Maximum dierence between INu and u as a function of the degree of the approximation N

-1 -05 0 05 1x

-3

-2

-1

0

1

2

ursquoIN

(ursquo)

(IN

u)rsquo

N=4

-1 -05 0 05 1x

-3

-2

-1

0

1

2

ursquoIN

(ursquo)

(IN

u)rsquo

N=8

FIG 4 First derivative of u = cos3 (x2) (x + 1)3 8 interpolant of the derivative and derivative of the interpolant forN = 4 and N = 8

coecient space Indeed one can simply approximate u by the derivative of the interpolant

u (x) [INu] =N

n=0

unpn (x) (8)

Such an approximation only requires the knowledge of the coecients of u and how the basis polynomials are derivedLet us note that the obtained polynomial even if it is a good approximation of u is not the interpolant of u Inother terms the interpolation and the derivation are two operations that do not commute (INu) = IN (u) This isclearly illustrated on Fig 4 where the derivative of u = cos3 (x2) (x + 1)3 8 is plotted along with the functions(INu) and IN (u) In particular one can note that the functions used to represent u ie (INu) does not coincidewith u at the collocation points However even with N = 8 only the three functions can not be distinguished byeye

The maximum dierence between u and (INu) as a function of N is shown on Fig 5 Once again the convergenceis exponential

Gibbs phenomena9

-1 -05 0 05 1x

-02

0

02

04

06

08

1

FIG 8 Step function and some interpolants for N = 7 N = 15 and N = 31 As N increases the maximum dierence staysconstant and the number of oscillations increases

4 5 6 7 8 9 10 20 30 40 50 60 70 8090

N

10-2

10-1

100

-1

1 |I N

u -

u| dx

FIG 9

1

1|INu u| dx as a function of N u being a step function The convergence obeys a power-law which decays slower

than N1

B A test problem

We propose to solve the equation

d2u

dx2 4

du

dx+ 4u = exp (x) + C (25)

with x [1 1] and C = 4e

1 + e2

As boundary conditions we simply ask that the solution is zero at theboundaries

u (1) = 0 and u (1) = 0 (26)

Representations

can be represented by either or

for algebraic operators use

for differential operators use

Differential equations

solve for then recover

Spectral methods in Numerical Relativity

Black hole evolution by spectral methods

Lawrence E Kidder Mark A Scheel and Saul A TeukolskyCenter for Radiophysics and Space Research Cornell University Ithaca New York 14853

Eric D Carlson and Gregory B CookDepartment of Physics Wake Forest University Winston-Salem North Carolina 27109

Received 15 May 2000 published 26 September 2000

Current methods of evolving a spacetime containing one or more black holes are plagued by instabilities that

prohibit long-term evolution Some of these instabilities may be due to the numerical method used tradition-

ally finite differencing In this paper we explore the use of a pseudospectral collocation PSC method for theevolution of a spherically symmetric black hole spacetime in one dimension using a hyperbolic formulation of

Einsteinrsquos equations We demonstrate that our PSC method is able to evolve a spherically symmetric black

hole spacetime forever without enforcing constraints even if we add dynamics via a Klein-Gordon scalar field

We find that in contrast with finite-differencing methods black hole excision is a trivial operation using PSC

applied to a hyperbolic formulation of Einsteinrsquos equations We discuss the extension of this method to three

spatial dimensions

PACS numbers 0425Dm 0270Hm

I INTRODUCTION AND SUMMARY

A major thrust of research in classical general relativity in

the past decade has been to devise algorithms to solve Ein-

steinrsquos equations numerically Despite advances in our ana-

lytic understanding of general relativity we still do not knowwhat all the features of the theory really are Numerical so-lutions will continue to provide fresh insights into the theoryas they have in the past for example the critical behavior inblack hole formation 1$ and the formation of toroidal blackholes 2$New urgency has been injected into numerical relativity

by the imminent deployment of the Laser InterferometricGravitational Wave Observatory LIGO The prime targetfor LIGO is coalescence of binary neutron star and blackhole systems The waveform is reasonably well predicted bythe post-Newtonian approximation when the binary compo-nents are at large separation However extracting the mostimportant physics requires us to be able to deal with fullynon-linear general relativity as the system spirals togetherand coalesces Moreover a number of people believe thatthere is a significant event rate for the coalescence of mas-sive black hole systems (20M) 3$ In this case LIGO ismost sensitive to waves emitted from the strong field regimeIndeed without some theoretical guidance as to what to ex-pect from this regime it is possible we may miss theseevents entirely 4$However the goal of developing a general algorithm that

can solve Einsteinrsquos equations for two black holes has re-mained elusive All attempts to date have been plagued byinstabilities These instabilities are caused by an interplay ofthree factors 1 Einsteinrsquos equations are an overdeterminedsystem with the evolution equations subject to constraintsSo if for example you choose to solve only the evolutionequations then there can be unstable solutions that are in factsolutions of the evolution equations but do not satisfy theconstraints Small numerical errors may cause these solu-tions to appear and swamp the true solution lsquolsquoconstraint-

violating modesrsquorsquo 2 The coordinate freedom inherent in

the theory means that it is very easy to impose coordinate

conditions that lead to numerical instabilities lsquolsquogaugemodesrsquorsquo 3 Experience has shown that the kind of bound-ary conditions we choose and how we implement them can

affect the stability of an algorithm enormously

Similar instabilities have hampered efforts to solve therelated problem of binary neutron stars only very recently56$ has there been some success in finding stable algo-rithms However black hole evolutions face an additionalobstacle that is absent in the case of neutron stars for neu-tron stars the gravitational field is everywhere regular but forblack holes one must somehow deal with the physical singu-larity that lurks inside each holeThere are two main approaches for handling these singu-

larities The first is to use gauge conditions eg maximalslicing that avoid the singularities altogether Such condi-tions however lead to large gradients in the gravitationalfield variables near the horizon These grow exponentially intime and ultimately cannot be resolved by the numerical evo-lution causing the code to crash The alternative approach isto excise the region containing the singularity from the com-putational domain and evolve only the exterior region If theexcision boundary is placed inside the horizon of the blackhole causality assures us that we do not need to impose aphysical boundary condition thereHowever black hole excision is only known to be math-

ematically well posed if the evolution equations are hyper-bolic with characteristic speeds less than or equal to c In thiscase the structure of the equations guarantees that even un-physical modes present in the solution gauge modesconstraint-violating modes behave causally and cannotpropagate out of the horizon For many representations ofgeneral relativity such as the usual Arnowitt-Deser-MisnerADM 7$ formulation the evolution equations are of nomathematical type for which well-posedness has beenproved so the suitability of these formulations for black holeexcision must be determined empirically on a case-by-case

PHYSICAL REVIEW D VOLUME 62 084032

0556-2821200062808403220$1500 copy2000 The American Physical Society62 084032-1

Stable evolutions in 1+1 d

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

Pros and cons

No dissipation

Exponential convergence

Superb results for little effort

Gibbs phenomena

Discrete equations are fully coupled

Lack of dissipation allows high frequency errors to remain

Can not freely choose location of nodes

Regge calculus

Regge calculus

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is piecewise flat

Curvature is a distribution

Defect angle at each vertex

2d example

2d example

2d example

2d example

2d example

3d example

3d example

3d example

3d example

3d example

4d example

Field equations

Some questions

Can smooth metrics be accurately approximated by Regge lattices

Do the Regge equations reduce to the Einstein equations in some suitable limit

Do solutions of the Regge equations converge to solutions of the Einstein equations

Some answers

Regge

Reggesolutions

Yes

Yes as an average

Allendorfer Weyl

Cheeger Muller Schrader

Continuum

Einsteinsolutions

Applications

FRW cosmologies

Brill wave initial data

Quantum gravityKasner cosmologies

Misner initial data

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 12: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Bowen-York initial data

Bowen-York initial data

Bowen-York initial data

Pros and cons

Free to place nodes wherever we like

Exponential convergence

Must solve exceedingly ill-conditioned system

Requires care for asymptoticly flat geometries

Only two papers with results volunteers most welcome

Very accurate for very steep functions

Living Rev Relativity 12 (2009) 1httpwwwlivingreviewsorglrr-2009-1

Spectral Methods for Numerical Relativity

Philippe GrandclementLaboratoire Univers et Theories

UMR 8102 du CNRS Observatoire de ParisF-92195 Meudon Cedex France

email PhilippeGrandclementobspmfrhttpwwwluthobspmfrminisitephpnom=Grandclement

Jerome NovakLaboratoire Univers et Theories

UMR 8102 du CNRS Observatoire de ParisF-92195 Meudon Cedex Franceemail JeromeNovakobspmfr

httpwwwluthobspmfrminisitephpnom=Novak

Living Reviews in RelativityISSN 1433-8351

Accepted on 23 October 2008Published on 9 January 2009

Abstract

Equations arising in general relativity are usually too complicated to be solved analyticallyand one must rely on numerical methods to solve sets of coupled partial dierential equationsAmong the possible choices this paper focuses on a class called spectral methods in whichtypically the various functions are expanded in sets of orthogonal polynomials or functionsFirst a theoretical introduction of spectral expansion is given with a particular emphasis on thefast convergence of the spectral approximation We then present dierent approaches to solvingpartial dierential equations first limiting ourselves to the one-dimensional case with one ormore domains Generalization to more dimensions is then discussed In particular the case oftime evolutions is carefully studied and the stability of such evolutions investigated We thenpresent results obtained by various groups in the field of general relativity by means of spectralmethods Work which does not involve explicit time-evolutions is discussed going fromrapidly-rotating strange stars to the computation of black-holendashbinary initial data Finallythe evolution of various systems of astrophysical interest are presented from supernovae corecollapse to black-holendashbinary mergers

This review is licensed under a Creative CommonsAttribution-Non-Commercial-NoDerivs 30 Germany Licensehttpcreativecommonsorglicensesby-nc-nd30de

Spectral methods

A gentle introduction

Spectral interpolants

is a function

is a polynomial approximation of degree to

Convergence3

-1 -05 0 05 1x

-15

-1

-05

0

05

1

u = cos3(x2) - (x+1)

38

Pu

Iu

N=4

-1 -05 0 05 1x

-15

-1

-05

0

05

1

u = cos3(x2) - (x+1)

38

Pu

Iu

N=8

FIG 2 Same as Fig 1 with also the interpolant of u The collocation points are denoted by the circles

3 Interpolation

If one applies the Gauss quadratures to approximate the coecient of the expansion one obtains

un =1

n

N

j=0

u (xj) pn (xj)wj with n =N

j=0

p2n (xj)wj (5)

Let us precise that this is not exact in the sense that un = un However the computation of u only requires toevaluate u at the N + 1 collocation points The interpolant of u is then defined as the following polynomial

INu =N

n=0

unpn (x) (6)

The dierence between Inu and Pnu is called the aliasing error The interpolant of u is the spectral approximate ofu and one can show that it is the only polynomials of degree N that coincides with u at each collocation point

[Inu] (xi) = u (xi) i N (7)

Figure 2 shows the same function as Fig1 but the interpolant is also plotted One can see that indeed INu coincideswith u at the collocation points that are indicated by the circles Once again even with as few points as N = 8 nodierence can be seen between the various functions

Figure 3 shows the maximum dierence between INu and u on as a function of the degree of the approximation N We can observe the very general feature of spectral methods that the error decreases exponentially until one reachesthe machine accuracy (here 1014 the computation being done in double precision) This very fast convergenceexplains why spectral methods are so ecient especially compared to finite dierence ones where the error followsonly a power-law in terms of N We will later be more quantitative about the convergence properties of the spectralapproximation

Let us note that a function u can be described either by its value at each collocation point u (xi) or by the coecientsof the interpolant ui If the values at collocation points are known one is working in the configuration space and inthe coecient space if u is given in terms of its coecients There is a bijection between the two descriptions and onesimply goes from one space to another by using

bull un =1

n

N

j=0

u (xj) pn (xj) wj (configuration $ coecient)

bull u (xn) =N

j=0

ujpj (xn) (coecient $ configuration)

Depending on the operation one has to perform one choice of space is usually more suited than the other Forinstance let us assume that one wants to compute the derivative of u This is easily done if u is known in the

Exponential convergence 4

0 5 10 15 20 25 30 35

Number of coefficients

1e-15

1e-12

1e-09

1e-06

1e-03

1e+00m

ax |I N

u -

u|

FIG 3 Maximum dierence between INu and u as a function of the degree of the approximation N

-1 -05 0 05 1x

-3

-2

-1

0

1

2

ursquoIN

(ursquo)

(IN

u)rsquo

N=4

-1 -05 0 05 1x

-3

-2

-1

0

1

2

ursquoIN

(ursquo)

(IN

u)rsquo

N=8

FIG 4 First derivative of u = cos3 (x2) (x + 1)3 8 interpolant of the derivative and derivative of the interpolant forN = 4 and N = 8

coecient space Indeed one can simply approximate u by the derivative of the interpolant

u (x) [INu] =N

n=0

unpn (x) (8)

Such an approximation only requires the knowledge of the coecients of u and how the basis polynomials are derivedLet us note that the obtained polynomial even if it is a good approximation of u is not the interpolant of u Inother terms the interpolation and the derivation are two operations that do not commute (INu) = IN (u) This isclearly illustrated on Fig 4 where the derivative of u = cos3 (x2) (x + 1)3 8 is plotted along with the functions(INu) and IN (u) In particular one can note that the functions used to represent u ie (INu) does not coincidewith u at the collocation points However even with N = 8 only the three functions can not be distinguished byeye

The maximum dierence between u and (INu) as a function of N is shown on Fig 5 Once again the convergenceis exponential

Gibbs phenomena9

-1 -05 0 05 1x

-02

0

02

04

06

08

1

FIG 8 Step function and some interpolants for N = 7 N = 15 and N = 31 As N increases the maximum dierence staysconstant and the number of oscillations increases

4 5 6 7 8 9 10 20 30 40 50 60 70 8090

N

10-2

10-1

100

-1

1 |I N

u -

u| dx

FIG 9

1

1|INu u| dx as a function of N u being a step function The convergence obeys a power-law which decays slower

than N1

B A test problem

We propose to solve the equation

d2u

dx2 4

du

dx+ 4u = exp (x) + C (25)

with x [1 1] and C = 4e

1 + e2

As boundary conditions we simply ask that the solution is zero at theboundaries

u (1) = 0 and u (1) = 0 (26)

Representations

can be represented by either or

for algebraic operators use

for differential operators use

Differential equations

solve for then recover

Spectral methods in Numerical Relativity

Black hole evolution by spectral methods

Lawrence E Kidder Mark A Scheel and Saul A TeukolskyCenter for Radiophysics and Space Research Cornell University Ithaca New York 14853

Eric D Carlson and Gregory B CookDepartment of Physics Wake Forest University Winston-Salem North Carolina 27109

Received 15 May 2000 published 26 September 2000

Current methods of evolving a spacetime containing one or more black holes are plagued by instabilities that

prohibit long-term evolution Some of these instabilities may be due to the numerical method used tradition-

ally finite differencing In this paper we explore the use of a pseudospectral collocation PSC method for theevolution of a spherically symmetric black hole spacetime in one dimension using a hyperbolic formulation of

Einsteinrsquos equations We demonstrate that our PSC method is able to evolve a spherically symmetric black

hole spacetime forever without enforcing constraints even if we add dynamics via a Klein-Gordon scalar field

We find that in contrast with finite-differencing methods black hole excision is a trivial operation using PSC

applied to a hyperbolic formulation of Einsteinrsquos equations We discuss the extension of this method to three

spatial dimensions

PACS numbers 0425Dm 0270Hm

I INTRODUCTION AND SUMMARY

A major thrust of research in classical general relativity in

the past decade has been to devise algorithms to solve Ein-

steinrsquos equations numerically Despite advances in our ana-

lytic understanding of general relativity we still do not knowwhat all the features of the theory really are Numerical so-lutions will continue to provide fresh insights into the theoryas they have in the past for example the critical behavior inblack hole formation 1$ and the formation of toroidal blackholes 2$New urgency has been injected into numerical relativity

by the imminent deployment of the Laser InterferometricGravitational Wave Observatory LIGO The prime targetfor LIGO is coalescence of binary neutron star and blackhole systems The waveform is reasonably well predicted bythe post-Newtonian approximation when the binary compo-nents are at large separation However extracting the mostimportant physics requires us to be able to deal with fullynon-linear general relativity as the system spirals togetherand coalesces Moreover a number of people believe thatthere is a significant event rate for the coalescence of mas-sive black hole systems (20M) 3$ In this case LIGO ismost sensitive to waves emitted from the strong field regimeIndeed without some theoretical guidance as to what to ex-pect from this regime it is possible we may miss theseevents entirely 4$However the goal of developing a general algorithm that

can solve Einsteinrsquos equations for two black holes has re-mained elusive All attempts to date have been plagued byinstabilities These instabilities are caused by an interplay ofthree factors 1 Einsteinrsquos equations are an overdeterminedsystem with the evolution equations subject to constraintsSo if for example you choose to solve only the evolutionequations then there can be unstable solutions that are in factsolutions of the evolution equations but do not satisfy theconstraints Small numerical errors may cause these solu-tions to appear and swamp the true solution lsquolsquoconstraint-

violating modesrsquorsquo 2 The coordinate freedom inherent in

the theory means that it is very easy to impose coordinate

conditions that lead to numerical instabilities lsquolsquogaugemodesrsquorsquo 3 Experience has shown that the kind of bound-ary conditions we choose and how we implement them can

affect the stability of an algorithm enormously

Similar instabilities have hampered efforts to solve therelated problem of binary neutron stars only very recently56$ has there been some success in finding stable algo-rithms However black hole evolutions face an additionalobstacle that is absent in the case of neutron stars for neu-tron stars the gravitational field is everywhere regular but forblack holes one must somehow deal with the physical singu-larity that lurks inside each holeThere are two main approaches for handling these singu-

larities The first is to use gauge conditions eg maximalslicing that avoid the singularities altogether Such condi-tions however lead to large gradients in the gravitationalfield variables near the horizon These grow exponentially intime and ultimately cannot be resolved by the numerical evo-lution causing the code to crash The alternative approach isto excise the region containing the singularity from the com-putational domain and evolve only the exterior region If theexcision boundary is placed inside the horizon of the blackhole causality assures us that we do not need to impose aphysical boundary condition thereHowever black hole excision is only known to be math-

ematically well posed if the evolution equations are hyper-bolic with characteristic speeds less than or equal to c In thiscase the structure of the equations guarantees that even un-physical modes present in the solution gauge modesconstraint-violating modes behave causally and cannotpropagate out of the horizon For many representations ofgeneral relativity such as the usual Arnowitt-Deser-MisnerADM 7$ formulation the evolution equations are of nomathematical type for which well-posedness has beenproved so the suitability of these formulations for black holeexcision must be determined empirically on a case-by-case

PHYSICAL REVIEW D VOLUME 62 084032

0556-2821200062808403220$1500 copy2000 The American Physical Society62 084032-1

Stable evolutions in 1+1 d

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

Pros and cons

No dissipation

Exponential convergence

Superb results for little effort

Gibbs phenomena

Discrete equations are fully coupled

Lack of dissipation allows high frequency errors to remain

Can not freely choose location of nodes

Regge calculus

Regge calculus

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is piecewise flat

Curvature is a distribution

Defect angle at each vertex

2d example

2d example

2d example

2d example

2d example

3d example

3d example

3d example

3d example

3d example

4d example

Field equations

Some questions

Can smooth metrics be accurately approximated by Regge lattices

Do the Regge equations reduce to the Einstein equations in some suitable limit

Do solutions of the Regge equations converge to solutions of the Einstein equations

Some answers

Regge

Reggesolutions

Yes

Yes as an average

Allendorfer Weyl

Cheeger Muller Schrader

Continuum

Einsteinsolutions

Applications

FRW cosmologies

Brill wave initial data

Quantum gravityKasner cosmologies

Misner initial data

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 13: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Bowen-York initial data

Bowen-York initial data

Pros and cons

Free to place nodes wherever we like

Exponential convergence

Must solve exceedingly ill-conditioned system

Requires care for asymptoticly flat geometries

Only two papers with results volunteers most welcome

Very accurate for very steep functions

Living Rev Relativity 12 (2009) 1httpwwwlivingreviewsorglrr-2009-1

Spectral Methods for Numerical Relativity

Philippe GrandclementLaboratoire Univers et Theories

UMR 8102 du CNRS Observatoire de ParisF-92195 Meudon Cedex France

email PhilippeGrandclementobspmfrhttpwwwluthobspmfrminisitephpnom=Grandclement

Jerome NovakLaboratoire Univers et Theories

UMR 8102 du CNRS Observatoire de ParisF-92195 Meudon Cedex Franceemail JeromeNovakobspmfr

httpwwwluthobspmfrminisitephpnom=Novak

Living Reviews in RelativityISSN 1433-8351

Accepted on 23 October 2008Published on 9 January 2009

Abstract

Equations arising in general relativity are usually too complicated to be solved analyticallyand one must rely on numerical methods to solve sets of coupled partial dierential equationsAmong the possible choices this paper focuses on a class called spectral methods in whichtypically the various functions are expanded in sets of orthogonal polynomials or functionsFirst a theoretical introduction of spectral expansion is given with a particular emphasis on thefast convergence of the spectral approximation We then present dierent approaches to solvingpartial dierential equations first limiting ourselves to the one-dimensional case with one ormore domains Generalization to more dimensions is then discussed In particular the case oftime evolutions is carefully studied and the stability of such evolutions investigated We thenpresent results obtained by various groups in the field of general relativity by means of spectralmethods Work which does not involve explicit time-evolutions is discussed going fromrapidly-rotating strange stars to the computation of black-holendashbinary initial data Finallythe evolution of various systems of astrophysical interest are presented from supernovae corecollapse to black-holendashbinary mergers

