Three-dimensional numerical general relativistic hydrodynamics: Formulations, methods, and code...

transcript

Three-dimensional general relativistic hydrodynamics II long-term dynamics ofsingle relativistic stars

Jose A Font(12) Tom Goodale(3) Sai Iyer(4) Mark Miller(4) Luciano Rezzolla(56) Edward Seidel(37)Nikolaos Stergioulas(8) Wai-Mo Suen(49) Malcolm Tobias(4)

(1)Max-Planck-Institut fur Astrophysik Karl-Schwarzschild-Str 1 D-85740 Garching Germany(2)Departamento de Astronomıa y Astrofısica Universidad de Valencia 46100 Burjassot (Valencia) Spain

(3)Max-Planck-Institut fur Gravitationsphysik Am Muhlenberg 1 D-14476 Golm Germany(4)McDonnell Center for the Space Sciences Department of Physics Washington University St Louis Missouri 63130

(5)SISSA International School for Advanced Studies Via Beirut 2-4 34014 Trieste Italy(6)INFN Department of Physics University of Trieste Via A Valerio 2 34127 Trieste Italy

(7)National Center for Supercomputing Applications BeckmanInstitute 405 N Mathews Ave Urbana IL 61801(8)Department of Physics Aristotle University of Thessaloniki Thessaloniki 54006 Greece

(9)Physics Department Chinese University of Hong Kong Shatin Hong Kong(February 6 2008)

This is the second in a series of papers on the construction and validation of a three-dimensional code for thesolution of the coupled system of the Einstein equations andof the general relativistic hydrodynamic equationsand on the application of this code to problems in general relativistic astrophysics In particular we report on theaccuracy of our code in the long-term dynamical evolution ofrelativistic stars and on some new physics resultsobtained in the process of code testing The following aspects of our code have been validated the generationof initial data representing perturbed general relativistic polytropic models (both rotating and non-rotating) thelong-term evolution of relativistic stellar models and the coupling of our evolution code to analysis modulesproviding for instance the detection of apparent horizons or the extraction of gravitational waveforms Thetests involve single non-rotating stars in stable equilibrium non-rotating stars undergoing radial and quadrupo-lar oscillations non-rotating stars on the unstable branch of the equilibrium configurations migrating to thestable branch non-rotating stars undergoing gravitational collapse to a black hole and rapidly rotating starsin stable equilibrium and undergoing quasi-radial oscillations We have carried out evolutions in full generalrelativity and compared the results to those obtained either with perturbation techniques or with lower dimen-sional numerical codes or in the Cowling approximation (inwhich all the perturbations of the spacetime areneglected) In all cases an excellent agreement has been found The numerical evolutions have been carried outusing different types of polytropic equations of state using either the rest-mass density only or the rest-massdensity and the internal energy as independent variables New variants of the spacetime evolution and new highresolution shock capturing (HRSC) treatments based on Riemann solvers and slope limiters have been imple-mented and the results compared with those obtained from previous methods In particular we have found theldquomonotonized central differencingrdquo (MC) limiter to be particularly effective in evolving the relativistic stellarmodels considered Finally we have obtained the first eigenfrequencies of rotating stars in full general relativityand rapid rotation A long standing problem such frequencies have not been obtained by other methods Over-all and to the best of our knowledge the results presented in this paper represent the most accurate long-termthree-dimensional evolutions of relativistic stars available to date

I INTRODUCTION

Computational general relativistic astrophysics is an in-creasingly important field of research Its development is be-ing driven by a number of factors Firstly the large amount ofobservational data by high-energy X-ray andγ-ray satellitessuch as Chandra XMM and others [1] Secondly the newgeneration of gravitational wave detectors coming online inthe next few years [2] and thirdly the rapid increase in com-puting power through massively parallel supercomputers andthe associated advance in software technologies which makelarge-scale multi-dimensional numerical simulations possi-ble Three-dimensional (3D) simulations of general relativis-tic astrophysical events such as stellar gravitational collapseor collisions of compact stars and black holes are needed to

fully understand the incoming wealth of observations fromhigh-energy astronomy and gravitational wave astronomy Itis thus not surprising that in recent years hydrodynamical sim-ulations of compact objects in numerical relativity has becomethe focus of several research groups [3ndash10]

In a previous paper [6] (hereafter paper I) we presenteda 3D general-relativistic hydrodynamics code (GRAstro)constructed for the NASA Neutron Star Grand ChallengeProject [11] The GRAstro code has been developed byWashington University and the Albert Einstein Institute andhas the capability of solving the coupled set of the Einsteinequations and the general relativistic hydrodynamic (GRHy-dro) equations [12] It has been built using the Cactus Com-putational Toolkit [13] constructed by the Albert EinsteinIn-stitute Washington University and other institutes Paper I

presented our formulation for the GRHydro equations coupledeither to the standard Arnowitt-Deser-Misner (ADM) [14] for-mulation of the Einstein equations or to a hyperbolic formu-lation of the equations [15] It demonstrated the consistencyand convergence of the code for a comprehensive sample oftestbeds having analytic solutions It gave a detailed analy-sis of twelve different combinations of spacetime and hydro-dynamics evolution methods including Roersquos and other ap-proximate Riemann solvers as well as their relative perfor-mance and comparisons when applied to the various testbedsThe code as described and validated in paper I has been ap-plied to various physical problems such as those discussedinRefs [71617] and is now freely available [12]

