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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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 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
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
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
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
Lattice
FRW dust interior Schwarzschild exterior
Evolution of the curvature
t=0-60m
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