This review is licensed under a Creative CommonsAttribution-Non-Commercial-NoDerivs 30 Germany Licensehttpcreativecommonsorglicensesby-nc-nd30de

Spectral methods

A gentle introduction

Spectral interpolants

is a function

is a polynomial approximation of degree to

Convergence3

-1 -05 0 05 1x

-15

-1

-05

0

05

1

u = cos3(x2) - (x+1)

38

Pu

Iu

N=4

-1 -05 0 05 1x

-15

-1

-05

0

05

1

u = cos3(x2) - (x+1)

38

Pu

Iu

N=8

FIG 2 Same as Fig 1 with also the interpolant of u The collocation points are denoted by the circles

3 Interpolation

If one applies the Gauss quadratures to approximate the coecient of the expansion one obtains

un =1

n

N

j=0

u (xj) pn (xj)wj with n =N

j=0

p2n (xj)wj (5)

Let us precise that this is not exact in the sense that un = un However the computation of u only requires toevaluate u at the N + 1 collocation points The interpolant of u is then defined as the following polynomial

INu =N

n=0

unpn (x) (6)

The dierence between Inu and Pnu is called the aliasing error The interpolant of u is the spectral approximate ofu and one can show that it is the only polynomials of degree N that coincides with u at each collocation point

[Inu] (xi) = u (xi) i N (7)

Figure 2 shows the same function as Fig1 but the interpolant is also plotted One can see that indeed INu coincideswith u at the collocation points that are indicated by the circles Once again even with as few points as N = 8 nodierence can be seen between the various functions

Figure 3 shows the maximum dierence between INu and u on as a function of the degree of the approximation N We can observe the very general feature of spectral methods that the error decreases exponentially until one reachesthe machine accuracy (here 1014 the computation being done in double precision) This very fast convergenceexplains why spectral methods are so ecient especially compared to finite dierence ones where the error followsonly a power-law in terms of N We will later be more quantitative about the convergence properties of the spectralapproximation

Let us note that a function u can be described either by its value at each collocation point u (xi) or by the coecientsof the interpolant ui If the values at collocation points are known one is working in the configuration space and inthe coecient space if u is given in terms of its coecients There is a bijection between the two descriptions and onesimply goes from one space to another by using

bull un =1

n

N

j=0

u (xj) pn (xj) wj (configuration $ coecient)

bull u (xn) =N

j=0

ujpj (xn) (coecient $ configuration)

Depending on the operation one has to perform one choice of space is usually more suited than the other Forinstance let us assume that one wants to compute the derivative of u This is easily done if u is known in the

Exponential convergence 4

0 5 10 15 20 25 30 35

Number of coefficients

1e-15

1e-12

1e-09

1e-06

1e-03

1e+00m

ax |I N

u -

u|

FIG 3 Maximum dierence between INu and u as a function of the degree of the approximation N

-1 -05 0 05 1x

-3

-2

-1

0

1

2

ursquoIN

(ursquo)

(IN

u)rsquo

N=4

-1 -05 0 05 1x

-3

-2

-1

0

1

2

ursquoIN

(ursquo)

(IN

u)rsquo

N=8

FIG 4 First derivative of u = cos3 (x2) (x + 1)3 8 interpolant of the derivative and derivative of the interpolant forN = 4 and N = 8

coecient space Indeed one can simply approximate u by the derivative of the interpolant

u (x) [INu] =N

n=0

unpn (x) (8)

Such an approximation only requires the knowledge of the coecients of u and how the basis polynomials are derivedLet us note that the obtained polynomial even if it is a good approximation of u is not the interpolant of u Inother terms the interpolation and the derivation are two operations that do not commute (INu) = IN (u) This isclearly illustrated on Fig 4 where the derivative of u = cos3 (x2) (x + 1)3 8 is plotted along with the functions(INu) and IN (u) In particular one can note that the functions used to represent u ie (INu) does not coincidewith u at the collocation points However even with N = 8 only the three functions can not be distinguished byeye

The maximum dierence between u and (INu) as a function of N is shown on Fig 5 Once again the convergenceis exponential

Gibbs phenomena9

-1 -05 0 05 1x

-02

0

02

04

06

08

1

FIG 8 Step function and some interpolants for N = 7 N = 15 and N = 31 As N increases the maximum dierence staysconstant and the number of oscillations increases

4 5 6 7 8 9 10 20 30 40 50 60 70 8090

N

10-2

10-1

100

-1

1 |I N

u -

u| dx

FIG 9

1

1|INu u| dx as a function of N u being a step function The convergence obeys a power-law which decays slower

than N1

B A test problem

We propose to solve the equation

d2u

dx2 4

du

dx+ 4u = exp (x) + C (25)

with x [1 1] and C = 4e

1 + e2

As boundary conditions we simply ask that the solution is zero at theboundaries

u (1) = 0 and u (1) = 0 (26)

Representations

can be represented by either or

for algebraic operators use

for differential operators use

Differential equations

solve for then recover

Spectral methods in Numerical Relativity

Black hole evolution by spectral methods

Lawrence E Kidder Mark A Scheel and Saul A TeukolskyCenter for Radiophysics and Space Research Cornell University Ithaca New York 14853

Eric D Carlson and Gregory B CookDepartment of Physics Wake Forest University Winston-Salem North Carolina 27109

Received 15 May 2000 published 26 September 2000

Current methods of evolving a spacetime containing one or more black holes are plagued by instabilities that

prohibit long-term evolution Some of these instabilities may be due to the numerical method used tradition-

ally finite differencing In this paper we explore the use of a pseudospectral collocation PSC method for theevolution of a spherically symmetric black hole spacetime in one dimension using a hyperbolic formulation of

Einsteinrsquos equations We demonstrate that our PSC method is able to evolve a spherically symmetric black

hole spacetime forever without enforcing constraints even if we add dynamics via a Klein-Gordon scalar field

We find that in contrast with finite-differencing methods black hole excision is a trivial operation using PSC

applied to a hyperbolic formulation of Einsteinrsquos equations We discuss the extension of this method to three

spatial dimensions

PACS numbers 0425Dm 0270Hm

I INTRODUCTION AND SUMMARY

A major thrust of research in classical general relativity in

the past decade has been to devise algorithms to solve Ein-

steinrsquos equations numerically Despite advances in our ana-

lytic understanding of general relativity we still do not knowwhat all the features of the theory really are Numerical so-lutions will continue to provide fresh insights into the theoryas they have in the past for example the critical behavior inblack hole formation 1$ and the formation of toroidal blackholes 2$New urgency has been injected into numerical relativity

by the imminent deployment of the Laser InterferometricGravitational Wave Observatory LIGO The prime targetfor LIGO is coalescence of binary neutron star and blackhole systems The waveform is reasonably well predicted bythe post-Newtonian approximation when the binary compo-nents are at large separation However extracting the mostimportant physics requires us to be able to deal with fullynon-linear general relativity as the system spirals togetherand coalesces Moreover a number of people believe thatthere is a significant event rate for the coalescence of mas-sive black hole systems (20M) 3$ In this case LIGO ismost sensitive to waves emitted from the strong field regimeIndeed without some theoretical guidance as to what to ex-pect from this regime it is possible we may miss theseevents entirely 4$However the goal of developing a general algorithm that

can solve Einsteinrsquos equations for two black holes has re-mained elusive All attempts to date have been plagued byinstabilities These instabilities are caused by an interplay ofthree factors 1 Einsteinrsquos equations are an overdeterminedsystem with the evolution equations subject to constraintsSo if for example you choose to solve only the evolutionequations then there can be unstable solutions that are in factsolutions of the evolution equations but do not satisfy theconstraints Small numerical errors may cause these solu-tions to appear and swamp the true solution lsquolsquoconstraint-

violating modesrsquorsquo 2 The coordinate freedom inherent in

the theory means that it is very easy to impose coordinate

conditions that lead to numerical instabilities lsquolsquogaugemodesrsquorsquo 3 Experience has shown that the kind of bound-ary conditions we choose and how we implement them can

affect the stability of an algorithm enormously

Similar instabilities have hampered efforts to solve therelated problem of binary neutron stars only very recently56$ has there been some success in finding stable algo-rithms However black hole evolutions face an additionalobstacle that is absent in the case of neutron stars for neu-tron stars the gravitational field is everywhere regular but forblack holes one must somehow deal with the physical singu-larity that lurks inside each holeThere are two main approaches for handling these singu-

larities The first is to use gauge conditions eg maximalslicing that avoid the singularities altogether Such condi-tions however lead to large gradients in the gravitationalfield variables near the horizon These grow exponentially intime and ultimately cannot be resolved by the numerical evo-lution causing the code to crash The alternative approach isto excise the region containing the singularity from the com-putational domain and evolve only the exterior region If theexcision boundary is placed inside the horizon of the blackhole causality assures us that we do not need to impose aphysical boundary condition thereHowever black hole excision is only known to be math-

ematically well posed if the evolution equations are hyper-bolic with characteristic speeds less than or equal to c In thiscase the structure of the equations guarantees that even un-physical modes present in the solution gauge modesconstraint-violating modes behave causally and cannotpropagate out of the horizon For many representations ofgeneral relativity such as the usual Arnowitt-Deser-MisnerADM 7$ formulation the evolution equations are of nomathematical type for which well-posedness has beenproved so the suitability of these formulations for black holeexcision must be determined empirically on a case-by-case

PHYSICAL REVIEW D VOLUME 62 084032

0556-2821200062808403220$1500 copy2000 The American Physical Society62 084032-1

Stable evolutions in 1+1 d

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

Pros and cons

No dissipation

Exponential convergence

Superb results for little effort

Gibbs phenomena

Discrete equations are fully coupled

Lack of dissipation allows high frequency errors to remain

Can not freely choose location of nodes

Regge calculus

Regge calculus

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is piecewise flat

Curvature is a distribution

Defect angle at each vertex

2d example

2d example

2d example

2d example

2d example

3d example

3d example

3d example

3d example

3d example

4d example

Field equations

Some questions

Can smooth metrics be accurately approximated by Regge lattices

Do the Regge equations reduce to the Einstein equations in some suitable limit

Do solutions of the Regge equations converge to solutions of the Einstein equations

Some answers

Regge

Reggesolutions

Yes

Yes as an average

Allendorfer Weyl

Cheeger Muller Schrader

Continuum

Einsteinsolutions

Applications

FRW cosmologies

Brill wave initial data

Quantum gravityKasner cosmologies

Misner initial data

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 14: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Bowen-York initial data

Pros and cons

Free to place nodes wherever we like

Exponential convergence

Must solve exceedingly ill-conditioned system

Requires care for asymptoticly flat geometries

Only two papers with results volunteers most welcome

Very accurate for very steep functions

Living Rev Relativity 12 (2009) 1httpwwwlivingreviewsorglrr-2009-1

Spectral Methods for Numerical Relativity

Philippe GrandclementLaboratoire Univers et Theories

UMR 8102 du CNRS Observatoire de ParisF-92195 Meudon Cedex France

email PhilippeGrandclementobspmfrhttpwwwluthobspmfrminisitephpnom=Grandclement

Jerome NovakLaboratoire Univers et Theories

UMR 8102 du CNRS Observatoire de ParisF-92195 Meudon Cedex Franceemail JeromeNovakobspmfr

httpwwwluthobspmfrminisitephpnom=Novak

Living Reviews in RelativityISSN 1433-8351

Accepted on 23 October 2008Published on 9 January 2009

Abstract

Equations arising in general relativity are usually too complicated to be solved analyticallyand one must rely on numerical methods to solve sets of coupled partial dierential equationsAmong the possible choices this paper focuses on a class called spectral methods in whichtypically the various functions are expanded in sets of orthogonal polynomials or functionsFirst a theoretical introduction of spectral expansion is given with a particular emphasis on thefast convergence of the spectral approximation We then present dierent approaches to solvingpartial dierential equations first limiting ourselves to the one-dimensional case with one ormore domains Generalization to more dimensions is then discussed In particular the case oftime evolutions is carefully studied and the stability of such evolutions investigated We thenpresent results obtained by various groups in the field of general relativity by means of spectralmethods Work which does not involve explicit time-evolutions is discussed going fromrapidly-rotating strange stars to the computation of black-holendashbinary initial data Finallythe evolution of various systems of astrophysical interest are presented from supernovae corecollapse to black-holendashbinary mergers

This review is licensed under a Creative CommonsAttribution-Non-Commercial-NoDerivs 30 Germany Licensehttpcreativecommonsorglicensesby-nc-nd30de

Spectral methods

A gentle introduction

Spectral interpolants

is a function

is a polynomial approximation of degree to

Convergence3

-1 -05 0 05 1x

-15

-1

-05

0

05

1

u = cos3(x2) - (x+1)

38

Pu

Iu

N=4

-1 -05 0 05 1x

-15

-1

-05

0

05

1

u = cos3(x2) - (x+1)

38

Pu

Iu

N=8

FIG 2 Same as Fig 1 with also the interpolant of u The collocation points are denoted by the circles

3 Interpolation

If one applies the Gauss quadratures to approximate the coecient of the expansion one obtains

un =1

n

N

j=0

u (xj) pn (xj)wj with n =N

j=0

p2n (xj)wj (5)

Let us precise that this is not exact in the sense that un = un However the computation of u only requires toevaluate u at the N + 1 collocation points The interpolant of u is then defined as the following polynomial

INu =N

n=0

unpn (x) (6)

The dierence between Inu and Pnu is called the aliasing error The interpolant of u is the spectral approximate ofu and one can show that it is the only polynomials of degree N that coincides with u at each collocation point

[Inu] (xi) = u (xi) i N (7)

Figure 2 shows the same function as Fig1 but the interpolant is also plotted One can see that indeed INu coincideswith u at the collocation points that are indicated by the circles Once again even with as few points as N = 8 nodierence can be seen between the various functions

Figure 3 shows the maximum dierence between INu and u on as a function of the degree of the approximation N We can observe the very general feature of spectral methods that the error decreases exponentially until one reachesthe machine accuracy (here 1014 the computation being done in double precision) This very fast convergenceexplains why spectral methods are so ecient especially compared to finite dierence ones where the error followsonly a power-law in terms of N We will later be more quantitative about the convergence properties of the spectralapproximation

Let us note that a function u can be described either by its value at each collocation point u (xi) or by the coecientsof the interpolant ui If the values at collocation points are known one is working in the configuration space and inthe coecient space if u is given in terms of its coecients There is a bijection between the two descriptions and onesimply goes from one space to another by using

bull un =1

n

N

j=0

u (xj) pn (xj) wj (configuration $ coecient)

bull u (xn) =N

j=0

ujpj (xn) (coecient $ configuration)

Depending on the operation one has to perform one choice of space is usually more suited than the other Forinstance let us assume that one wants to compute the derivative of u This is easily done if u is known in the

Exponential convergence 4

0 5 10 15 20 25 30 35

Number of coefficients

1e-15

1e-12

1e-09

1e-06

1e-03

1e+00m

ax |I N

u -

u|

FIG 3 Maximum dierence between INu and u as a function of the degree of the approximation N

-1 -05 0 05 1x

-3

-2

-1

0

1

2

ursquoIN

(ursquo)

(IN

u)rsquo

N=4

-1 -05 0 05 1x

-3

-2

-1

0

1

2

ursquoIN

(ursquo)

(IN

u)rsquo

N=8

FIG 4 First derivative of u = cos3 (x2) (x + 1)3 8 interpolant of the derivative and derivative of the interpolant forN = 4 and N = 8

coecient space Indeed one can simply approximate u by the derivative of the interpolant

u (x) [INu] =N

n=0

unpn (x) (8)

Such an approximation only requires the knowledge of the coecients of u and how the basis polynomials are derivedLet us note that the obtained polynomial even if it is a good approximation of u is not the interpolant of u Inother terms the interpolation and the derivation are two operations that do not commute (INu) = IN (u) This isclearly illustrated on Fig 4 where the derivative of u = cos3 (x2) (x + 1)3 8 is plotted along with the functions(INu) and IN (u) In particular one can note that the functions used to represent u ie (INu) does not coincidewith u at the collocation points However even with N = 8 only the three functions can not be distinguished byeye

The maximum dierence between u and (INu) as a function of N is shown on Fig 5 Once again the convergenceis exponential

Gibbs phenomena9

-1 -05 0 05 1x

-02

0

02

04

06

08

1

FIG 8 Step function and some interpolants for N = 7 N = 15 and N = 31 As N increases the maximum dierence staysconstant and the number of oscillations increases

4 5 6 7 8 9 10 20 30 40 50 60 70 8090

N

10-2

10-1

100

-1

1 |I N

u -

u| dx

FIG 9

1

1|INu u| dx as a function of N u being a step function The convergence obeys a power-law which decays slower

than N1

B A test problem

We propose to solve the equation

d2u

dx2 4

du

dx+ 4u = exp (x) + C (25)

with x [1 1] and C = 4e

1 + e2

As boundary conditions we simply ask that the solution is zero at theboundaries

u (1) = 0 and u (1) = 0 (26)

Representations

can be represented by either or

for algebraic operators use

for differential operators use

Differential equations

solve for then recover

Spectral methods in Numerical Relativity

Black hole evolution by spectral methods

Lawrence E Kidder Mark A Scheel and Saul A TeukolskyCenter for Radiophysics and Space Research Cornell University Ithaca New York 14853

Eric D Carlson and Gregory B CookDepartment of Physics Wake Forest University Winston-Salem North Carolina 27109

Received 15 May 2000 published 26 September 2000

Current methods of evolving a spacetime containing one or more black holes are plagued by instabilities that

prohibit long-term evolution Some of these instabilities may be due to the numerical method used tradition-

ally finite differencing In this paper we explore the use of a pseudospectral collocation PSC method for theevolution of a spherically symmetric black hole spacetime in one dimension using a hyperbolic formulation of

Einsteinrsquos equations We demonstrate that our PSC method is able to evolve a spherically symmetric black

hole spacetime forever without enforcing constraints even if we add dynamics via a Klein-Gordon scalar field

We find that in contrast with finite-differencing methods black hole excision is a trivial operation using PSC

applied to a hyperbolic formulation of Einsteinrsquos equations We discuss the extension of this method to three

spatial dimensions

PACS numbers 0425Dm 0270Hm

I INTRODUCTION AND SUMMARY

A major thrust of research in classical general relativity in

the past decade has been to devise algorithms to solve Ein-

steinrsquos equations numerically Despite advances in our ana-

lytic understanding of general relativity we still do not knowwhat all the features of the theory really are Numerical so-lutions will continue to provide fresh insights into the theoryas they have in the past for example the critical behavior inblack hole formation 1$ and the formation of toroidal blackholes 2$New urgency has been injected into numerical relativity

by the imminent deployment of the Laser InterferometricGravitational Wave Observatory LIGO The prime targetfor LIGO is coalescence of binary neutron star and blackhole systems The waveform is reasonably well predicted bythe post-Newtonian approximation when the binary compo-nents are at large separation However extracting the mostimportant physics requires us to be able to deal with fullynon-linear general relativity as the system spirals togetherand coalesces Moreover a number of people believe thatthere is a significant event rate for the coalescence of mas-sive black hole systems (20M) 3$ In this case LIGO ismost sensitive to waves emitted from the strong field regimeIndeed without some theoretical guidance as to what to ex-pect from this regime it is possible we may miss theseevents entirely 4$However the goal of developing a general algorithm that

can solve Einsteinrsquos equations for two black holes has re-mained elusive All attempts to date have been plagued byinstabilities These instabilities are caused by an interplay ofthree factors 1 Einsteinrsquos equations are an overdeterminedsystem with the evolution equations subject to constraintsSo if for example you choose to solve only the evolutionequations then there can be unstable solutions that are in factsolutions of the evolution equations but do not satisfy theconstraints Small numerical errors may cause these solu-tions to appear and swamp the true solution lsquolsquoconstraint-

violating modesrsquorsquo 2 The coordinate freedom inherent in

the theory means that it is very easy to impose coordinate

conditions that lead to numerical instabilities lsquolsquogaugemodesrsquorsquo 3 Experience has shown that the kind of bound-ary conditions we choose and how we implement them can

affect the stability of an algorithm enormously

Similar instabilities have hampered efforts to solve therelated problem of binary neutron stars only very recently56$ has there been some success in finding stable algo-rithms However black hole evolutions face an additionalobstacle that is absent in the case of neutron stars for neu-tron stars the gravitational field is everywhere regular but forblack holes one must somehow deal with the physical singu-larity that lurks inside each holeThere are two main approaches for handling these singu-

larities The first is to use gauge conditions eg maximalslicing that avoid the singularities altogether Such condi-tions however lead to large gradients in the gravitationalfield variables near the horizon These grow exponentially intime and ultimately cannot be resolved by the numerical evo-lution causing the code to crash The alternative approach isto excise the region containing the singularity from the com-putational domain and evolve only the exterior region If theexcision boundary is placed inside the horizon of the blackhole causality assures us that we do not need to impose aphysical boundary condition thereHowever black hole excision is only known to be math-

ematically well posed if the evolution equations are hyper-bolic with characteristic speeds less than or equal to c In thiscase the structure of the equations guarantees that even un-physical modes present in the solution gauge modesconstraint-violating modes behave causally and cannotpropagate out of the horizon For many representations ofgeneral relativity such as the usual Arnowitt-Deser-MisnerADM 7$ formulation the evolution equations are of nomathematical type for which well-posedness has beenproved so the suitability of these formulations for black holeexcision must be determined empirically on a case-by-case

PHYSICAL REVIEW D VOLUME 62 084032

0556-2821200062808403220$1500 copy2000 The American Physical Society62 084032-1

Stable evolutions in 1+1 d

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

Pros and cons

No dissipation

Exponential convergence

Superb results for little effort

Gibbs phenomena

Discrete equations are fully coupled

Lack of dissipation allows high frequency errors to remain

Can not freely choose location of nodes

Regge calculus

Regge calculus

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is piecewise flat

Curvature is a distribution

Defect angle at each vertex

2d example

2d example

2d example

2d example

2d example

3d example

3d example

3d example

3d example

3d example

4d example

Field equations

Some questions

Can smooth metrics be accurately approximated by Regge lattices

Do the Regge equations reduce to the Einstein equations in some suitable limit

Do solutions of the Regge equations converge to solutions of the Einstein equations

Some answers

Regge

Reggesolutions

Yes

Yes as an average

Allendorfer Weyl

Cheeger Muller Schrader

Continuum

Einsteinsolutions

Applications

FRW cosmologies

Brill wave initial data

Quantum gravityKasner cosmologies

Misner initial data

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 15: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Pros and cons

Free to place nodes wherever we like

Exponential convergence

Must solve exceedingly ill-conditioned system

Requires care for asymptoticly flat geometries

Only two papers with results volunteers most welcome

Very accurate for very steep functions

Living Rev Relativity 12 (2009) 1httpwwwlivingreviewsorglrr-2009-1

Spectral Methods for Numerical Relativity

Philippe GrandclementLaboratoire Univers et Theories

UMR 8102 du CNRS Observatoire de ParisF-92195 Meudon Cedex France

email PhilippeGrandclementobspmfrhttpwwwluthobspmfrminisitephpnom=Grandclement

Jerome NovakLaboratoire Univers et Theories

UMR 8102 du CNRS Observatoire de ParisF-92195 Meudon Cedex Franceemail JeromeNovakobspmfr

httpwwwluthobspmfrminisitephpnom=Novak

Living Reviews in RelativityISSN 1433-8351

Accepted on 23 October 2008Published on 9 January 2009

Abstract

Equations arising in general relativity are usually too complicated to be solved analyticallyand one must rely on numerical methods to solve sets of coupled partial dierential equationsAmong the possible choices this paper focuses on a class called spectral methods in whichtypically the various functions are expanded in sets of orthogonal polynomials or functionsFirst a theoretical introduction of spectral expansion is given with a particular emphasis on thefast convergence of the spectral approximation We then present dierent approaches to solvingpartial dierential equations first limiting ourselves to the one-dimensional case with one ormore domains Generalization to more dimensions is then discussed In particular the case oftime evolutions is carefully studied and the stability of such evolutions investigated We thenpresent results obtained by various groups in the field of general relativity by means of spectralmethods Work which does not involve explicit time-evolutions is discussed going fromrapidly-rotating strange stars to the computation of black-holendashbinary initial data Finallythe evolution of various systems of astrophysical interest are presented from supernovae corecollapse to black-holendashbinary mergers

This review is licensed under a Creative CommonsAttribution-Non-Commercial-NoDerivs 30 Germany Licensehttpcreativecommonsorglicensesby-nc-nd30de

Spectral methods

A gentle introduction

Spectral interpolants

is a function

is a polynomial approximation of degree to

Convergence3

-1 -05 0 05 1x

-15

-1

-05

0

05

1

u = cos3(x2) - (x+1)

38

Pu

Iu

N=4

-1 -05 0 05 1x

-15

-1

-05

0

05

1

u = cos3(x2) - (x+1)

38

Pu

Iu

N=8

FIG 2 Same as Fig 1 with also the interpolant of u The collocation points are denoted by the circles

3 Interpolation

If one applies the Gauss quadratures to approximate the coecient of the expansion one obtains

un =1

n

N

j=0

u (xj) pn (xj)wj with n =N

j=0

p2n (xj)wj (5)

Let us precise that this is not exact in the sense that un = un However the computation of u only requires toevaluate u at the N + 1 collocation points The interpolant of u is then defined as the following polynomial

INu =N

n=0

unpn (x) (6)

The dierence between Inu and Pnu is called the aliasing error The interpolant of u is the spectral approximate ofu and one can show that it is the only polynomials of degree N that coincides with u at each collocation point

[Inu] (xi) = u (xi) i N (7)

Figure 2 shows the same function as Fig1 but the interpolant is also plotted One can see that indeed INu coincideswith u at the collocation points that are indicated by the circles Once again even with as few points as N = 8 nodierence can be seen between the various functions

Figure 3 shows the maximum dierence between INu and u on as a function of the degree of the approximation N We can observe the very general feature of spectral methods that the error decreases exponentially until one reachesthe machine accuracy (here 1014 the computation being done in double precision) This very fast convergenceexplains why spectral methods are so ecient especially compared to finite dierence ones where the error followsonly a power-law in terms of N We will later be more quantitative about the convergence properties of the spectralapproximation

Let us note that a function u can be described either by its value at each collocation point u (xi) or by the coecientsof the interpolant ui If the values at collocation points are known one is working in the configuration space and inthe coecient space if u is given in terms of its coecients There is a bijection between the two descriptions and onesimply goes from one space to another by using

bull un =1

n

N

j=0

u (xj) pn (xj) wj (configuration $ coecient)

bull u (xn) =N

j=0

ujpj (xn) (coecient $ configuration)

Depending on the operation one has to perform one choice of space is usually more suited than the other Forinstance let us assume that one wants to compute the derivative of u This is easily done if u is known in the

Exponential convergence 4

0 5 10 15 20 25 30 35

Number of coefficients

1e-15

1e-12

1e-09

1e-06

1e-03

1e+00m

ax |I N

u -

u|

FIG 3 Maximum dierence between INu and u as a function of the degree of the approximation N

-1 -05 0 05 1x

-3

-2

-1

0

1

2

ursquoIN

(ursquo)

(IN

u)rsquo

N=4

-1 -05 0 05 1x

-3

-2

-1

0

1

2

ursquoIN

(ursquo)

(IN

u)rsquo

N=8

FIG 4 First derivative of u = cos3 (x2) (x + 1)3 8 interpolant of the derivative and derivative of the interpolant forN = 4 and N = 8

coecient space Indeed one can simply approximate u by the derivative of the interpolant

u (x) [INu] =N

n=0

unpn (x) (8)

Such an approximation only requires the knowledge of the coecients of u and how the basis polynomials are derivedLet us note that the obtained polynomial even if it is a good approximation of u is not the interpolant of u Inother terms the interpolation and the derivation are two operations that do not commute (INu) = IN (u) This isclearly illustrated on Fig 4 where the derivative of u = cos3 (x2) (x + 1)3 8 is plotted along with the functions(INu) and IN (u) In particular one can note that the functions used to represent u ie (INu) does not coincidewith u at the collocation points However even with N = 8 only the three functions can not be distinguished byeye

The maximum dierence between u and (INu) as a function of N is shown on Fig 5 Once again the convergenceis exponential

Gibbs phenomena9

-1 -05 0 05 1x

-02

0

02

04

06

08

1

FIG 8 Step function and some interpolants for N = 7 N = 15 and N = 31 As N increases the maximum dierence staysconstant and the number of oscillations increases

4 5 6 7 8 9 10 20 30 40 50 60 70 8090

N

10-2

10-1

100

-1

1 |I N

u -

u| dx

FIG 9

1

1|INu u| dx as a function of N u being a step function The convergence obeys a power-law which decays slower

than N1

B A test problem

We propose to solve the equation

d2u

dx2 4

du

dx+ 4u = exp (x) + C (25)

with x [1 1] and C = 4e

1 + e2

As boundary conditions we simply ask that the solution is zero at theboundaries

u (1) = 0 and u (1) = 0 (26)

Representations

can be represented by either or

for algebraic operators use

for differential operators use

Differential equations

solve for then recover

Spectral methods in Numerical Relativity

Black hole evolution by spectral methods

Lawrence E Kidder Mark A Scheel and Saul A TeukolskyCenter for Radiophysics and Space Research Cornell University Ithaca New York 14853

Eric D Carlson and Gregory B CookDepartment of Physics Wake Forest University Winston-Salem North Carolina 27109

Received 15 May 2000 published 26 September 2000

Current methods of evolving a spacetime containing one or more black holes are plagued by instabilities that

prohibit long-term evolution Some of these instabilities may be due to the numerical method used tradition-

ally finite differencing In this paper we explore the use of a pseudospectral collocation PSC method for theevolution of a spherically symmetric black hole spacetime in one dimension using a hyperbolic formulation of

Einsteinrsquos equations We demonstrate that our PSC method is able to evolve a spherically symmetric black

hole spacetime forever without enforcing constraints even if we add dynamics via a Klein-Gordon scalar field

We find that in contrast with finite-differencing methods black hole excision is a trivial operation using PSC

applied to a hyperbolic formulation of Einsteinrsquos equations We discuss the extension of this method to three

spatial dimensions

PACS numbers 0425Dm 0270Hm

I INTRODUCTION AND SUMMARY

A major thrust of research in classical general relativity in

the past decade has been to devise algorithms to solve Ein-

steinrsquos equations numerically Despite advances in our ana-

lytic understanding of general relativity we still do not knowwhat all the features of the theory really are Numerical so-lutions will continue to provide fresh insights into the theoryas they have in the past for example the critical behavior inblack hole formation 1$ and the formation of toroidal blackholes 2$New urgency has been injected into numerical relativity

by the imminent deployment of the Laser InterferometricGravitational Wave Observatory LIGO The prime targetfor LIGO is coalescence of binary neutron star and blackhole systems The waveform is reasonably well predicted bythe post-Newtonian approximation when the binary compo-nents are at large separation However extracting the mostimportant physics requires us to be able to deal with fullynon-linear general relativity as the system spirals togetherand coalesces Moreover a number of people believe thatthere is a significant event rate for the coalescence of mas-sive black hole systems (20M) 3$ In this case LIGO ismost sensitive to waves emitted from the strong field regimeIndeed without some theoretical guidance as to what to ex-pect from this regime it is possible we may miss theseevents entirely 4$However the goal of developing a general algorithm that

can solve Einsteinrsquos equations for two black holes has re-mained elusive All attempts to date have been plagued byinstabilities These instabilities are caused by an interplay ofthree factors 1 Einsteinrsquos equations are an overdeterminedsystem with the evolution equations subject to constraintsSo if for example you choose to solve only the evolutionequations then there can be unstable solutions that are in factsolutions of the evolution equations but do not satisfy theconstraints Small numerical errors may cause these solu-tions to appear and swamp the true solution lsquolsquoconstraint-

violating modesrsquorsquo 2 The coordinate freedom inherent in

the theory means that it is very easy to impose coordinate

conditions that lead to numerical instabilities lsquolsquogaugemodesrsquorsquo 3 Experience has shown that the kind of bound-ary conditions we choose and how we implement them can

affect the stability of an algorithm enormously

Similar instabilities have hampered efforts to solve therelated problem of binary neutron stars only very recently56$ has there been some success in finding stable algo-rithms However black hole evolutions face an additionalobstacle that is absent in the case of neutron stars for neu-tron stars the gravitational field is everywhere regular but forblack holes one must somehow deal with the physical singu-larity that lurks inside each holeThere are two main approaches for handling these singu-

larities The first is to use gauge conditions eg maximalslicing that avoid the singularities altogether Such condi-tions however lead to large gradients in the gravitationalfield variables near the horizon These grow exponentially intime and ultimately cannot be resolved by the numerical evo-lution causing the code to crash The alternative approach isto excise the region containing the singularity from the com-putational domain and evolve only the exterior region If theexcision boundary is placed inside the horizon of the blackhole causality assures us that we do not need to impose aphysical boundary condition thereHowever black hole excision is only known to be math-

ematically well posed if the evolution equations are hyper-bolic with characteristic speeds less than or equal to c In thiscase the structure of the equations guarantees that even un-physical modes present in the solution gauge modesconstraint-violating modes behave causally and cannotpropagate out of the horizon For many representations ofgeneral relativity such as the usual Arnowitt-Deser-MisnerADM 7$ formulation the evolution equations are of nomathematical type for which well-posedness has beenproved so the suitability of these formulations for black holeexcision must be determined empirically on a case-by-case

PHYSICAL REVIEW D VOLUME 62 084032

0556-2821200062808403220$1500 copy2000 The American Physical Society62 084032-1

Stable evolutions in 1+1 d

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

Pros and cons

No dissipation

Exponential convergence

Superb results for little effort

Gibbs phenomena

Discrete equations are fully coupled

Lack of dissipation allows high frequency errors to remain

Can not freely choose location of nodes

Regge calculus

Regge calculus

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is piecewise flat

Curvature is a distribution

Defect angle at each vertex

2d example

2d example

2d example

2d example

2d example

3d example

3d example

3d example

3d example

3d example

4d example

Field equations

Some questions

Can smooth metrics be accurately approximated by Regge lattices

Do the Regge equations reduce to the Einstein equations in some suitable limit

Do solutions of the Regge equations converge to solutions of the Einstein equations

Some answers

Regge

Reggesolutions

Yes

Yes as an average

Allendorfer Weyl

Cheeger Muller Schrader

Continuum

Einsteinsolutions

Applications

FRW cosmologies

Brill wave initial data

Quantum gravityKasner cosmologies

Misner initial data

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 16: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Living Rev Relativity 12 (2009) 1httpwwwlivingreviewsorglrr-2009-1

Spectral Methods for Numerical Relativity

Philippe GrandclementLaboratoire Univers et Theories

UMR 8102 du CNRS Observatoire de ParisF-92195 Meudon Cedex France

email PhilippeGrandclementobspmfrhttpwwwluthobspmfrminisitephpnom=Grandclement

Jerome NovakLaboratoire Univers et Theories

UMR 8102 du CNRS Observatoire de ParisF-92195 Meudon Cedex Franceemail JeromeNovakobspmfr

httpwwwluthobspmfrminisitephpnom=Novak

Living Reviews in RelativityISSN 1433-8351

Accepted on 23 October 2008Published on 9 January 2009

Abstract

Equations arising in general relativity are usually too complicated to be solved analyticallyand one must rely on numerical methods to solve sets of coupled partial dierential equationsAmong the possible choices this paper focuses on a class called spectral methods in whichtypically the various functions are expanded in sets of orthogonal polynomials or functionsFirst a theoretical introduction of spectral expansion is given with a particular emphasis on thefast convergence of the spectral approximation We then present dierent approaches to solvingpartial dierential equations first limiting ourselves to the one-dimensional case with one ormore domains Generalization to more dimensions is then discussed In particular the case oftime evolutions is carefully studied and the stability of such evolutions investigated We thenpresent results obtained by various groups in the field of general relativity by means of spectralmethods Work which does not involve explicit time-evolutions is discussed going fromrapidly-rotating strange stars to the computation of black-holendashbinary initial data Finallythe evolution of various systems of astrophysical interest are presented from supernovae corecollapse to black-holendashbinary mergers

This review is licensed under a Creative CommonsAttribution-Non-Commercial-NoDerivs 30 Germany Licensehttpcreativecommonsorglicensesby-nc-nd30de

Spectral methods

A gentle introduction

Spectral interpolants

is a function

is a polynomial approximation of degree to

Convergence3

-1 -05 0 05 1x

-15

-1

-05

0

05

1

u = cos3(x2) - (x+1)

38

Pu

Iu

N=4

-1 -05 0 05 1x

-15

-1

-05

0

05

1

u = cos3(x2) - (x+1)

38

Pu

Iu

N=8

FIG 2 Same as Fig 1 with also the interpolant of u The collocation points are denoted by the circles

3 Interpolation

If one applies the Gauss quadratures to approximate the coecient of the expansion one obtains

un =1

n

N

j=0

u (xj) pn (xj)wj with n =N

j=0

p2n (xj)wj (5)

Let us precise that this is not exact in the sense that un = un However the computation of u only requires toevaluate u at the N + 1 collocation points The interpolant of u is then defined as the following polynomial

INu =N

n=0

unpn (x) (6)

The dierence between Inu and Pnu is called the aliasing error The interpolant of u is the spectral approximate ofu and one can show that it is the only polynomials of degree N that coincides with u at each collocation point

[Inu] (xi) = u (xi) i N (7)

Figure 2 shows the same function as Fig1 but the interpolant is also plotted One can see that indeed INu coincideswith u at the collocation points that are indicated by the circles Once again even with as few points as N = 8 nodierence can be seen between the various functions

Figure 3 shows the maximum dierence between INu and u on as a function of the degree of the approximation N We can observe the very general feature of spectral methods that the error decreases exponentially until one reachesthe machine accuracy (here 1014 the computation being done in double precision) This very fast convergenceexplains why spectral methods are so ecient especially compared to finite dierence ones where the error followsonly a power-law in terms of N We will later be more quantitative about the convergence properties of the spectralapproximation

Let us note that a function u can be described either by its value at each collocation point u (xi) or by the coecientsof the interpolant ui If the values at collocation points are known one is working in the configuration space and inthe coecient space if u is given in terms of its coecients There is a bijection between the two descriptions and onesimply goes from one space to another by using

bull un =1

n

N

j=0

u (xj) pn (xj) wj (configuration $ coecient)

bull u (xn) =N

j=0

ujpj (xn) (coecient $ configuration)

Depending on the operation one has to perform one choice of space is usually more suited than the other Forinstance let us assume that one wants to compute the derivative of u This is easily done if u is known in the

Exponential convergence 4

0 5 10 15 20 25 30 35

Number of coefficients

1e-15

1e-12

1e-09

1e-06

1e-03

1e+00m

ax |I N

u -

u|

FIG 3 Maximum dierence between INu and u as a function of the degree of the approximation N

-1 -05 0 05 1x

-3

-2

-1

0

1

2

ursquoIN

(ursquo)

(IN

u)rsquo

N=4

-1 -05 0 05 1x

-3

-2

-1

0

1

2

ursquoIN

(ursquo)

(IN

u)rsquo

N=8

FIG 4 First derivative of u = cos3 (x2) (x + 1)3 8 interpolant of the derivative and derivative of the interpolant forN = 4 and N = 8

coecient space Indeed one can simply approximate u by the derivative of the interpolant

u (x) [INu] =N

n=0

unpn (x) (8)

Such an approximation only requires the knowledge of the coecients of u and how the basis polynomials are derivedLet us note that the obtained polynomial even if it is a good approximation of u is not the interpolant of u Inother terms the interpolation and the derivation are two operations that do not commute (INu) = IN (u) This isclearly illustrated on Fig 4 where the derivative of u = cos3 (x2) (x + 1)3 8 is plotted along with the functions(INu) and IN (u) In particular one can note that the functions used to represent u ie (INu) does not coincidewith u at the collocation points However even with N = 8 only the three functions can not be distinguished byeye

The maximum dierence between u and (INu) as a function of N is shown on Fig 5 Once again the convergenceis exponential

Gibbs phenomena9

-1 -05 0 05 1x

-02

0

02

04

06

08

1

FIG 8 Step function and some interpolants for N = 7 N = 15 and N = 31 As N increases the maximum dierence staysconstant and the number of oscillations increases

4 5 6 7 8 9 10 20 30 40 50 60 70 8090

N

10-2

10-1

100

-1

1 |I N

u -

u| dx

FIG 9

1

1|INu u| dx as a function of N u being a step function The convergence obeys a power-law which decays slower

than N1

B A test problem

We propose to solve the equation

d2u

dx2 4

du

dx+ 4u = exp (x) + C (25)

with x [1 1] and C = 4e

1 + e2

As boundary conditions we simply ask that the solution is zero at theboundaries

u (1) = 0 and u (1) = 0 (26)

Representations

can be represented by either or

for algebraic operators use

for differential operators use

Differential equations

solve for then recover

Spectral methods in Numerical Relativity

Black hole evolution by spectral methods

Lawrence E Kidder Mark A Scheel and Saul A TeukolskyCenter for Radiophysics and Space Research Cornell University Ithaca New York 14853

Eric D Carlson and Gregory B CookDepartment of Physics Wake Forest University Winston-Salem North Carolina 27109

Received 15 May 2000 published 26 September 2000

Current methods of evolving a spacetime containing one or more black holes are plagued by instabilities that

prohibit long-term evolution Some of these instabilities may be due to the numerical method used tradition-

ally finite differencing In this paper we explore the use of a pseudospectral collocation PSC method for theevolution of a spherically symmetric black hole spacetime in one dimension using a hyperbolic formulation of

Einsteinrsquos equations We demonstrate that our PSC method is able to evolve a spherically symmetric black

hole spacetime forever without enforcing constraints even if we add dynamics via a Klein-Gordon scalar field

We find that in contrast with finite-differencing methods black hole excision is a trivial operation using PSC

applied to a hyperbolic formulation of Einsteinrsquos equations We discuss the extension of this method to three

spatial dimensions

PACS numbers 0425Dm 0270Hm

I INTRODUCTION AND SUMMARY

A major thrust of research in classical general relativity in

the past decade has been to devise algorithms to solve Ein-

steinrsquos equations numerically Despite advances in our ana-

lytic understanding of general relativity we still do not knowwhat all the features of the theory really are Numerical so-lutions will continue to provide fresh insights into the theoryas they have in the past for example the critical behavior inblack hole formation 1$ and the formation of toroidal blackholes 2$New urgency has been injected into numerical relativity

by the imminent deployment of the Laser InterferometricGravitational Wave Observatory LIGO The prime targetfor LIGO is coalescence of binary neutron star and blackhole systems The waveform is reasonably well predicted bythe post-Newtonian approximation when the binary compo-nents are at large separation However extracting the mostimportant physics requires us to be able to deal with fullynon-linear general relativity as the system spirals togetherand coalesces Moreover a number of people believe thatthere is a significant event rate for the coalescence of mas-sive black hole systems (20M) 3$ In this case LIGO ismost sensitive to waves emitted from the strong field regimeIndeed without some theoretical guidance as to what to ex-pect from this regime it is possible we may miss theseevents entirely 4$However the goal of developing a general algorithm that

can solve Einsteinrsquos equations for two black holes has re-mained elusive All attempts to date have been plagued byinstabilities These instabilities are caused by an interplay ofthree factors 1 Einsteinrsquos equations are an overdeterminedsystem with the evolution equations subject to constraintsSo if for example you choose to solve only the evolutionequations then there can be unstable solutions that are in factsolutions of the evolution equations but do not satisfy theconstraints Small numerical errors may cause these solu-tions to appear and swamp the true solution lsquolsquoconstraint-

violating modesrsquorsquo 2 The coordinate freedom inherent in

the theory means that it is very easy to impose coordinate

conditions that lead to numerical instabilities lsquolsquogaugemodesrsquorsquo 3 Experience has shown that the kind of bound-ary conditions we choose and how we implement them can

affect the stability of an algorithm enormously

Similar instabilities have hampered efforts to solve therelated problem of binary neutron stars only very recently56$ has there been some success in finding stable algo-rithms However black hole evolutions face an additionalobstacle that is absent in the case of neutron stars for neu-tron stars the gravitational field is everywhere regular but forblack holes one must somehow deal with the physical singu-larity that lurks inside each holeThere are two main approaches for handling these singu-

larities The first is to use gauge conditions eg maximalslicing that avoid the singularities altogether Such condi-tions however lead to large gradients in the gravitationalfield variables near the horizon These grow exponentially intime and ultimately cannot be resolved by the numerical evo-lution causing the code to crash The alternative approach isto excise the region containing the singularity from the com-putational domain and evolve only the exterior region If theexcision boundary is placed inside the horizon of the blackhole causality assures us that we do not need to impose aphysical boundary condition thereHowever black hole excision is only known to be math-

ematically well posed if the evolution equations are hyper-bolic with characteristic speeds less than or equal to c In thiscase the structure of the equations guarantees that even un-physical modes present in the solution gauge modesconstraint-violating modes behave causally and cannotpropagate out of the horizon For many representations ofgeneral relativity such as the usual Arnowitt-Deser-MisnerADM 7$ formulation the evolution equations are of nomathematical type for which well-posedness has beenproved so the suitability of these formulations for black holeexcision must be determined empirically on a case-by-case

PHYSICAL REVIEW D VOLUME 62 084032

0556-2821200062808403220$1500 copy2000 The American Physical Society62 084032-1

Stable evolutions in 1+1 d

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

Pros and cons

No dissipation

Exponential convergence

Superb results for little effort

Gibbs phenomena

Discrete equations are fully coupled

Lack of dissipation allows high frequency errors to remain

Can not freely choose location of nodes

Regge calculus

Regge calculus

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is piecewise flat

Curvature is a distribution

Defect angle at each vertex

2d example

2d example

2d example

2d example

2d example

3d example

3d example

3d example

3d example

3d example

4d example

Field equations

Some questions

Can smooth metrics be accurately approximated by Regge lattices

Do the Regge equations reduce to the Einstein equations in some suitable limit

Do solutions of the Regge equations converge to solutions of the Einstein equations

Some answers

Regge

Reggesolutions

Yes

Yes as an average

Allendorfer Weyl

Cheeger Muller Schrader

Continuum

Einsteinsolutions

Applications

FRW cosmologies

Brill wave initial data

Quantum gravityKasner cosmologies

Misner initial data

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 17: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

A gentle introduction

Spectral interpolants

is a function

is a polynomial approximation of degree to

Convergence3

-1 -05 0 05 1x

-15

-1

-05

0

05

1

u = cos3(x2) - (x+1)

38

Pu

Iu

N=4

-1 -05 0 05 1x

-15

-1

-05

0

05

1

u = cos3(x2) - (x+1)

38

Pu

Iu

N=8

FIG 2 Same as Fig 1 with also the interpolant of u The collocation points are denoted by the circles

3 Interpolation

If one applies the Gauss quadratures to approximate the coecient of the expansion one obtains

un =1

n

N

j=0

u (xj) pn (xj)wj with n =N

j=0

p2n (xj)wj (5)

Let us precise that this is not exact in the sense that un = un However the computation of u only requires toevaluate u at the N + 1 collocation points The interpolant of u is then defined as the following polynomial

INu =N

n=0

unpn (x) (6)

The dierence between Inu and Pnu is called the aliasing error The interpolant of u is the spectral approximate ofu and one can show that it is the only polynomials of degree N that coincides with u at each collocation point

[Inu] (xi) = u (xi) i N (7)

Figure 2 shows the same function as Fig1 but the interpolant is also plotted One can see that indeed INu coincideswith u at the collocation points that are indicated by the circles Once again even with as few points as N = 8 nodierence can be seen between the various functions

Figure 3 shows the maximum dierence between INu and u on as a function of the degree of the approximation N We can observe the very general feature of spectral methods that the error decreases exponentially until one reachesthe machine accuracy (here 1014 the computation being done in double precision) This very fast convergenceexplains why spectral methods are so ecient especially compared to finite dierence ones where the error followsonly a power-law in terms of N We will later be more quantitative about the convergence properties of the spectralapproximation

Let us note that a function u can be described either by its value at each collocation point u (xi) or by the coecientsof the interpolant ui If the values at collocation points are known one is working in the configuration space and inthe coecient space if u is given in terms of its coecients There is a bijection between the two descriptions and onesimply goes from one space to another by using

bull un =1

n

N

j=0

u (xj) pn (xj) wj (configuration $ coecient)

bull u (xn) =N

j=0

ujpj (xn) (coecient $ configuration)

Depending on the operation one has to perform one choice of space is usually more suited than the other Forinstance let us assume that one wants to compute the derivative of u This is easily done if u is known in the

Exponential convergence 4

0 5 10 15 20 25 30 35

Number of coefficients

1e-15

1e-12

1e-09

1e-06

1e-03

1e+00m

ax |I N

u -

u|

FIG 3 Maximum dierence between INu and u as a function of the degree of the approximation N

-1 -05 0 05 1x

-3

-2

-1

0

1

2

ursquoIN

(ursquo)

(IN

u)rsquo

N=4

-1 -05 0 05 1x

-3

-2

-1

0

1

2

ursquoIN

(ursquo)

(IN

u)rsquo

N=8

FIG 4 First derivative of u = cos3 (x2) (x + 1)3 8 interpolant of the derivative and derivative of the interpolant forN = 4 and N = 8

coecient space Indeed one can simply approximate u by the derivative of the interpolant

u (x) [INu] =N

n=0

unpn (x) (8)

Such an approximation only requires the knowledge of the coecients of u and how the basis polynomials are derivedLet us note that the obtained polynomial even if it is a good approximation of u is not the interpolant of u Inother terms the interpolation and the derivation are two operations that do not commute (INu) = IN (u) This isclearly illustrated on Fig 4 where the derivative of u = cos3 (x2) (x + 1)3 8 is plotted along with the functions(INu) and IN (u) In particular one can note that the functions used to represent u ie (INu) does not coincidewith u at the collocation points However even with N = 8 only the three functions can not be distinguished byeye

The maximum dierence between u and (INu) as a function of N is shown on Fig 5 Once again the convergenceis exponential

Gibbs phenomena9

-1 -05 0 05 1x

-02

0

02

04

06

08

1

FIG 8 Step function and some interpolants for N = 7 N = 15 and N = 31 As N increases the maximum dierence staysconstant and the number of oscillations increases

4 5 6 7 8 9 10 20 30 40 50 60 70 8090

N

10-2

10-1

100

-1

1 |I N

u -

u| dx

FIG 9

1

1|INu u| dx as a function of N u being a step function The convergence obeys a power-law which decays slower

than N1

B A test problem

We propose to solve the equation

d2u

dx2 4

du

dx+ 4u = exp (x) + C (25)

with x [1 1] and C = 4e

1 + e2

As boundary conditions we simply ask that the solution is zero at theboundaries

u (1) = 0 and u (1) = 0 (26)

Representations

can be represented by either or

for algebraic operators use

for differential operators use

Differential equations

solve for then recover

Spectral methods in Numerical Relativity

Black hole evolution by spectral methods

Lawrence E Kidder Mark A Scheel and Saul A TeukolskyCenter for Radiophysics and Space Research Cornell University Ithaca New York 14853

Eric D Carlson and Gregory B CookDepartment of Physics Wake Forest University Winston-Salem North Carolina 27109

Received 15 May 2000 published 26 September 2000

Current methods of evolving a spacetime containing one or more black holes are plagued by instabilities that

prohibit long-term evolution Some of these instabilities may be due to the numerical method used tradition-

ally finite differencing In this paper we explore the use of a pseudospectral collocation PSC method for theevolution of a spherically symmetric black hole spacetime in one dimension using a hyperbolic formulation of

Einsteinrsquos equations We demonstrate that our PSC method is able to evolve a spherically symmetric black

hole spacetime forever without enforcing constraints even if we add dynamics via a Klein-Gordon scalar field

We find that in contrast with finite-differencing methods black hole excision is a trivial operation using PSC

applied to a hyperbolic formulation of Einsteinrsquos equations We discuss the extension of this method to three

spatial dimensions

PACS numbers 0425Dm 0270Hm

I INTRODUCTION AND SUMMARY

A major thrust of research in classical general relativity in

the past decade has been to devise algorithms to solve Ein-

steinrsquos equations numerically Despite advances in our ana-

lytic understanding of general relativity we still do not knowwhat all the features of the theory really are Numerical so-lutions will continue to provide fresh insights into the theoryas they have in the past for example the critical behavior inblack hole formation 1$ and the formation of toroidal blackholes 2$New urgency has been injected into numerical relativity

by the imminent deployment of the Laser InterferometricGravitational Wave Observatory LIGO The prime targetfor LIGO is coalescence of binary neutron star and blackhole systems The waveform is reasonably well predicted bythe post-Newtonian approximation when the binary compo-nents are at large separation However extracting the mostimportant physics requires us to be able to deal with fullynon-linear general relativity as the system spirals togetherand coalesces Moreover a number of people believe thatthere is a significant event rate for the coalescence of mas-sive black hole systems (20M) 3$ In this case LIGO ismost sensitive to waves emitted from the strong field regimeIndeed without some theoretical guidance as to what to ex-pect from this regime it is possible we may miss theseevents entirely 4$However the goal of developing a general algorithm that

can solve Einsteinrsquos equations for two black holes has re-mained elusive All attempts to date have been plagued byinstabilities These instabilities are caused by an interplay ofthree factors 1 Einsteinrsquos equations are an overdeterminedsystem with the evolution equations subject to constraintsSo if for example you choose to solve only the evolutionequations then there can be unstable solutions that are in factsolutions of the evolution equations but do not satisfy theconstraints Small numerical errors may cause these solu-tions to appear and swamp the true solution lsquolsquoconstraint-

violating modesrsquorsquo 2 The coordinate freedom inherent in

the theory means that it is very easy to impose coordinate

conditions that lead to numerical instabilities lsquolsquogaugemodesrsquorsquo 3 Experience has shown that the kind of bound-ary conditions we choose and how we implement them can

affect the stability of an algorithm enormously

Similar instabilities have hampered efforts to solve therelated problem of binary neutron stars only very recently56$ has there been some success in finding stable algo-rithms However black hole evolutions face an additionalobstacle that is absent in the case of neutron stars for neu-tron stars the gravitational field is everywhere regular but forblack holes one must somehow deal with the physical singu-larity that lurks inside each holeThere are two main approaches for handling these singu-

larities The first is to use gauge conditions eg maximalslicing that avoid the singularities altogether Such condi-tions however lead to large gradients in the gravitationalfield variables near the horizon These grow exponentially intime and ultimately cannot be resolved by the numerical evo-lution causing the code to crash The alternative approach isto excise the region containing the singularity from the com-putational domain and evolve only the exterior region If theexcision boundary is placed inside the horizon of the blackhole causality assures us that we do not need to impose aphysical boundary condition thereHowever black hole excision is only known to be math-

ematically well posed if the evolution equations are hyper-bolic with characteristic speeds less than or equal to c In thiscase the structure of the equations guarantees that even un-physical modes present in the solution gauge modesconstraint-violating modes behave causally and cannotpropagate out of the horizon For many representations ofgeneral relativity such as the usual Arnowitt-Deser-MisnerADM 7$ formulation the evolution equations are of nomathematical type for which well-posedness has beenproved so the suitability of these formulations for black holeexcision must be determined empirically on a case-by-case

PHYSICAL REVIEW D VOLUME 62 084032

0556-2821200062808403220$1500 copy2000 The American Physical Society62 084032-1

Stable evolutions in 1+1 d

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

Pros and cons

No dissipation

Exponential convergence

Superb results for little effort

Gibbs phenomena

Discrete equations are fully coupled

Lack of dissipation allows high frequency errors to remain

Can not freely choose location of nodes

Regge calculus

Regge calculus

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is piecewise flat

Curvature is a distribution

Defect angle at each vertex

2d example

2d example

2d example

2d example

2d example

3d example

3d example

3d example

3d example

3d example

4d example

Field equations

Some questions

Can smooth metrics be accurately approximated by Regge lattices

Do the Regge equations reduce to the Einstein equations in some suitable limit

Do solutions of the Regge equations converge to solutions of the Einstein equations

Some answers

Regge

Reggesolutions

Yes

Yes as an average

Allendorfer Weyl

Cheeger Muller Schrader

Continuum

Einsteinsolutions

Applications

FRW cosmologies

Brill wave initial data

Quantum gravityKasner cosmologies

Misner initial data

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 18: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Spectral interpolants

is a function

is a polynomial approximation of degree to

Convergence3

-1 -05 0 05 1x

-15

-1

-05

0

05

1

u = cos3(x2) - (x+1)

38

Pu

Iu

N=4

-1 -05 0 05 1x

-15

-1

-05

0

05

1

u = cos3(x2) - (x+1)

38

Pu

Iu

N=8

FIG 2 Same as Fig 1 with also the interpolant of u The collocation points are denoted by the circles

3 Interpolation

If one applies the Gauss quadratures to approximate the coecient of the expansion one obtains

un =1

n

N

j=0

u (xj) pn (xj)wj with n =N

j=0

p2n (xj)wj (5)

Let us precise that this is not exact in the sense that un = un However the computation of u only requires toevaluate u at the N + 1 collocation points The interpolant of u is then defined as the following polynomial

INu =N

n=0

unpn (x) (6)

The dierence between Inu and Pnu is called the aliasing error The interpolant of u is the spectral approximate ofu and one can show that it is the only polynomials of degree N that coincides with u at each collocation point

[Inu] (xi) = u (xi) i N (7)

Figure 2 shows the same function as Fig1 but the interpolant is also plotted One can see that indeed INu coincideswith u at the collocation points that are indicated by the circles Once again even with as few points as N = 8 nodierence can be seen between the various functions

Figure 3 shows the maximum dierence between INu and u on as a function of the degree of the approximation N We can observe the very general feature of spectral methods that the error decreases exponentially until one reachesthe machine accuracy (here 1014 the computation being done in double precision) This very fast convergenceexplains why spectral methods are so ecient especially compared to finite dierence ones where the error followsonly a power-law in terms of N We will later be more quantitative about the convergence properties of the spectralapproximation

Let us note that a function u can be described either by its value at each collocation point u (xi) or by the coecientsof the interpolant ui If the values at collocation points are known one is working in the configuration space and inthe coecient space if u is given in terms of its coecients There is a bijection between the two descriptions and onesimply goes from one space to another by using

bull un =1

n

N

j=0

u (xj) pn (xj) wj (configuration $ coecient)

bull u (xn) =N

j=0

ujpj (xn) (coecient $ configuration)

Depending on the operation one has to perform one choice of space is usually more suited than the other Forinstance let us assume that one wants to compute the derivative of u This is easily done if u is known in the

Exponential convergence 4

0 5 10 15 20 25 30 35

Number of coefficients

1e-15

1e-12

1e-09

1e-06

1e-03

1e+00m

ax |I N

u -

u|

FIG 3 Maximum dierence between INu and u as a function of the degree of the approximation N

-1 -05 0 05 1x

-3

-2

-1

0

1

2

ursquoIN

(ursquo)

(IN

u)rsquo

N=4

-1 -05 0 05 1x

-3

-2

-1

0

1

2

ursquoIN

(ursquo)

(IN

u)rsquo

N=8

FIG 4 First derivative of u = cos3 (x2) (x + 1)3 8 interpolant of the derivative and derivative of the interpolant forN = 4 and N = 8

coecient space Indeed one can simply approximate u by the derivative of the interpolant

u (x) [INu] =N

n=0

unpn (x) (8)

Such an approximation only requires the knowledge of the coecients of u and how the basis polynomials are derivedLet us note that the obtained polynomial even if it is a good approximation of u is not the interpolant of u Inother terms the interpolation and the derivation are two operations that do not commute (INu) = IN (u) This isclearly illustrated on Fig 4 where the derivative of u = cos3 (x2) (x + 1)3 8 is plotted along with the functions(INu) and IN (u) In particular one can note that the functions used to represent u ie (INu) does not coincidewith u at the collocation points However even with N = 8 only the three functions can not be distinguished byeye

The maximum dierence between u and (INu) as a function of N is shown on Fig 5 Once again the convergenceis exponential

Gibbs phenomena9

-1 -05 0 05 1x

-02

0

02

04

06

08

1

FIG 8 Step function and some interpolants for N = 7 N = 15 and N = 31 As N increases the maximum dierence staysconstant and the number of oscillations increases

4 5 6 7 8 9 10 20 30 40 50 60 70 8090

N

10-2

10-1

100

-1

1 |I N

u -

u| dx

FIG 9

1

1|INu u| dx as a function of N u being a step function The convergence obeys a power-law which decays slower

than N1

B A test problem

We propose to solve the equation

d2u

dx2 4

du

dx+ 4u = exp (x) + C (25)

with x [1 1] and C = 4e

1 + e2

As boundary conditions we simply ask that the solution is zero at theboundaries

u (1) = 0 and u (1) = 0 (26)

Representations

can be represented by either or

for algebraic operators use

for differential operators use

Differential equations

solve for then recover

Spectral methods in Numerical Relativity

Black hole evolution by spectral methods

Lawrence E Kidder Mark A Scheel and Saul A TeukolskyCenter for Radiophysics and Space Research Cornell University Ithaca New York 14853

Eric D Carlson and Gregory B CookDepartment of Physics Wake Forest University Winston-Salem North Carolina 27109

Received 15 May 2000 published 26 September 2000

Current methods of evolving a spacetime containing one or more black holes are plagued by instabilities that

prohibit long-term evolution Some of these instabilities may be due to the numerical method used tradition-

ally finite differencing In this paper we explore the use of a pseudospectral collocation PSC method for theevolution of a spherically symmetric black hole spacetime in one dimension using a hyperbolic formulation of

Einsteinrsquos equations We demonstrate that our PSC method is able to evolve a spherically symmetric black

hole spacetime forever without enforcing constraints even if we add dynamics via a Klein-Gordon scalar field

We find that in contrast with finite-differencing methods black hole excision is a trivial operation using PSC

applied to a hyperbolic formulation of Einsteinrsquos equations We discuss the extension of this method to three

spatial dimensions

PACS numbers 0425Dm 0270Hm

I INTRODUCTION AND SUMMARY

A major thrust of research in classical general relativity in

the past decade has been to devise algorithms to solve Ein-

steinrsquos equations numerically Despite advances in our ana-

lytic understanding of general relativity we still do not knowwhat all the features of the theory really are Numerical so-lutions will continue to provide fresh insights into the theoryas they have in the past for example the critical behavior inblack hole formation 1$ and the formation of toroidal blackholes 2$New urgency has been injected into numerical relativity

by the imminent deployment of the Laser InterferometricGravitational Wave Observatory LIGO The prime targetfor LIGO is coalescence of binary neutron star and blackhole systems The waveform is reasonably well predicted bythe post-Newtonian approximation when the binary compo-nents are at large separation However extracting the mostimportant physics requires us to be able to deal with fullynon-linear general relativity as the system spirals togetherand coalesces Moreover a number of people believe thatthere is a significant event rate for the coalescence of mas-sive black hole systems (20M) 3$ In this case LIGO ismost sensitive to waves emitted from the strong field regimeIndeed without some theoretical guidance as to what to ex-pect from this regime it is possible we may miss theseevents entirely 4$However the goal of developing a general algorithm that

can solve Einsteinrsquos equations for two black holes has re-mained elusive All attempts to date have been plagued byinstabilities These instabilities are caused by an interplay ofthree factors 1 Einsteinrsquos equations are an overdeterminedsystem with the evolution equations subject to constraintsSo if for example you choose to solve only the evolutionequations then there can be unstable solutions that are in factsolutions of the evolution equations but do not satisfy theconstraints Small numerical errors may cause these solu-tions to appear and swamp the true solution lsquolsquoconstraint-

violating modesrsquorsquo 2 The coordinate freedom inherent in

the theory means that it is very easy to impose coordinate

conditions that lead to numerical instabilities lsquolsquogaugemodesrsquorsquo 3 Experience has shown that the kind of bound-ary conditions we choose and how we implement them can

affect the stability of an algorithm enormously

Similar instabilities have hampered efforts to solve therelated problem of binary neutron stars only very recently56$ has there been some success in finding stable algo-rithms However black hole evolutions face an additionalobstacle that is absent in the case of neutron stars for neu-tron stars the gravitational field is everywhere regular but forblack holes one must somehow deal with the physical singu-larity that lurks inside each holeThere are two main approaches for handling these singu-

larities The first is to use gauge conditions eg maximalslicing that avoid the singularities altogether Such condi-tions however lead to large gradients in the gravitationalfield variables near the horizon These grow exponentially intime and ultimately cannot be resolved by the numerical evo-lution causing the code to crash The alternative approach isto excise the region containing the singularity from the com-putational domain and evolve only the exterior region If theexcision boundary is placed inside the horizon of the blackhole causality assures us that we do not need to impose aphysical boundary condition thereHowever black hole excision is only known to be math-

ematically well posed if the evolution equations are hyper-bolic with characteristic speeds less than or equal to c In thiscase the structure of the equations guarantees that even un-physical modes present in the solution gauge modesconstraint-violating modes behave causally and cannotpropagate out of the horizon For many representations ofgeneral relativity such as the usual Arnowitt-Deser-MisnerADM 7$ formulation the evolution equations are of nomathematical type for which well-posedness has beenproved so the suitability of these formulations for black holeexcision must be determined empirically on a case-by-case

PHYSICAL REVIEW D VOLUME 62 084032

0556-2821200062808403220$1500 copy2000 The American Physical Society62 084032-1

Stable evolutions in 1+1 d

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

Pros and cons

No dissipation

Exponential convergence

Superb results for little effort

Gibbs phenomena

Discrete equations are fully coupled

Lack of dissipation allows high frequency errors to remain

Can not freely choose location of nodes

Regge calculus

Regge calculus

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is piecewise flat

Curvature is a distribution

Defect angle at each vertex

2d example

2d example

2d example

2d example

2d example

3d example

3d example

3d example

3d example

3d example

4d example

Field equations

Some questions

Can smooth metrics be accurately approximated by Regge lattices

Do the Regge equations reduce to the Einstein equations in some suitable limit

Do solutions of the Regge equations converge to solutions of the Einstein equations

Some answers

Regge

Reggesolutions

Yes

Yes as an average

Allendorfer Weyl

Cheeger Muller Schrader

Continuum

Einsteinsolutions

Applications

FRW cosmologies

Brill wave initial data

Quantum gravityKasner cosmologies

Misner initial data

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 19: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Convergence3

-1 -05 0 05 1x

-15

-1

-05

0

05

1

u = cos3(x2) - (x+1)

38

Pu

Iu

N=4

-1 -05 0 05 1x

-15

-1

-05

0

05

1

u = cos3(x2) - (x+1)

38

Pu

Iu

N=8

FIG 2 Same as Fig 1 with also the interpolant of u The collocation points are denoted by the circles

3 Interpolation

If one applies the Gauss quadratures to approximate the coecient of the expansion one obtains

un =1

n

N

j=0

u (xj) pn (xj)wj with n =N

j=0

p2n (xj)wj (5)

Let us precise that this is not exact in the sense that un = un However the computation of u only requires toevaluate u at the N + 1 collocation points The interpolant of u is then defined as the following polynomial

INu =N

n=0

unpn (x) (6)

The dierence between Inu and Pnu is called the aliasing error The interpolant of u is the spectral approximate ofu and one can show that it is the only polynomials of degree N that coincides with u at each collocation point

[Inu] (xi) = u (xi) i N (7)

Figure 2 shows the same function as Fig1 but the interpolant is also plotted One can see that indeed INu coincideswith u at the collocation points that are indicated by the circles Once again even with as few points as N = 8 nodierence can be seen between the various functions

Figure 3 shows the maximum dierence between INu and u on as a function of the degree of the approximation N We can observe the very general feature of spectral methods that the error decreases exponentially until one reachesthe machine accuracy (here 1014 the computation being done in double precision) This very fast convergenceexplains why spectral methods are so ecient especially compared to finite dierence ones where the error followsonly a power-law in terms of N We will later be more quantitative about the convergence properties of the spectralapproximation

Let us note that a function u can be described either by its value at each collocation point u (xi) or by the coecientsof the interpolant ui If the values at collocation points are known one is working in the configuration space and inthe coecient space if u is given in terms of its coecients There is a bijection between the two descriptions and onesimply goes from one space to another by using

bull un =1

n

N

j=0

u (xj) pn (xj) wj (configuration $ coecient)

bull u (xn) =N

j=0

ujpj (xn) (coecient $ configuration)

Depending on the operation one has to perform one choice of space is usually more suited than the other Forinstance let us assume that one wants to compute the derivative of u This is easily done if u is known in the

Exponential convergence 4

0 5 10 15 20 25 30 35

Number of coefficients

1e-15

1e-12

1e-09

1e-06

1e-03

1e+00m

ax |I N

u -

u|

FIG 3 Maximum dierence between INu and u as a function of the degree of the approximation N

-1 -05 0 05 1x

-3

-2

-1

0

1

2

ursquoIN

(ursquo)

(IN

u)rsquo

N=4

-1 -05 0 05 1x

-3

-2

-1

0

1

2

ursquoIN

(ursquo)

(IN

u)rsquo

N=8

FIG 4 First derivative of u = cos3 (x2) (x + 1)3 8 interpolant of the derivative and derivative of the interpolant forN = 4 and N = 8

coecient space Indeed one can simply approximate u by the derivative of the interpolant

u (x) [INu] =N

n=0

unpn (x) (8)

Such an approximation only requires the knowledge of the coecients of u and how the basis polynomials are derivedLet us note that the obtained polynomial even if it is a good approximation of u is not the interpolant of u Inother terms the interpolation and the derivation are two operations that do not commute (INu) = IN (u) This isclearly illustrated on Fig 4 where the derivative of u = cos3 (x2) (x + 1)3 8 is plotted along with the functions(INu) and IN (u) In particular one can note that the functions used to represent u ie (INu) does not coincidewith u at the collocation points However even with N = 8 only the three functions can not be distinguished byeye

The maximum dierence between u and (INu) as a function of N is shown on Fig 5 Once again the convergenceis exponential

Gibbs phenomena9

-1 -05 0 05 1x

-02

0

02

04

06

08

1

FIG 8 Step function and some interpolants for N = 7 N = 15 and N = 31 As N increases the maximum dierence staysconstant and the number of oscillations increases

4 5 6 7 8 9 10 20 30 40 50 60 70 8090

N

10-2

10-1

100

-1

1 |I N

u -

u| dx

FIG 9

1

1|INu u| dx as a function of N u being a step function The convergence obeys a power-law which decays slower

than N1

B A test problem

We propose to solve the equation

d2u

dx2 4

du

dx+ 4u = exp (x) + C (25)

with x [1 1] and C = 4e

1 + e2

As boundary conditions we simply ask that the solution is zero at theboundaries

u (1) = 0 and u (1) = 0 (26)

Representations

can be represented by either or

for algebraic operators use

for differential operators use

Differential equations

solve for then recover

Spectral methods in Numerical Relativity

Black hole evolution by spectral methods

Lawrence E Kidder Mark A Scheel and Saul A TeukolskyCenter for Radiophysics and Space Research Cornell University Ithaca New York 14853

Eric D Carlson and Gregory B CookDepartment of Physics Wake Forest University Winston-Salem North Carolina 27109

Received 15 May 2000 published 26 September 2000

Current methods of evolving a spacetime containing one or more black holes are plagued by instabilities that

prohibit long-term evolution Some of these instabilities may be due to the numerical method used tradition-

ally finite differencing In this paper we explore the use of a pseudospectral collocation PSC method for theevolution of a spherically symmetric black hole spacetime in one dimension using a hyperbolic formulation of

Einsteinrsquos equations We demonstrate that our PSC method is able to evolve a spherically symmetric black

hole spacetime forever without enforcing constraints even if we add dynamics via a Klein-Gordon scalar field

We find that in contrast with finite-differencing methods black hole excision is a trivial operation using PSC

applied to a hyperbolic formulation of Einsteinrsquos equations We discuss the extension of this method to three

spatial dimensions

PACS numbers 0425Dm 0270Hm

I INTRODUCTION AND SUMMARY

A major thrust of research in classical general relativity in

the past decade has been to devise algorithms to solve Ein-

steinrsquos equations numerically Despite advances in our ana-

lytic understanding of general relativity we still do not knowwhat all the features of the theory really are Numerical so-lutions will continue to provide fresh insights into the theoryas they have in the past for example the critical behavior inblack hole formation 1$ and the formation of toroidal blackholes 2$New urgency has been injected into numerical relativity

by the imminent deployment of the Laser InterferometricGravitational Wave Observatory LIGO The prime targetfor LIGO is coalescence of binary neutron star and blackhole systems The waveform is reasonably well predicted bythe post-Newtonian approximation when the binary compo-nents are at large separation However extracting the mostimportant physics requires us to be able to deal with fullynon-linear general relativity as the system spirals togetherand coalesces Moreover a number of people believe thatthere is a significant event rate for the coalescence of mas-sive black hole systems (20M) 3$ In this case LIGO ismost sensitive to waves emitted from the strong field regimeIndeed without some theoretical guidance as to what to ex-pect from this regime it is possible we may miss theseevents entirely 4$However the goal of developing a general algorithm that

can solve Einsteinrsquos equations for two black holes has re-mained elusive All attempts to date have been plagued byinstabilities These instabilities are caused by an interplay ofthree factors 1 Einsteinrsquos equations are an overdeterminedsystem with the evolution equations subject to constraintsSo if for example you choose to solve only the evolutionequations then there can be unstable solutions that are in factsolutions of the evolution equations but do not satisfy theconstraints Small numerical errors may cause these solu-tions to appear and swamp the true solution lsquolsquoconstraint-

violating modesrsquorsquo 2 The coordinate freedom inherent in

the theory means that it is very easy to impose coordinate

conditions that lead to numerical instabilities lsquolsquogaugemodesrsquorsquo 3 Experience has shown that the kind of bound-ary conditions we choose and how we implement them can

affect the stability of an algorithm enormously

Similar instabilities have hampered efforts to solve therelated problem of binary neutron stars only very recently56$ has there been some success in finding stable algo-rithms However black hole evolutions face an additionalobstacle that is absent in the case of neutron stars for neu-tron stars the gravitational field is everywhere regular but forblack holes one must somehow deal with the physical singu-larity that lurks inside each holeThere are two main approaches for handling these singu-

larities The first is to use gauge conditions eg maximalslicing that avoid the singularities altogether Such condi-tions however lead to large gradients in the gravitationalfield variables near the horizon These grow exponentially intime and ultimately cannot be resolved by the numerical evo-lution causing the code to crash The alternative approach isto excise the region containing the singularity from the com-putational domain and evolve only the exterior region If theexcision boundary is placed inside the horizon of the blackhole causality assures us that we do not need to impose aphysical boundary condition thereHowever black hole excision is only known to be math-

ematically well posed if the evolution equations are hyper-bolic with characteristic speeds less than or equal to c In thiscase the structure of the equations guarantees that even un-physical modes present in the solution gauge modesconstraint-violating modes behave causally and cannotpropagate out of the horizon For many representations ofgeneral relativity such as the usual Arnowitt-Deser-MisnerADM 7$ formulation the evolution equations are of nomathematical type for which well-posedness has beenproved so the suitability of these formulations for black holeexcision must be determined empirically on a case-by-case

PHYSICAL REVIEW D VOLUME 62 084032

0556-2821200062808403220$1500 copy2000 The American Physical Society62 084032-1

Stable evolutions in 1+1 d

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

Pros and cons

No dissipation

Exponential convergence

Superb results for little effort

Gibbs phenomena

Discrete equations are fully coupled

Lack of dissipation allows high frequency errors to remain

Can not freely choose location of nodes

Regge calculus

Regge calculus

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is piecewise flat

Curvature is a distribution

Defect angle at each vertex

2d example

2d example

2d example

2d example

2d example

3d example

3d example

3d example

3d example

3d example

4d example

Field equations

Some questions

Can smooth metrics be accurately approximated by Regge lattices

Do the Regge equations reduce to the Einstein equations in some suitable limit

Do solutions of the Regge equations converge to solutions of the Einstein equations

Some answers

Regge

Reggesolutions

Yes

Yes as an average

Allendorfer Weyl

Cheeger Muller Schrader

Continuum

Einsteinsolutions

Applications

FRW cosmologies

Brill wave initial data

Quantum gravityKasner cosmologies

Misner initial data

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 20: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Exponential convergence 4

0 5 10 15 20 25 30 35

Number of coefficients

1e-15

1e-12

1e-09

1e-06

1e-03

1e+00m

ax |I N

u -

u|

FIG 3 Maximum dierence between INu and u as a function of the degree of the approximation N

-1 -05 0 05 1x

-3

-2

-1

0

1

2

ursquoIN

(ursquo)

(IN

u)rsquo

N=4

-1 -05 0 05 1x

-3

-2

-1

0

1

2

ursquoIN

(ursquo)

(IN

u)rsquo

N=8

FIG 4 First derivative of u = cos3 (x2) (x + 1)3 8 interpolant of the derivative and derivative of the interpolant forN = 4 and N = 8

coecient space Indeed one can simply approximate u by the derivative of the interpolant

u (x) [INu] =N

n=0

unpn (x) (8)

Such an approximation only requires the knowledge of the coecients of u and how the basis polynomials are derivedLet us note that the obtained polynomial even if it is a good approximation of u is not the interpolant of u Inother terms the interpolation and the derivation are two operations that do not commute (INu) = IN (u) This isclearly illustrated on Fig 4 where the derivative of u = cos3 (x2) (x + 1)3 8 is plotted along with the functions(INu) and IN (u) In particular one can note that the functions used to represent u ie (INu) does not coincidewith u at the collocation points However even with N = 8 only the three functions can not be distinguished byeye

The maximum dierence between u and (INu) as a function of N is shown on Fig 5 Once again the convergenceis exponential

Gibbs phenomena9

-1 -05 0 05 1x

-02

0

02

04

06

08

1

FIG 8 Step function and some interpolants for N = 7 N = 15 and N = 31 As N increases the maximum dierence staysconstant and the number of oscillations increases

4 5 6 7 8 9 10 20 30 40 50 60 70 8090

N

10-2

10-1

100

-1

1 |I N

u -

u| dx

FIG 9

1

1|INu u| dx as a function of N u being a step function The convergence obeys a power-law which decays slower

than N1

B A test problem

We propose to solve the equation

d2u

dx2 4

du

dx+ 4u = exp (x) + C (25)

with x [1 1] and C = 4e

1 + e2

As boundary conditions we simply ask that the solution is zero at theboundaries

u (1) = 0 and u (1) = 0 (26)

Representations

can be represented by either or

for algebraic operators use

for differential operators use

Differential equations

solve for then recover

Spectral methods in Numerical Relativity

Black hole evolution by spectral methods

Lawrence E Kidder Mark A Scheel and Saul A TeukolskyCenter for Radiophysics and Space Research Cornell University Ithaca New York 14853

Eric D Carlson and Gregory B CookDepartment of Physics Wake Forest University Winston-Salem North Carolina 27109

Received 15 May 2000 published 26 September 2000

Current methods of evolving a spacetime containing one or more black holes are plagued by instabilities that

prohibit long-term evolution Some of these instabilities may be due to the numerical method used tradition-

ally finite differencing In this paper we explore the use of a pseudospectral collocation PSC method for theevolution of a spherically symmetric black hole spacetime in one dimension using a hyperbolic formulation of

Einsteinrsquos equations We demonstrate that our PSC method is able to evolve a spherically symmetric black

hole spacetime forever without enforcing constraints even if we add dynamics via a Klein-Gordon scalar field

We find that in contrast with finite-differencing methods black hole excision is a trivial operation using PSC

applied to a hyperbolic formulation of Einsteinrsquos equations We discuss the extension of this method to three

spatial dimensions

PACS numbers 0425Dm 0270Hm

I INTRODUCTION AND SUMMARY

A major thrust of research in classical general relativity in

the past decade has been to devise algorithms to solve Ein-

steinrsquos equations numerically Despite advances in our ana-

lytic understanding of general relativity we still do not knowwhat all the features of the theory really are Numerical so-lutions will continue to provide fresh insights into the theoryas they have in the past for example the critical behavior inblack hole formation 1$ and the formation of toroidal blackholes 2$New urgency has been injected into numerical relativity

by the imminent deployment of the Laser InterferometricGravitational Wave Observatory LIGO The prime targetfor LIGO is coalescence of binary neutron star and blackhole systems The waveform is reasonably well predicted bythe post-Newtonian approximation when the binary compo-nents are at large separation However extracting the mostimportant physics requires us to be able to deal with fullynon-linear general relativity as the system spirals togetherand coalesces Moreover a number of people believe thatthere is a significant event rate for the coalescence of mas-sive black hole systems (20M) 3$ In this case LIGO ismost sensitive to waves emitted from the strong field regimeIndeed without some theoretical guidance as to what to ex-pect from this regime it is possible we may miss theseevents entirely 4$However the goal of developing a general algorithm that

can solve Einsteinrsquos equations for two black holes has re-mained elusive All attempts to date have been plagued byinstabilities These instabilities are caused by an interplay ofthree factors 1 Einsteinrsquos equations are an overdeterminedsystem with the evolution equations subject to constraintsSo if for example you choose to solve only the evolutionequations then there can be unstable solutions that are in factsolutions of the evolution equations but do not satisfy theconstraints Small numerical errors may cause these solu-tions to appear and swamp the true solution lsquolsquoconstraint-

violating modesrsquorsquo 2 The coordinate freedom inherent in

the theory means that it is very easy to impose coordinate

conditions that lead to numerical instabilities lsquolsquogaugemodesrsquorsquo 3 Experience has shown that the kind of bound-ary conditions we choose and how we implement them can

affect the stability of an algorithm enormously

Similar instabilities have hampered efforts to solve therelated problem of binary neutron stars only very recently56$ has there been some success in finding stable algo-rithms However black hole evolutions face an additionalobstacle that is absent in the case of neutron stars for neu-tron stars the gravitational field is everywhere regular but forblack holes one must somehow deal with the physical singu-larity that lurks inside each holeThere are two main approaches for handling these singu-

larities The first is to use gauge conditions eg maximalslicing that avoid the singularities altogether Such condi-tions however lead to large gradients in the gravitationalfield variables near the horizon These grow exponentially intime and ultimately cannot be resolved by the numerical evo-lution causing the code to crash The alternative approach isto excise the region containing the singularity from the com-putational domain and evolve only the exterior region If theexcision boundary is placed inside the horizon of the blackhole causality assures us that we do not need to impose aphysical boundary condition thereHowever black hole excision is only known to be math-

ematically well posed if the evolution equations are hyper-bolic with characteristic speeds less than or equal to c In thiscase the structure of the equations guarantees that even un-physical modes present in the solution gauge modesconstraint-violating modes behave causally and cannotpropagate out of the horizon For many representations ofgeneral relativity such as the usual Arnowitt-Deser-MisnerADM 7$ formulation the evolution equations are of nomathematical type for which well-posedness has beenproved so the suitability of these formulations for black holeexcision must be determined empirically on a case-by-case

PHYSICAL REVIEW D VOLUME 62 084032

0556-2821200062808403220$1500 copy2000 The American Physical Society62 084032-1

Stable evolutions in 1+1 d

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

Pros and cons

No dissipation

Exponential convergence

Superb results for little effort

Gibbs phenomena

Discrete equations are fully coupled

Lack of dissipation allows high frequency errors to remain

Can not freely choose location of nodes

Regge calculus

Regge calculus

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is piecewise flat

Curvature is a distribution

Defect angle at each vertex

2d example

2d example

2d example

2d example

2d example

3d example

3d example

3d example

3d example

3d example

4d example

Field equations

Some questions

Can smooth metrics be accurately approximated by Regge lattices

Do the Regge equations reduce to the Einstein equations in some suitable limit

Do solutions of the Regge equations converge to solutions of the Einstein equations

Some answers

Regge

Reggesolutions

Yes

Yes as an average

Allendorfer Weyl

Cheeger Muller Schrader

Continuum

Einsteinsolutions

Applications

FRW cosmologies

Brill wave initial data

Quantum gravityKasner cosmologies

Misner initial data

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 21: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Gibbs phenomena9

-1 -05 0 05 1x

-02

0

02

04

06

08

1

FIG 8 Step function and some interpolants for N = 7 N = 15 and N = 31 As N increases the maximum dierence staysconstant and the number of oscillations increases

4 5 6 7 8 9 10 20 30 40 50 60 70 8090

N

10-2

10-1

100

-1

1 |I N

u -

u| dx

FIG 9

1

1|INu u| dx as a function of N u being a step function The convergence obeys a power-law which decays slower

than N1

B A test problem

We propose to solve the equation

d2u

dx2 4

du

dx+ 4u = exp (x) + C (25)

with x [1 1] and C = 4e

1 + e2

As boundary conditions we simply ask that the solution is zero at theboundaries

u (1) = 0 and u (1) = 0 (26)

Representations

can be represented by either or

for algebraic operators use

for differential operators use

Differential equations

solve for then recover

Spectral methods in Numerical Relativity

Black hole evolution by spectral methods

Lawrence E Kidder Mark A Scheel and Saul A TeukolskyCenter for Radiophysics and Space Research Cornell University Ithaca New York 14853

Eric D Carlson and Gregory B CookDepartment of Physics Wake Forest University Winston-Salem North Carolina 27109

Received 15 May 2000 published 26 September 2000

Current methods of evolving a spacetime containing one or more black holes are plagued by instabilities that

prohibit long-term evolution Some of these instabilities may be due to the numerical method used tradition-

ally finite differencing In this paper we explore the use of a pseudospectral collocation PSC method for theevolution of a spherically symmetric black hole spacetime in one dimension using a hyperbolic formulation of

Einsteinrsquos equations We demonstrate that our PSC method is able to evolve a spherically symmetric black

hole spacetime forever without enforcing constraints even if we add dynamics via a Klein-Gordon scalar field

We find that in contrast with finite-differencing methods black hole excision is a trivial operation using PSC

applied to a hyperbolic formulation of Einsteinrsquos equations We discuss the extension of this method to three

spatial dimensions

PACS numbers 0425Dm 0270Hm

I INTRODUCTION AND SUMMARY

A major thrust of research in classical general relativity in

the past decade has been to devise algorithms to solve Ein-

steinrsquos equations numerically Despite advances in our ana-

lytic understanding of general relativity we still do not knowwhat all the features of the theory really are Numerical so-lutions will continue to provide fresh insights into the theoryas they have in the past for example the critical behavior inblack hole formation 1$ and the formation of toroidal blackholes 2$New urgency has been injected into numerical relativity

by the imminent deployment of the Laser InterferometricGravitational Wave Observatory LIGO The prime targetfor LIGO is coalescence of binary neutron star and blackhole systems The waveform is reasonably well predicted bythe post-Newtonian approximation when the binary compo-nents are at large separation However extracting the mostimportant physics requires us to be able to deal with fullynon-linear general relativity as the system spirals togetherand coalesces Moreover a number of people believe thatthere is a significant event rate for the coalescence of mas-sive black hole systems (20M) 3$ In this case LIGO ismost sensitive to waves emitted from the strong field regimeIndeed without some theoretical guidance as to what to ex-pect from this regime it is possible we may miss theseevents entirely 4$However the goal of developing a general algorithm that

can solve Einsteinrsquos equations for two black holes has re-mained elusive All attempts to date have been plagued byinstabilities These instabilities are caused by an interplay ofthree factors 1 Einsteinrsquos equations are an overdeterminedsystem with the evolution equations subject to constraintsSo if for example you choose to solve only the evolutionequations then there can be unstable solutions that are in factsolutions of the evolution equations but do not satisfy theconstraints Small numerical errors may cause these solu-tions to appear and swamp the true solution lsquolsquoconstraint-

violating modesrsquorsquo 2 The coordinate freedom inherent in

the theory means that it is very easy to impose coordinate

conditions that lead to numerical instabilities lsquolsquogaugemodesrsquorsquo 3 Experience has shown that the kind of bound-ary conditions we choose and how we implement them can

affect the stability of an algorithm enormously

Similar instabilities have hampered efforts to solve therelated problem of binary neutron stars only very recently56$ has there been some success in finding stable algo-rithms However black hole evolutions face an additionalobstacle that is absent in the case of neutron stars for neu-tron stars the gravitational field is everywhere regular but forblack holes one must somehow deal with the physical singu-larity that lurks inside each holeThere are two main approaches for handling these singu-

larities The first is to use gauge conditions eg maximalslicing that avoid the singularities altogether Such condi-tions however lead to large gradients in the gravitationalfield variables near the horizon These grow exponentially intime and ultimately cannot be resolved by the numerical evo-lution causing the code to crash The alternative approach isto excise the region containing the singularity from the com-putational domain and evolve only the exterior region If theexcision boundary is placed inside the horizon of the blackhole causality assures us that we do not need to impose aphysical boundary condition thereHowever black hole excision is only known to be math-

ematically well posed if the evolution equations are hyper-bolic with characteristic speeds less than or equal to c In thiscase the structure of the equations guarantees that even un-physical modes present in the solution gauge modesconstraint-violating modes behave causally and cannotpropagate out of the horizon For many representations ofgeneral relativity such as the usual Arnowitt-Deser-MisnerADM 7$ formulation the evolution equations are of nomathematical type for which well-posedness has beenproved so the suitability of these formulations for black holeexcision must be determined empirically on a case-by-case

PHYSICAL REVIEW D VOLUME 62 084032

0556-2821200062808403220$1500 copy2000 The American Physical Society62 084032-1

Stable evolutions in 1+1 d

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

Pros and cons

No dissipation

Exponential convergence

Superb results for little effort

Gibbs phenomena

Discrete equations are fully coupled

Lack of dissipation allows high frequency errors to remain

Can not freely choose location of nodes

Regge calculus

Regge calculus

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is piecewise flat

Curvature is a distribution

Defect angle at each vertex

2d example

2d example

2d example

2d example

2d example

3d example

3d example

3d example

3d example

3d example

4d example

Field equations

Some questions

Can smooth metrics be accurately approximated by Regge lattices

Do the Regge equations reduce to the Einstein equations in some suitable limit

Do solutions of the Regge equations converge to solutions of the Einstein equations

Some answers

Regge

Reggesolutions

Yes

Yes as an average

Allendorfer Weyl

Cheeger Muller Schrader

Continuum

Einsteinsolutions

Applications

FRW cosmologies

Brill wave initial data

Quantum gravityKasner cosmologies

Misner initial data

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 22: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Representations

can be represented by either or

for algebraic operators use

for differential operators use

Differential equations

solve for then recover

Spectral methods in Numerical Relativity

Black hole evolution by spectral methods

Lawrence E Kidder Mark A Scheel and Saul A TeukolskyCenter for Radiophysics and Space Research Cornell University Ithaca New York 14853

Eric D Carlson and Gregory B CookDepartment of Physics Wake Forest University Winston-Salem North Carolina 27109

Received 15 May 2000 published 26 September 2000

Current methods of evolving a spacetime containing one or more black holes are plagued by instabilities that

prohibit long-term evolution Some of these instabilities may be due to the numerical method used tradition-

ally finite differencing In this paper we explore the use of a pseudospectral collocation PSC method for theevolution of a spherically symmetric black hole spacetime in one dimension using a hyperbolic formulation of

Einsteinrsquos equations We demonstrate that our PSC method is able to evolve a spherically symmetric black

hole spacetime forever without enforcing constraints even if we add dynamics via a Klein-Gordon scalar field

We find that in contrast with finite-differencing methods black hole excision is a trivial operation using PSC

applied to a hyperbolic formulation of Einsteinrsquos equations We discuss the extension of this method to three

spatial dimensions

PACS numbers 0425Dm 0270Hm

I INTRODUCTION AND SUMMARY

A major thrust of research in classical general relativity in

the past decade has been to devise algorithms to solve Ein-

steinrsquos equations numerically Despite advances in our ana-

lytic understanding of general relativity we still do not knowwhat all the features of the theory really are Numerical so-lutions will continue to provide fresh insights into the theoryas they have in the past for example the critical behavior inblack hole formation 1$ and the formation of toroidal blackholes 2$New urgency has been injected into numerical relativity

by the imminent deployment of the Laser InterferometricGravitational Wave Observatory LIGO The prime targetfor LIGO is coalescence of binary neutron star and blackhole systems The waveform is reasonably well predicted bythe post-Newtonian approximation when the binary compo-nents are at large separation However extracting the mostimportant physics requires us to be able to deal with fullynon-linear general relativity as the system spirals togetherand coalesces Moreover a number of people believe thatthere is a significant event rate for the coalescence of mas-sive black hole systems (20M) 3$ In this case LIGO ismost sensitive to waves emitted from the strong field regimeIndeed without some theoretical guidance as to what to ex-pect from this regime it is possible we may miss theseevents entirely 4$However the goal of developing a general algorithm that

can solve Einsteinrsquos equations for two black holes has re-mained elusive All attempts to date have been plagued byinstabilities These instabilities are caused by an interplay ofthree factors 1 Einsteinrsquos equations are an overdeterminedsystem with the evolution equations subject to constraintsSo if for example you choose to solve only the evolutionequations then there can be unstable solutions that are in factsolutions of the evolution equations but do not satisfy theconstraints Small numerical errors may cause these solu-tions to appear and swamp the true solution lsquolsquoconstraint-

violating modesrsquorsquo 2 The coordinate freedom inherent in

the theory means that it is very easy to impose coordinate

conditions that lead to numerical instabilities lsquolsquogaugemodesrsquorsquo 3 Experience has shown that the kind of bound-ary conditions we choose and how we implement them can

affect the stability of an algorithm enormously

Similar instabilities have hampered efforts to solve therelated problem of binary neutron stars only very recently56$ has there been some success in finding stable algo-rithms However black hole evolutions face an additionalobstacle that is absent in the case of neutron stars for neu-tron stars the gravitational field is everywhere regular but forblack holes one must somehow deal with the physical singu-larity that lurks inside each holeThere are two main approaches for handling these singu-

larities The first is to use gauge conditions eg maximalslicing that avoid the singularities altogether Such condi-tions however lead to large gradients in the gravitationalfield variables near the horizon These grow exponentially intime and ultimately cannot be resolved by the numerical evo-lution causing the code to crash The alternative approach isto excise the region containing the singularity from the com-putational domain and evolve only the exterior region If theexcision boundary is placed inside the horizon of the blackhole causality assures us that we do not need to impose aphysical boundary condition thereHowever black hole excision is only known to be math-

ematically well posed if the evolution equations are hyper-bolic with characteristic speeds less than or equal to c In thiscase the structure of the equations guarantees that even un-physical modes present in the solution gauge modesconstraint-violating modes behave causally and cannotpropagate out of the horizon For many representations ofgeneral relativity such as the usual Arnowitt-Deser-MisnerADM 7$ formulation the evolution equations are of nomathematical type for which well-posedness has beenproved so the suitability of these formulations for black holeexcision must be determined empirically on a case-by-case

PHYSICAL REVIEW D VOLUME 62 084032

0556-2821200062808403220$1500 copy2000 The American Physical Society62 084032-1

Stable evolutions in 1+1 d

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

Pros and cons

No dissipation

Exponential convergence

Superb results for little effort

Gibbs phenomena

Discrete equations are fully coupled

Lack of dissipation allows high frequency errors to remain

Can not freely choose location of nodes

Regge calculus

Regge calculus

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is piecewise flat

Curvature is a distribution

Defect angle at each vertex

2d example

2d example

2d example

2d example

2d example

3d example

3d example

3d example

3d example

3d example

4d example

Field equations

Some questions

Can smooth metrics be accurately approximated by Regge lattices

Do the Regge equations reduce to the Einstein equations in some suitable limit

Do solutions of the Regge equations converge to solutions of the Einstein equations

Some answers

Regge

Reggesolutions

Yes

Yes as an average

Allendorfer Weyl

Cheeger Muller Schrader

Continuum

Einsteinsolutions

Applications

FRW cosmologies

Brill wave initial data

Quantum gravityKasner cosmologies

Misner initial data

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 23: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Differential equations

solve for then recover

Spectral methods in Numerical Relativity

Black hole evolution by spectral methods

Lawrence E Kidder Mark A Scheel and Saul A TeukolskyCenter for Radiophysics and Space Research Cornell University Ithaca New York 14853

Eric D Carlson and Gregory B CookDepartment of Physics Wake Forest University Winston-Salem North Carolina 27109

Received 15 May 2000 published 26 September 2000

Current methods of evolving a spacetime containing one or more black holes are plagued by instabilities that

prohibit long-term evolution Some of these instabilities may be due to the numerical method used tradition-

ally finite differencing In this paper we explore the use of a pseudospectral collocation PSC method for theevolution of a spherically symmetric black hole spacetime in one dimension using a hyperbolic formulation of

Einsteinrsquos equations We demonstrate that our PSC method is able to evolve a spherically symmetric black

hole spacetime forever without enforcing constraints even if we add dynamics via a Klein-Gordon scalar field

We find that in contrast with finite-differencing methods black hole excision is a trivial operation using PSC

applied to a hyperbolic formulation of Einsteinrsquos equations We discuss the extension of this method to three

spatial dimensions

PACS numbers 0425Dm 0270Hm

I INTRODUCTION AND SUMMARY

A major thrust of research in classical general relativity in

the past decade has been to devise algorithms to solve Ein-

steinrsquos equations numerically Despite advances in our ana-

lytic understanding of general relativity we still do not knowwhat all the features of the theory really are Numerical so-lutions will continue to provide fresh insights into the theoryas they have in the past for example the critical behavior inblack hole formation 1$ and the formation of toroidal blackholes 2$New urgency has been injected into numerical relativity

by the imminent deployment of the Laser InterferometricGravitational Wave Observatory LIGO The prime targetfor LIGO is coalescence of binary neutron star and blackhole systems The waveform is reasonably well predicted bythe post-Newtonian approximation when the binary compo-nents are at large separation However extracting the mostimportant physics requires us to be able to deal with fullynon-linear general relativity as the system spirals togetherand coalesces Moreover a number of people believe thatthere is a significant event rate for the coalescence of mas-sive black hole systems (20M) 3$ In this case LIGO ismost sensitive to waves emitted from the strong field regimeIndeed without some theoretical guidance as to what to ex-pect from this regime it is possible we may miss theseevents entirely 4$However the goal of developing a general algorithm that

can solve Einsteinrsquos equations for two black holes has re-mained elusive All attempts to date have been plagued byinstabilities These instabilities are caused by an interplay ofthree factors 1 Einsteinrsquos equations are an overdeterminedsystem with the evolution equations subject to constraintsSo if for example you choose to solve only the evolutionequations then there can be unstable solutions that are in factsolutions of the evolution equations but do not satisfy theconstraints Small numerical errors may cause these solu-tions to appear and swamp the true solution lsquolsquoconstraint-

violating modesrsquorsquo 2 The coordinate freedom inherent in

the theory means that it is very easy to impose coordinate

conditions that lead to numerical instabilities lsquolsquogaugemodesrsquorsquo 3 Experience has shown that the kind of bound-ary conditions we choose and how we implement them can

affect the stability of an algorithm enormously

Similar instabilities have hampered efforts to solve therelated problem of binary neutron stars only very recently56$ has there been some success in finding stable algo-rithms However black hole evolutions face an additionalobstacle that is absent in the case of neutron stars for neu-tron stars the gravitational field is everywhere regular but forblack holes one must somehow deal with the physical singu-larity that lurks inside each holeThere are two main approaches for handling these singu-

larities The first is to use gauge conditions eg maximalslicing that avoid the singularities altogether Such condi-tions however lead to large gradients in the gravitationalfield variables near the horizon These grow exponentially intime and ultimately cannot be resolved by the numerical evo-lution causing the code to crash The alternative approach isto excise the region containing the singularity from the com-putational domain and evolve only the exterior region If theexcision boundary is placed inside the horizon of the blackhole causality assures us that we do not need to impose aphysical boundary condition thereHowever black hole excision is only known to be math-

ematically well posed if the evolution equations are hyper-bolic with characteristic speeds less than or equal to c In thiscase the structure of the equations guarantees that even un-physical modes present in the solution gauge modesconstraint-violating modes behave causally and cannotpropagate out of the horizon For many representations ofgeneral relativity such as the usual Arnowitt-Deser-MisnerADM 7$ formulation the evolution equations are of nomathematical type for which well-posedness has beenproved so the suitability of these formulations for black holeexcision must be determined empirically on a case-by-case

PHYSICAL REVIEW D VOLUME 62 084032

0556-2821200062808403220$1500 copy2000 The American Physical Society62 084032-1

Stable evolutions in 1+1 d

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

Pros and cons

No dissipation

Exponential convergence

Superb results for little effort

Gibbs phenomena

Discrete equations are fully coupled

Lack of dissipation allows high frequency errors to remain

Can not freely choose location of nodes

Regge calculus

Regge calculus

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is piecewise flat

Curvature is a distribution

Defect angle at each vertex

2d example

2d example

2d example

2d example

2d example

3d example

3d example

3d example

3d example

3d example

4d example

Field equations

Some questions

Can smooth metrics be accurately approximated by Regge lattices

Do the Regge equations reduce to the Einstein equations in some suitable limit

Do solutions of the Regge equations converge to solutions of the Einstein equations

Some answers

Regge

Reggesolutions

Yes

Yes as an average

Allendorfer Weyl

Cheeger Muller Schrader

Continuum

Einsteinsolutions

Applications

FRW cosmologies

Brill wave initial data

Quantum gravityKasner cosmologies

Misner initial data

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 24: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Spectral methods in Numerical Relativity

Black hole evolution by spectral methods

Lawrence E Kidder Mark A Scheel and Saul A TeukolskyCenter for Radiophysics and Space Research Cornell University Ithaca New York 14853

Eric D Carlson and Gregory B CookDepartment of Physics Wake Forest University Winston-Salem North Carolina 27109

Received 15 May 2000 published 26 September 2000

Current methods of evolving a spacetime containing one or more black holes are plagued by instabilities that

prohibit long-term evolution Some of these instabilities may be due to the numerical method used tradition-

ally finite differencing In this paper we explore the use of a pseudospectral collocation PSC method for theevolution of a spherically symmetric black hole spacetime in one dimension using a hyperbolic formulation of

Einsteinrsquos equations We demonstrate that our PSC method is able to evolve a spherically symmetric black

hole spacetime forever without enforcing constraints even if we add dynamics via a Klein-Gordon scalar field

We find that in contrast with finite-differencing methods black hole excision is a trivial operation using PSC

applied to a hyperbolic formulation of Einsteinrsquos equations We discuss the extension of this method to three

spatial dimensions

PACS numbers 0425Dm 0270Hm

I INTRODUCTION AND SUMMARY

A major thrust of research in classical general relativity in

the past decade has been to devise algorithms to solve Ein-

steinrsquos equations numerically Despite advances in our ana-

lytic understanding of general relativity we still do not knowwhat all the features of the theory really are Numerical so-lutions will continue to provide fresh insights into the theoryas they have in the past for example the critical behavior inblack hole formation 1$ and the formation of toroidal blackholes 2$New urgency has been injected into numerical relativity

by the imminent deployment of the Laser InterferometricGravitational Wave Observatory LIGO The prime targetfor LIGO is coalescence of binary neutron star and blackhole systems The waveform is reasonably well predicted bythe post-Newtonian approximation when the binary compo-nents are at large separation However extracting the mostimportant physics requires us to be able to deal with fullynon-linear general relativity as the system spirals togetherand coalesces Moreover a number of people believe thatthere is a significant event rate for the coalescence of mas-sive black hole systems (20M) 3$ In this case LIGO ismost sensitive to waves emitted from the strong field regimeIndeed without some theoretical guidance as to what to ex-pect from this regime it is possible we may miss theseevents entirely 4$However the goal of developing a general algorithm that

can solve Einsteinrsquos equations for two black holes has re-mained elusive All attempts to date have been plagued byinstabilities These instabilities are caused by an interplay ofthree factors 1 Einsteinrsquos equations are an overdeterminedsystem with the evolution equations subject to constraintsSo if for example you choose to solve only the evolutionequations then there can be unstable solutions that are in factsolutions of the evolution equations but do not satisfy theconstraints Small numerical errors may cause these solu-tions to appear and swamp the true solution lsquolsquoconstraint-

violating modesrsquorsquo 2 The coordinate freedom inherent in

the theory means that it is very easy to impose coordinate

conditions that lead to numerical instabilities lsquolsquogaugemodesrsquorsquo 3 Experience has shown that the kind of bound-ary conditions we choose and how we implement them can

affect the stability of an algorithm enormously

Similar instabilities have hampered efforts to solve therelated problem of binary neutron stars only very recently56$ has there been some success in finding stable algo-rithms However black hole evolutions face an additionalobstacle that is absent in the case of neutron stars for neu-tron stars the gravitational field is everywhere regular but forblack holes one must somehow deal with the physical singu-larity that lurks inside each holeThere are two main approaches for handling these singu-

larities The first is to use gauge conditions eg maximalslicing that avoid the singularities altogether Such condi-tions however lead to large gradients in the gravitationalfield variables near the horizon These grow exponentially intime and ultimately cannot be resolved by the numerical evo-lution causing the code to crash The alternative approach isto excise the region containing the singularity from the com-putational domain and evolve only the exterior region If theexcision boundary is placed inside the horizon of the blackhole causality assures us that we do not need to impose aphysical boundary condition thereHowever black hole excision is only known to be math-

ematically well posed if the evolution equations are hyper-bolic with characteristic speeds less than or equal to c In thiscase the structure of the equations guarantees that even un-physical modes present in the solution gauge modesconstraint-violating modes behave causally and cannotpropagate out of the horizon For many representations ofgeneral relativity such as the usual Arnowitt-Deser-MisnerADM 7$ formulation the evolution equations are of nomathematical type for which well-posedness has beenproved so the suitability of these formulations for black holeexcision must be determined empirically on a case-by-case

PHYSICAL REVIEW D VOLUME 62 084032

0556-2821200062808403220$1500 copy2000 The American Physical Society62 084032-1

Stable evolutions in 1+1 d

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

Pros and cons

No dissipation

Exponential convergence

Superb results for little effort

Gibbs phenomena

Discrete equations are fully coupled

Lack of dissipation allows high frequency errors to remain

Can not freely choose location of nodes

Regge calculus

Regge calculus

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is piecewise flat

Curvature is a distribution

Defect angle at each vertex

2d example

2d example

2d example

2d example

2d example

3d example

3d example

3d example

3d example

3d example

4d example

Field equations

Some questions

Can smooth metrics be accurately approximated by Regge lattices

Do the Regge equations reduce to the Einstein equations in some suitable limit

Do solutions of the Regge equations converge to solutions of the Einstein equations

Some answers

Regge

Reggesolutions

Yes

Yes as an average

Allendorfer Weyl

Cheeger Muller Schrader

Continuum

Einsteinsolutions

Applications

FRW cosmologies

Brill wave initial data

Quantum gravityKasner cosmologies

Misner initial data

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 25: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Stable evolutions in 1+1 d

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

fields Ur0 Ut

0 Ur and UT

One can suppress the

constraint-violating modes seen in Fig 4 by replacing the

freezing boundary conditions on Ur0 Ut

0 and UT with con-

straint boundary conditions as discussed in Sec II D The

resulting evolutions are shown in Figs 5 and 6 Except for

the evolution with Nr32 discussed below the Hamiltonianconstraint C settles to a steady state that converges exponen-tially to zero The same is true for the other three constraints

CrT Crrr and Cr However the metric quantities and otherfundamental variables grow approximately quadratically

with time eventually causing the simulations to terminate

Because the constraints remain satisfied we attribute this

quadratic growth to a gauge mode

The Nr32 case shown in Figs 5 and 6 suffers fromhigh-frequency noise that grows exponentially in time We

have experimented with various methods of damping this

noise including filtering the fundamental variables after each

time step and adding numerical dissipation terms to the equa-

tions However we have obtained best results by changing

our fourth order Runge-Kutta time-stepping algorithm to an

implicit backwards Euler scheme which is much more dis-

sipative Figures 7 and 8 show the results of this modifica-

tion The evolution now satisfies the constraints at late times

for sufficiently fine resolution but still suffers from a qua-

dratically growing gauge mode that causes the coarser reso-

lution runs to crash This gauge mode can be suppressed by

applying active gauge conditions as shown in Sec IV A 2

below Evolutions of fully harmonic initial data A4 pro-duce results similar to those shown in Figs 4ndash8

We note that even with analytic gauge conditions and

freezing outer boundary conditions evolutions of harmonic

and fully harmonic initial data such as those shown in Fig 4

become stable when the outer boundary is moved sufficiently

close to the black hole see runs 12 and 23 A similar de-

pendence on the outer boundary location has also been re-

ported by others 2127$ A possible explanation for this is

discussed briefly in 21$ For a nonzero shift vector any

TABLE II Input parameters for selected evolutions using ellip-

tic gauge conditions For each evolution we list the initial data type

ID the outer boundary condition on UT OBT the outer bound-

ary condition on both Ur0 and Ut

0 OB0 the inner and outer bound-ary conditions on the stationary mean curvature densitized lapse

LapseBC the shift equation used Shift the outer boundary con-dition on the shift ShiftOB the time stepping algorithm TS andthe result of the evolution Res The domain is (09M 109M ) forfully harmonic initial data and (19M 119M ) for all other cases

The inner boundary condition on the shift is given by Eq 230and all inner boundary conditions on gauge variables are imposed at

the current location of the apparent horizon

Run IDa OBTb OB0b LapseBCc Shiftd ShiftOB TSe Resf

25 KS F F c1 c1 MD tgT0 R4 Exp

26 KS F F c1 c1 MS tgT0 R4 Exp

27 KS C F c1 c1 MS tgT0 R4 LG

28 KS F C c1 c1 MS tgT0 R4 Exp

29 KS C C c1 c1 MS tgT0 R4 LG

30 KS C C c1 c1 MD tgT0 R4 LG

31 KS C C c1 c1 MS tgT0 BE LG

32 PG F F c2 F MS Robin R4 Exp

33 PG C F c2 F MS Robin R4 Exp

34 PG F C c2 F MS Robin R4 LG

35 PG C C c2 F MS Robin R4 LG

36 PG C C c2 F MS Robin BE LG

37 H C C c12 F MS tgT0 BE LG

38 H C C c12 F MS tgT0 R4 LG

39 FH C C c12 F MS tgT0 BE LG

aPG Painleve-Gullstrand KS Kerr-Schild H harmonic time FH

fully harmonicbF freezing C constraintcF freezing c(x) Equation 232 with cx dMS minimal strain MD minimal distortioneR4 4th-order Runge-Kutta BE backward EulerfExp exponential growth LG linearly growing gauge mode

FIG 1 Long-term stability of the evolution of Kerr-Schild ini-

tial data run 1 from Table I Plotted is the l2 norm of the Hamil-

tonian constraint 220a in units of M2 as a function of time for

several spatial resolutions The number of spectral coefficients Nr

for each plot starting at the top is 12 16 20 24 27 32 36 40

45 48 54 and 60

FIG 2 Norm of the error in grr as a function of time for the

same evolutions shown in Fig 1

BLACK HOLE EVOLUTION BY SPECTRAL METHODS PHYSICAL REVIEW D 62 084032

084032-13

Pros and cons

No dissipation

Exponential convergence

Superb results for little effort

Gibbs phenomena

Discrete equations are fully coupled

Lack of dissipation allows high frequency errors to remain

Can not freely choose location of nodes

Regge calculus

Regge calculus

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is piecewise flat

Curvature is a distribution

Defect angle at each vertex

2d example

2d example

2d example

2d example

2d example

3d example

3d example

3d example

3d example

3d example

4d example

Field equations

Some questions

Can smooth metrics be accurately approximated by Regge lattices

Do the Regge equations reduce to the Einstein equations in some suitable limit

Do solutions of the Regge equations converge to solutions of the Einstein equations

Some answers

Regge

Reggesolutions

Yes

Yes as an average

Allendorfer Weyl

Cheeger Muller Schrader

Continuum

Einsteinsolutions

Applications

FRW cosmologies

Brill wave initial data

Quantum gravityKasner cosmologies

Misner initial data

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 26: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Pros and cons

No dissipation

Exponential convergence

Superb results for little effort

Gibbs phenomena

Discrete equations are fully coupled

Lack of dissipation allows high frequency errors to remain

Can not freely choose location of nodes

Regge calculus

Regge calculus

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is piecewise flat

Curvature is a distribution

Defect angle at each vertex

2d example

2d example

2d example

2d example

2d example

3d example

3d example

3d example

3d example

3d example

4d example

Field equations

Some questions

Can smooth metrics be accurately approximated by Regge lattices

Do the Regge equations reduce to the Einstein equations in some suitable limit

Do solutions of the Regge equations converge to solutions of the Einstein equations

Some answers

Regge

Reggesolutions

Yes

Yes as an average

Allendorfer Weyl

Cheeger Muller Schrader

Continuum

Einsteinsolutions

Applications

FRW cosmologies

Brill wave initial data

Quantum gravityKasner cosmologies

Misner initial data

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 27: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Regge calculus

Regge calculus

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is piecewise flat

Curvature is a distribution

Defect angle at each vertex

2d example

2d example

2d example

2d example

2d example

3d example

3d example

3d example

3d example

3d example

4d example

Field equations

Some questions

Can smooth metrics be accurately approximated by Regge lattices

Do the Regge equations reduce to the Einstein equations in some suitable limit

Do solutions of the Regge equations converge to solutions of the Einstein equations

Some answers

Regge

Reggesolutions

Yes

Yes as an average

Allendorfer Weyl

Cheeger Muller Schrader

Continuum

Einsteinsolutions

Applications

FRW cosmologies

Brill wave initial data

Quantum gravityKasner cosmologies

Misner initial data

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 28: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Regge calculus

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is piecewise flat

Curvature is a distribution

Defect angle at each vertex

2d example

2d example

2d example

2d example

2d example

3d example

3d example

3d example

3d example

3d example

4d example

Field equations

Some questions

Can smooth metrics be accurately approximated by Regge lattices

Do the Regge equations reduce to the Einstein equations in some suitable limit

Do solutions of the Regge equations converge to solutions of the Einstein equations

Some answers

Regge

Reggesolutions

Yes

Yes as an average

Allendorfer Weyl

Cheeger Muller Schrader

Continuum

Einsteinsolutions

Applications

FRW cosmologies

Brill wave initial data

Quantum gravityKasner cosmologies

Misner initial data

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 29: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is piecewise flat

Curvature is a distribution

Defect angle at each vertex

2d example

2d example

2d example

2d example

2d example

3d example

3d example

3d example

3d example

3d example

4d example

Field equations

Some questions

Can smooth metrics be accurately approximated by Regge lattices

Do the Regge equations reduce to the Einstein equations in some suitable limit

Do solutions of the Regge equations converge to solutions of the Einstein equations

Some answers

Regge

Reggesolutions

Yes

Yes as an average

Allendorfer Weyl

Cheeger Muller Schrader

Continuum

Einsteinsolutions

Applications

FRW cosmologies

Brill wave initial data

Quantum gravityKasner cosmologies

Misner initial data

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 30: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is piecewise flat

Curvature is a distribution

Defect angle at each vertex

2d example

2d example

2d example

2d example

2d example

3d example

3d example

3d example

3d example

3d example

4d example

Field equations

Some questions

Can smooth metrics be accurately approximated by Regge lattices

Do the Regge equations reduce to the Einstein equations in some suitable limit

Do solutions of the Regge equations converge to solutions of the Einstein equations

Some answers

Regge

Reggesolutions

Yes

Yes as an average

Allendorfer Weyl

Cheeger Muller Schrader

Continuum

Einsteinsolutions

Applications

FRW cosmologies

Brill wave initial data

Quantum gravityKasner cosmologies

Misner initial data

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 31: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

2d example

Metric is piecewise flat

Curvature is a distribution

Defect angle at each vertex

2d example

2d example

2d example

2d example

2d example

3d example

3d example

3d example

3d example

3d example

4d example

Field equations

Some questions

Can smooth metrics be accurately approximated by Regge lattices

Do the Regge equations reduce to the Einstein equations in some suitable limit

Do solutions of the Regge equations converge to solutions of the Einstein equations

Some answers

Regge

Reggesolutions

Yes

Yes as an average

Allendorfer Weyl

Cheeger Muller Schrader

Continuum

Einsteinsolutions

Applications

FRW cosmologies

Brill wave initial data

Quantum gravityKasner cosmologies

Misner initial data

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 32: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

2d example

2d example

2d example

2d example

2d example

3d example

3d example

3d example

3d example

3d example

4d example

Field equations

Some questions

Can smooth metrics be accurately approximated by Regge lattices

Do the Regge equations reduce to the Einstein equations in some suitable limit

Do solutions of the Regge equations converge to solutions of the Einstein equations

Some answers

Regge

Reggesolutions

Yes

Yes as an average

Allendorfer Weyl

Cheeger Muller Schrader

Continuum

Einsteinsolutions

Applications

FRW cosmologies

Brill wave initial data

Quantum gravityKasner cosmologies

Misner initial data

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 33: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

2d example

2d example

2d example

2d example

3d example

3d example

3d example

3d example

3d example

4d example

Field equations

Some questions

Can smooth metrics be accurately approximated by Regge lattices

Do the Regge equations reduce to the Einstein equations in some suitable limit

Do solutions of the Regge equations converge to solutions of the Einstein equations

Some answers

Regge

Reggesolutions

Yes

Yes as an average

Allendorfer Weyl

Cheeger Muller Schrader

Continuum

Einsteinsolutions

Applications

FRW cosmologies

Brill wave initial data

Quantum gravityKasner cosmologies

Misner initial data

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 34: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

2d example

2d example

2d example

3d example

3d example

3d example

3d example

3d example

4d example

Field equations

Some questions

Can smooth metrics be accurately approximated by Regge lattices

Do the Regge equations reduce to the Einstein equations in some suitable limit

Do solutions of the Regge equations converge to solutions of the Einstein equations

Some answers

Regge

Reggesolutions

Yes

Yes as an average

Allendorfer Weyl

Cheeger Muller Schrader

Continuum

Einsteinsolutions

Applications

FRW cosmologies

Brill wave initial data

Quantum gravityKasner cosmologies

Misner initial data

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 35: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

2d example

2d example

3d example

3d example

3d example

3d example

3d example

4d example

Field equations

Some questions

Can smooth metrics be accurately approximated by Regge lattices

Do the Regge equations reduce to the Einstein equations in some suitable limit

Do solutions of the Regge equations converge to solutions of the Einstein equations

Some answers

Regge

Reggesolutions

Yes

Yes as an average

Allendorfer Weyl

Cheeger Muller Schrader

Continuum

Einsteinsolutions

Applications

FRW cosmologies

Brill wave initial data

Quantum gravityKasner cosmologies

Misner initial data

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 36: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

2d example

3d example

3d example

3d example

3d example

3d example

4d example

Field equations

Some questions

Can smooth metrics be accurately approximated by Regge lattices

Do the Regge equations reduce to the Einstein equations in some suitable limit

Do solutions of the Regge equations converge to solutions of the Einstein equations

Some answers

Regge

Reggesolutions

Yes

Yes as an average

Allendorfer Weyl

Cheeger Muller Schrader

Continuum

Einsteinsolutions

Applications

FRW cosmologies

Brill wave initial data

Quantum gravityKasner cosmologies

Misner initial data

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 37: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

3d example

3d example

3d example

3d example

3d example

4d example

Field equations

Some questions

Can smooth metrics be accurately approximated by Regge lattices

Do the Regge equations reduce to the Einstein equations in some suitable limit

Do solutions of the Regge equations converge to solutions of the Einstein equations

Some answers

Regge

Reggesolutions

Yes

Yes as an average

Allendorfer Weyl

Cheeger Muller Schrader

Continuum

Einsteinsolutions

Applications

FRW cosmologies

Brill wave initial data

Quantum gravityKasner cosmologies

Misner initial data

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 38: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

3d example

3d example

3d example

3d example

4d example

Field equations

Some questions

Can smooth metrics be accurately approximated by Regge lattices

Do the Regge equations reduce to the Einstein equations in some suitable limit

Do solutions of the Regge equations converge to solutions of the Einstein equations

Some answers

Regge

Reggesolutions

Yes

Yes as an average

Allendorfer Weyl

Cheeger Muller Schrader

Continuum

Einsteinsolutions

Applications

FRW cosmologies

Brill wave initial data

Quantum gravityKasner cosmologies

Misner initial data

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 39: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

3d example

3d example

3d example

4d example

Field equations

Some questions

Can smooth metrics be accurately approximated by Regge lattices

Do the Regge equations reduce to the Einstein equations in some suitable limit

Do solutions of the Regge equations converge to solutions of the Einstein equations

Some answers

Regge

Reggesolutions

Yes

Yes as an average

Allendorfer Weyl

Cheeger Muller Schrader

Continuum

Einsteinsolutions

Applications

FRW cosmologies

Brill wave initial data

Quantum gravityKasner cosmologies

Misner initial data

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 40: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

3d example

3d example

4d example

Field equations

Some questions

Can smooth metrics be accurately approximated by Regge lattices

Do the Regge equations reduce to the Einstein equations in some suitable limit

Do solutions of the Regge equations converge to solutions of the Einstein equations

Some answers

Regge

Reggesolutions

Yes

Yes as an average

Allendorfer Weyl

Cheeger Muller Schrader

Continuum

Einsteinsolutions

Applications

FRW cosmologies

Brill wave initial data

Quantum gravityKasner cosmologies

Misner initial data

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 41: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

3d example

4d example

Field equations

Some questions

Can smooth metrics be accurately approximated by Regge lattices

Do the Regge equations reduce to the Einstein equations in some suitable limit

Do solutions of the Regge equations converge to solutions of the Einstein equations

Some answers

Regge

Reggesolutions

Yes

Yes as an average

Allendorfer Weyl

Cheeger Muller Schrader

Continuum

Einsteinsolutions

Applications

FRW cosmologies

Brill wave initial data

Quantum gravityKasner cosmologies

Misner initial data

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 42: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

4d example

Field equations

Some questions

Can smooth metrics be accurately approximated by Regge lattices

Do the Regge equations reduce to the Einstein equations in some suitable limit

Do solutions of the Regge equations converge to solutions of the Einstein equations

Some answers

Regge

Reggesolutions

Yes

Yes as an average

Allendorfer Weyl

Cheeger Muller Schrader

Continuum

Einsteinsolutions

Applications

FRW cosmologies

Brill wave initial data

Quantum gravityKasner cosmologies

Misner initial data

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 43: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Field equations

Some questions

Can smooth metrics be accurately approximated by Regge lattices

Do the Regge equations reduce to the Einstein equations in some suitable limit

Do solutions of the Regge equations converge to solutions of the Einstein equations

Some answers

Regge

Reggesolutions

Yes

Yes as an average

Allendorfer Weyl

Cheeger Muller Schrader

Continuum

Einsteinsolutions

Applications

FRW cosmologies

Brill wave initial data

Quantum gravityKasner cosmologies

Misner initial data

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 44: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Some questions

Can smooth metrics be accurately approximated by Regge lattices

Do the Regge equations reduce to the Einstein equations in some suitable limit

Do solutions of the Regge equations converge to solutions of the Einstein equations

Some answers

Regge

Reggesolutions

Yes

Yes as an average

Allendorfer Weyl

Cheeger Muller Schrader

Continuum

Einsteinsolutions

Applications

FRW cosmologies

Brill wave initial data

Quantum gravityKasner cosmologies

Misner initial data

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 45: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Some answers

Regge

Reggesolutions

Yes

Yes as an average

Allendorfer Weyl

Cheeger Muller Schrader

Continuum

Einsteinsolutions

Applications

FRW cosmologies

Brill wave initial data

Quantum gravityKasner cosmologies

Misner initial data

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 46: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Applications

FRW cosmologies

Brill wave initial data

Quantum gravityKasner cosmologies

Misner initial data

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 47: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Kasner cosmology

Class Quantum Grav 15 (1998) 389ndash405 Printed in the UK PII S0264-9381(98)84850-8

A fully (3+ 1)-dimensional Regge calculus model of theKasner cosmology

Adrian P Gentledagger and Warner A MillerDaggerTheoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM

87545 USA

Received 5 June 1997 in final form 23 October 1997

Abstract We describe the first discrete-time four-dimensional numerical application of Regge

calculus The spacetime is represented as a complex of four-dimensional simplices and the

geometry interior to each 4-simplex is flat Minkowski spacetime This simplicial spacetime is

constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built

from tetrahedra We implement a novel 2-surface initial-data prescription for Regge calculus

and provide the first fully four-dimensional application of an implicit decoupled evolution

scheme (the lsquoSorkin evolution schemersquo) We benchmark this code on the Kasner cosmologymdasha

cosmology which embodies generic features of the collapse of many cosmological models We

(i) reproduce the continuum solution with a fractional error in the 3-volume of 105 after 10 000evolution steps (ii) demonstrate stable evolution (iii) preserve the standard deviation of spatial

homogeneity to less than 1010 and (iv) explicitly display the existence of diffeomorphismfreedom in Regge calculus We also present the second-order convergence properties of the

solution to the continuum

PACS numbers 0425D 0420 0460N

1 Regge calculus as an independent tool in general relativity

In this paper we describe the first fully (3+ 1)-dimensional application of Regge calculus

[1 2] to general relativity We develop an initial-value prescription based on the standard

York formalism and implement a four-stage parallel evolution algorithm We benchmark

these on the Kasner cosmological model

We present three findings First that the Regge solution exhibits second-order

convergence of the physical variables to the continuum Kasner solution Secondly Regge

calculus appears to have a complete diffeomorphic structure in that we are free to specify

three shift and one lapse condition per vertex Furthermore the four corresponding

constraint equations are conserved to within a controllable tolerance throughout the

evolution Finally the recently developed decoupled parallel evolution scheme [3] (the

lsquoSorkin evolution schemersquo) yields stable evolution

Although we have taken just the first few steps in developing a numerical Regge calculus

programme every indication (both numerical and analytic) suggests that it will be a valuable

tool in the study of gravity Our numerical studies together with analytic results [4] should

dagger Permanent address Department of Mathematics Monash University Clayton Victoria 3168 Australia E-mail

address adriannewtonmathsmonasheduau

Dagger E-mail address wamlanlgov

0264-938198020389+17$1950 c 1998 IOP Publishing Ltd 389

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 48: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Kasner cosmology

Vacuum homogenous and isotropic

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 49: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Kasner lattice

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 50: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Kasner lattice

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 51: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Kasner lattice

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 52: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Kasner lattice

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 53: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Sorkin evolution

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 54: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Kasner cosmology

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 55: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Brill wave initial data

Class Quantum Grav 16 (1999) 1987ndash2003 Printed in the UK PII S0264-9381(99)00855-2

Simplicial Brill wave initial data

Adrian P Gentle

Department of Mathematics and Statistics Monash University Clayton Victoria 3168 Australia

and

Theoretical Division (T-6 MS B288) Los Alamos National Laboratory Los Alamos NM 87545

USA

E-mail adriannewtonmathsmonasheduau

Received 12 January 1999

Abstract Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at

a moment of time symmetry We argue that only a tetrahedral lattice can successfully reproduce the

continuum solution and develop a simplicial axisymmetric lattice based on the coordinate structure

of the continuum metric This is used to construct initial data for Brill waves in an otherwise flat

spacetime and for the distorted black hole spacetime of Bernstein These initial data sets are shown

to be second-order accurate approximations to the corresponding continuum solutions

PACS numbers 0420 0425D

1 Introduction

Since its inception in 1961 Regge calculus [1] has been studied extensively in highly symmetric

spacetimes forwhich corresponding exact solutions of Einsteinrsquos equations are often available

A complete review and bibliography of this early work is provided by Williams and Tuckey

[2]

Following the realization that a fully decoupled parallelizable (3 + 1)-dimensional

evolution scheme occurs naturally within Regge calculus [3ndash5] this simplicial approach to

gravity is now on the verge of tackling physically interesting and dynamic problems The

first tentative step along this path was completed recently with the successful application of

simplicial Regge calculus to the Kasner spacetime in (3 + 1) dimensions [6]

A vital precursor to the evolution problem in any general relativistic simulation is the

construction of consistent initial data Gentle and Miller [6] present a general prescription

for the calculation of 2-surface initial data for the Regge lattice although their approach has

only been applied to the Kasner cosmology As a prelude to the construction of simplicial

initial data for complex spacetimes in this paper we consider the restricted case of vacuum

axisymmetric initial data at a moment of time symmetry This is the first step towards our

goal and provides a benchmark against which future fully four-dimensional initial data may

be compared

We construct vacuum initial data for the pure Brill wave spacetime [7] and for Brill

waves in a black hole spacetimemdashthe lsquodistorted black holesrsquo first considered by Bernstein [8]

Standard finite-difference techniques have been used to study pure Brill wave spacetimes by

Eppley [9] Miyama [10] Holz et al [11] and Alcubierre et al [12] The axisymmetric Brill

0264-938199061987+17$1950 copy 1999 IOP Publishing Ltd 1987

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 56: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Brill wave initial data

One equation per point One equation per vertex

Time symmetric initial data

Given compute

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 57: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Brill wave initial data1994 A P Gentle

prism-based and a tetrahedral lattice is the absence of a diagonal edge on the z faces Each

such face in a prism-based lattice is required by construction to be flat However the pureBrill

wave metric (6) allows fluctuations across faces in the ndashz-plane through both the conformal

factor ( z) and the form function q( z) A prism-based lattice is unable to capture such

variations

The most natural solution to this problem is to abandon the prism-based lattice for one

constructed from simplices In the remainder of this paper we shall follow just such a path

and show that a tetrahedral lattice resolves all of the problems encountered by Dubal

5 A tetrahedral 3-geometry

In this section we construct an axisymmetric shear-free simplicial 3-geometry and use it to

build time-symmetric initial data using Regge calculus

The simplicial lattice may be obtained by subdividing the basic prism shown in figure 3

Each block is divided into six tetrahedra introducing three face diagonals and one body

diagonal per vertex The result is shown in figure 4 The body diagonal within the prism

is denoted by bijk whilst face diagonals spanning the - and -axes are written as dijk and so

forth

Figure 4 The rectangular prism shown in figure 3 subdivided into six tetrahedra This involves

adding a diagonal brace to each face of the prism together with a body diagonal

To obtain an axisymmetric approximation in the style of the preceding section we take

the limit as the prism is collapsed along the -axis This is a more complicated procedure than

in the previous case [14] since we must now deal with the limiting form of the extra edges

introduced in the simplicial approach together with their associated deficit angles

To aid in constructing the lattice we consider the lsquoprototypersquo axisymmetric metric

ds2 = d2 + dz2 + 2 d2 (19)

This non-physical metric is useful to frame our discussion of the axisymmetric lattice and

later we will map from this lsquoprototypersquo to metrics of physical interest Figure 4 shows the

subdivision of a coordinate block such that the l h and w edges are locally aligned with the

- z- and -axes respectively We enforce axisymmetry by explicitly setting

wijk = rijk$ (20)

and demanding that there be no variation in the edges along the -axis The redundant j index

is neglected in the remainder of this paper

Simplicial Brill wave initial data 1993

The principal advantages of the prism-based approach are the ability to closely model the

symmetries of the spacetime of interest and the computational simplicity when compared

with a simplicial lattice Since there are fewer edges in a prism-based lattice than in a

simplicial lattice there are fewer hinges about which curvature is concentrated and hence

fewer calculations are required to obtain the deficit angles within the lattice

It is largely for these reasons that the prism approach has dominated much previous work

in Regge calculus The major drawback of a prism-based lattice is that even once all edges

have been specified the lattice is still not rigid additional constraints must be provided to

constrain the space of possible edge lengths In the presence of high symmetry it is possible

that natural restrictions are available to specify the remaining degrees of freedom This has

been the case for most previous prism-based applications of Regge calculus

The simplicial approach which constructs 3-geometries from tetrahedra and 4-geometries

from 4-simplices is more natural if one wishes to model a complex spacetime Once all of the

edge lengths in a simplicial lattice have been specified the geometry is uniquely determined

Noting that a simplicial lattice may be obtained by subdivision of a prism-based lattice some

of the advantages of the prism construction may be carried over to the simplicial lattice

Figure 3 A section of the axisymmetric lattice used by Dubal [14] The relation to the global polar

coordinate system is shown together with the angles used to specify the extra degree of freedom

For the particular case of axisymmetric non-rotating Brill waves the symmetry suggests

that a prism-based lattice may be convenient Indeed Dubal [14] approximated the pure

Brill wave spacetime at a moment of time symmetry with a lattice constructed from the

coordinate blocks of the continuum a typical element of which is shown in figure 3 The

desired axisymmetry is built into the lattice by demanding that there is no variation along the

-axis The limit as tends to zero is then taken to obtain a purely axisymmetric Regge

initial value equation

Dubal fixed the remaining degrees of freedom in the prisms by introducing several angles

on each ndashz face as shown in figure 3 and specifying them in terms of the surrounding

edges The clever assignment chosen by Dubal allowed him to construct Regge solutions in

reasonable agreement with the continuum solution at least for low-amplitude Brill waves

Dubal noted several limitations to his approach including the relatively poor Regge mass

estimates for lsquolarge-amplitudersquo (a 12) Brill waves Another indication of problems was the

low convergence rate of the prism-based Regge solution to the continuum As Dubal notes

[14] these problems arise because of the approximation made in specifying the angles within

each prism To obtain a better solution an improved relation between the angles and prism

edges is required We argue that the best solution is to abandon the prisms altogether and

introduce a tetrahedral lattice

Once the axisymmetric limit ( 0) has been taken the major difference between a

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 58: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Brill wave conformal factor

Simplicial Brill wave initial data 1989

together with the condition that q = O(r2) as r to ensure that the hypersurface has

an asymptotically well defined mass With this choice of background metric the Hamiltonian

constraint (5) takes the form

$2 =

4

2q

2+

2q

z2

(7)

which is solved for ( z) once q( z) is given

To allow comparison with Eppley [9] we choose q( z) to be of the form

q = a2

1 + rnwhere r2 = 2 + z2 (8)

and the boundary conditions on q imply that n 4 In the remainder of this paper we set

n = 5 The single remaining parameter the wave amplitude a is arbitrary

To obtain an axisymmetric asymptotically flat solution which is reflection symmetric on

the z = 0 axis we use boundary conditions on of the form

(0 z) = 0

z( 0) = 0 (9)

together with a Robin outer boundary condition

r= 1

ras r (10)

Figure 1 shows the solution of equation (7) obtained using a centred finite-difference

approximation to both the equation and boundary conditions The solution was calculated on

a 601 601 grid with a Brill wave amplitude of a = 10

Figure 1 The conformal factor for Brill wave initial data calculated on a 601601 grid using acentred finite-difference approximation to equation (7) The Eppley form of q( z) (equation (8))was used with wave amplitude a = 10 and the outer boundaries were placed at = 20 and

z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

1998 A P Gentle

Figure 5 The conformal factor for a Brill wave of amplitude a = 10 calculated using the

simplicial Regge lattice The agreement between this and the continuum solution in figure 1 is

excellent The calculation was performed using a lattice consisting of 601 601 vertices with theouter boundaries at = 20 and z = 20

(This figure can be viewed in colour in the electronic version of the article seehttpwwwioporg)

We estimate the mass of the initial data shown in figures 1 and 5 by examining the fall off

of the conformal factor in the asymptotic region Assuming that the conformal factor takes the

form

= 1 +M

2r+ middot middot middot as r (37)

we perform a least-squares fit of this function to each numerical solution far from the region in

which the Brill wave is concentrated This asymptotic form for is guaranteed by the choice

of outer boundary conditionmdashonly the value of M is left undetermined The mass estimates

are shown in table 1 for various choices of Brill wave amplitude The masses calculated for

the simplicial Regge and continuum solutions agree remarkably well and also show excellent

agreement with previous calculations

Table 1 Mass estimates for the different initial data sets using the Eppley form function q( z)Themass is calculated by examining the fall off of in the asymptotic region for each data setmdashthesimplicial Regge (MS ) and continuum (MC ) solutions We also show the results of the previous

calculations by Alcubierre et al [12] which are in excellent agreement with both the continuum

and simplicial Regge solutions

a MS MC M (Alcubierre et al)

1 47 10$2 47 10$2 48 10$2

2 172 10$1 172 10$1 174 10$1

5 877 10$1 879 10$1 883 10$1

10 322 322 322

12 484 484 485

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 59: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Brill wave lattice convergence

Simplicial Brill wave initial data 2001

since a second-order accurate approximation to equation (11) was used to generate ik In

section 22 a convergence test for the pure black hole solution confirmed that the continuum

code is indeed second-order accurate

Now consider the simplicial Regge solution Rik which we also expect to differ from the

continuum solution by some amount dependent on the scale length By this reasoning we

see that the Regge solution should also differ from the numerical solution of the continuum

equations by some small amount If we assume that

Rik = ik +Kik

q + middot middot middot (45)

where Kij = O(1) we can write

Rik ik

ik

= Kik

ik

q + middot middot middot (46)

where we have neglected only terms smaller thanq Noting that the coefficient ofq scales

as order unity we define the lsquoerrorrsquo between the numerical continuum and Regge solutions as

eN = 1

N

n$

i=0

n

k=0

R

ik ik

ik

(47)

where N = (n$+1)(n +1) The error eN may be calculated directly from the two numerical

solutions and doing so for a variety of grid scales yields an estimate of the convergence rate

q

Figure 7 shows the behaviour of the error eN as the number of grid points is increased

(hence decreased) for both the pure Brill wave and the distorted black hole initial data sets

The figure suggests a value of q 2 That is the difference between the simplicial Regge

solution and the numerical solution ik decreases as approximately the second power of the

discretization scale Since the numerical solution ik itself differs from the exact solution

by terms of order 2 we conclude that the Regge solution is a second-order accurate

approximation to the underlying exact solution of the continuum equations

Figure 7 The averaged fractional difference eN between the simplicial and continuum solutions

shown as a function of the number of vertices (grid points) All calculations were performed on

an N N grid for the pure Brill wave space with a = 10 (triangles) and distorted black hole

initial data with parameters (1 2 1) (diamonds) In both cases the fractional difference betweenthe simplicial Regge and continuum solutions reduces as the second power of the grid spacing

From this we conclude that the simplicial Regge solution is a second-order accurate approximation

to the underlying continuum solution

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 60: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Pros and cons

Clear separation between topology and metric

Coordinate free

Local computations are trivial

Metric not smooth

Must treat curvature as a distribution

Analysis is hard very few theorems

Convergence is unclear

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 61: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Smooth lattice GR

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 62: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Smooth lattice GR

Chicken wire relativity

Originally called

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 63: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

2d example

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 64: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

2d example

Metric recorded by table of leg-lengths

Topology recorded byconnection matrix

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 65: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

2d example

Metric is smooth

Curvature is a point function

Legs are geodesic segments

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 66: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Riemann normal coordinates

etc

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 67: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Geodesic leg lengths

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 68: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

General relativity

Given solve for and

is now an algebraic system for the

Then impose the vacuum field equations

Solve the equations Et voila -- a numerical spacetime

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 69: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

ADM equations on a lattice

With zero shift and unit lapse

Applied to the lattice

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 70: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

A quick derivation

Differentiate twice and retain leading order terms

Alternatively can use 2nd variation of arc-length

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 71: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Examples

Flat Kasner cosmologies

Maximal amp geodesic slicing of Schwarzschild

Teukolsky spacetime (in progress)

Oppenheimer-Snyder dust collapse (with Jules Kajtar)

1+1 evolution

3+1 evolution

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 72: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Maximally sliced Schwarzschild

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class Quantum Grav 19 (2002) 429ndash455 PII S0264-9381(02)27278-0

Long term stable integration of a maximally slicedSchwarzschild black hole using a smoothlattice method

Leo Brewin

Department of Mathematics amp Statistics Monash University Clayton Victoria 3800 Australia

Received 27 July 2001Published 14 January 2002Online at stacksioporgCQG19429

AbstractWe will present results of a numerical integration of a maximally slicedSchwarzschild black hole using a smooth lattice method The results showno signs of any instability forming during the evolutions to t = 1000m Theprinciple features of our method are (i) the use of a lattice to record the geometry(ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADMequations to the lattice and (iii) the use of the Bianchi identities to assist inthe computation of the curvatures No other special techniques are used Theevolution is unconstrained and the ADM equations are used in their standardform

PACS numbers 0425D 0460N 0270

1 Introduction

Recent studies [1ndash7] have shown that the stability of numerical integrations of the Einsteinfield equations can depend on the formulation of the evolution equations Subtle changes in thestructure of the evolution equations have been shown to have a dramatic effect on the long termstability of the integrations These are relatively new investigations and thus at present there isno precise mathematical explanation as to what is the root cause of the instabilities or how bestthey can be avoided or minimized What we have at present is a growing set of examples whichsuggests that the standard ADM evolution equations may not be the most suitable equationsfor numerical relativity Consequently many people are looking at alternative formulationssuch as the hyperbolic formulations of Einsteinrsquos equations [8ndash10] and the conformal ADMequations of Shibita and Nakamura [6] and Baumgarte and Shapiro [3]

One alternative is the smooth lattice approach which we presented in two earlier papers[11 12] This is a method which uses a lattice similar to that used in the Regge calculusbut differing significantly in the way the field equations are imposed on the lattice In thesmooth lattice method we employ a series of local Riemann normal coordinates in which theconnection vanishes at the origin of each such frame Collectively these frames enable us

0264-938102030429+27$3000 copy 2002 IOP Publishing Ltd Printed in the UK 429

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 73: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Schwarzschild

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 74: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

The lattice

Spherical symmetry so only need one ladder

Edges of ladder are radial geodesics

Evolve the rungs and radial segments

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 75: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Evolution equations

Note now using 3d Riemann curvatures

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 76: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Curvature equations

Geodesic deviation

Bianchi identity

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 77: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Collapse of the lapse

Radial proper distance

N

8000700060005000400030002000100000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Radial proper distance

N

80070060050040030020010000

100e+00

800e-01

600e-01

400e-01

200e-01

000e+00

Maximal time

Centr

alla

pse

N

100090080070060050040030020010000

100e+00

100e-05

100e-10

100e-15

100e-20

100e-25

Maximal time

Centr

alla

pse

N

1200010000800060004000200000

100e+00

100e-50

100e-100

100e-150

100e-200

100e-250

t=10m-100m t=100m-1000m

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 78: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Oppenheimer-Snyder dust collapse

Smooth lattice construction of the Oppenheimer-Snyder spacetime

Leo Brewin and Jules Kajtardagger

School of Mathematical Sciences Monash University 3800 Australia(Received 23 May 2009 published 5 November 2009)

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime The

results are in excellent agreement with theory and numerical results from other authors

DOI 101103PhysRevD80104004 PACS numbers 0425D

I INTRODUCTION

In recent times many numerical relativists have goodreason to celebratemdashthe long battle to secure the holy grail[1] is over (though some might prefer to redraw the battlelines) The works of Pretorius [23] and others [45] haveopened a new era for computational general relativity Thishas spawned many new projects that directly address theneeds of the gravitational wave community Many groupsare now running detailed simulations of binary systems infull general relativity as a matter of course Does this meanthat the development of computational methods for generalrelativity is now over The experience in other fields wouldsuggest otherwise look for example at computationalfluid dynamics where a multitude of techniques are com-monly used including spectral methods finite elementmethods smooth particle hydrodynamics high resolutionshock capture methods and the list goes on The importantpoint to note is that one method does not solve all theproblems and thus in numerical relativity it is wise even inthe face of the current successes to seek other methods tosolve the Einstein equations It is in that spirit that we havebeen developing what we call the smooth lattice method[6ndash8] This is a fundamentally discrete approach to generalrelativity based on a large collection of short geodesicsegments connected to form a lattice representation ofspacetime The Einstein equations are cast as evolutionequations for the leg-lengths with the Riemann and energy-momentum tensors acting as sources Of course theRiemann tensor must be computed from the leg-lengthsand this can be done in a number of related ways such asby fitting a local Riemann normal coordinate expansion toa local cluster of legs or to use the geodesic deviationequation or and with more generality to use the secondvariation of arc-length Past applications of the methodhave included a full 3 1 simulation of the vacuumKasner cosmology [8] and a 1 1 maximally slicedSchwarzschild spacetime [6] In both cases the simulationswere stable and showed excellent agreement with theknown solutions while showing no signs of instabilities(the maximally sliced Schwarzschild solution ran for t gt1000m and was stopped only because there was no point inrunning the code any longer)

In this paper we report on our recent work using theOppenheimer-Snyder [9] spacetime as a benchmark for oursmooth lattice method [6ndash8] We chose this spacetime formany reasons it has been cited by many authors [10ndash16] asa standard benchmark for numerical codes (and thus com-parative results are available) the analytic solution isknown (in a number of time slicings) the equations aresimple and there are many simple diagnostics that can beused to check the accuracy of the results (as described inSecs XI and XII)In an impressive series of papers Shapiro and Teukolsky

([101114]) used the Oppenheimer-Snyder spacetime asthe first in a series of test cases They were motivated bycertain problems in relativistic stellar dynamics (such asthe formation of neutron stars and black holes from super-nova) and they developed a set of codes based on thestandard ADM equations adapted to spherical symmetryin both maximal and polar slicing and using an N-bodyparticle simulation for the hydrodynamics They madelimited use of the exact Schwarzschild solution to developan outer boundary condition for the lapse function whileusing both the Schwarzschild and FRW solutions to set theinitial data Though their discussion on the size of theirerrors is brief (for the Oppenheimer-Snyder test case) theydid note that the errors were of the order of a percent or so(for a system with 240 grid points and 1180 dust particles)In a later work Baumgarte et al [15] extended their workby expressing the metric and the equations in terms of anoutgoing null coordinate This leads to a slicing that coversall of the spacetime outside (and arbitrarily close to) theevent horizon In this version of their code Baumgarte et al[15] chose to solve only the equations for the dustball byusing the Schwarzschild solution as an outer boundaryconditionThis idea to replace the exterior equations with the

known Schwarzschild solution has been used byGourgoulhon [13] Schinder et al [12] and Romero et al[16] Gourgoulhon [13] used a radial gauge and polarslicing while solving the equations using a spectral methodand reported errors in the metric variables between 107 to105 However with the onset of the Gibbs phenomenathe code could only be run until the central lapse collapsedto around 2 103 Schinder et al [12] used the sameequations as Gourgoulhon [13] but with a discretizationbased on a standard finite difference scheme They re-

LeoBrewinscimonasheduaudaggerJulesKajtarscimonasheduau

PHYSICAL REVIEW D 80 104004 (2009)

1550-7998=2009=80(10)=104004(18) 104004-1 2009 The American Physical Society

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 79: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Lattice

FRW dust interior Schwarzschild exterior

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 80: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Evolution of the curvature

t=0-60m

>

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 81: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Evolution of the curvature

t=0-200m

>

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 82: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

Pros and cons

Metric is differentiable

All the pros of Regge calculus plus

Curvature is a pointwise function

Can use all the usual mathematical tools

Must solve a coupled system to compute curvatures

Coordinates not known apriori

How do we impose boundary conditions

Lacks empirical support but results are promising

No non-symmetric 3+1 resultsbut stay tuned to gr-qc

Page 83: Non-standard Computational Methods in Numerical RelativityLeo.pdf · Non-standard methods Numerical Relativity Regge calculus Multiquadrics Spectral methods Smooth lattices Tetrad

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