The main purpose of this paper is to examine and validateour code in long-term accurate simulations of the dynamicsof isolated stars in strong gravitational fields Single relativis-tic stars are indeed expected as the end-point of a number ofastrophysical scenarios (such as gravitational collapse and bi-nary neutron star merging) and should provide important in-formation about strong field physics both through electromag-netic and gravitational wave emissions A number of new nu-merical techniques have been incorporated in the present codeleading to a much improved ability to simulate relativisticstars These techniques concern both the evolution of the fieldequations for which we have implemented new conformal-traceless formulations of the Einstein equations and in theevolution of the hydrodynamical variables for which the useof the ldquomonotonized central differencingrdquo (MC) limiter hasprovided us with the small error growth-rates necessary forsimulations over several dynamical timescales

More precisely in this paper we focus on the accuracy ofthe code during long-term evolution of spherical and rapidlyrotating stellar models We also investigate the nonlineardy-namics of stellar models that are unstable to the fundamen-tal radial mode of pulsation Upon perturbation the unstablemodels will either collapse to a black hole or migrate to a con-figuration in the stable branch of equilibrium configurations(a behavior studied in the case of unstable boson stars [18])In the case of collapse we follow the evolution all the waydown to the formation of a black hole tracking the genera-tion of its apparent horizon In the case of migration to thestable branch on the other hand we are able to accuratelyfollow the nonlinear oscillations that accompany this processand that can give rise to strong shocks The ability to simu-late large amplitude oscillations is important as we expectaneutron star formed in a supernova core-collapse [1920] orin the accretion-induced collapse of a white dwarf to oscillateviolently in its early stages of life

Particularly important for their astrophysical implicationswe study the linear pulsations of spherical and rapidly rotatingstars The computed frequencies of radial quasi-radial andquadrupolar oscillations are compared with the correspondingfrequencies obtained with lower-dimensional numerical codesor with alternative techniques such as the Cowling approx-imation (in which the spacetime is held fixed and only theGRHydro equations are evolved) or relativistic perturbativemethods The comparison shows an excellent agreement con-firming the ability of the code to extract physically relevant

information from tiny perturbations The successful determi-nation of the eigenfrequencies for rapidly rotating stars com-puted with our code is noteworthy Such frequencies have notbeen obtained before with the system being too complicatedfor perturbative techniques

The simulations discussed here make use of two differentpolytropic equations of state (EOS) In addition to the stan-dard ldquoadiabaticrdquo EOS in which the pressure is expressed asa power law of the rest-mass density we have carried outsimulations implementing the ldquoideal fluidrdquo EOS in which thepressure is proportional to both the rest-mass density and thespecific internal energy density This latter choice increasesthe computational costs (there is one additional equation tobe solved) but allows for the modeling of non-adiabatic pro-cesses such as strong shocks and the conversion of bulk ki-netic energy into internal energy which are expected to ac-company relativistic astrophysical events

There are a number of reasons why we advocate the carefulvalidation of general relativistic astrophysics codes Firstlythe space of solutions of the coupled system of the Ein-stein and GRHydro equations is to a large extent unknownSecondly the numerical codes must solve a complicated setof coupled partial differential equations involving thousandsof terms and there are plenty of chances for coding errorsThirdly the complex computational infrastructure neededforthe use of the code in a massively parallel environment in-creases the risk of computational errors a risk that can only beminimized through meticulous tests such as those presentedhere as well as in paper I This paper however wants to bemore than a list of testbeds the results presented show thatour current numerical methods are mature enough for obtain-ing answers to new and outstanding problems in the physicsof relativistic stars

The organization of this paper is as follows the formulationof the differential equations for the spacetime and the hydro-dynamics is briefly reviewed in section II Section III givesashort description of the numerical methods with emphases onthe new schemes introduced in this paper (in addition to thosein paper I) Sections IVndash VI represent the core of the paperand there we present the main results of our simulations Insection IV in particular we focus our attention on the simula-tion of nonrotating relativistic stars In section V we considerthe evolution of rotating stars Section VI is dedicated to theextraction of gravitational waveforms generated by the non-radial pulsations of perturbed relativistic stars In section VIIwe summarize our results and conclusions We use a spacelikesignature(minus+++) and units in whichc = G = M⊙ = 1(geometric units based on solar mass) unless explicitly spec-ified Greek indices are taken to run from 0 to 3 and Latinindices from 1 to 3

II BASIC EQUATIONS

We give a brief overview of the system of equations in thissection We refer the reader to paper I for more details

A Field Equations

In general relativity the dynamics of the spacetime is de-scribed by the Einstein field equationsGmicroν = 8πTmicroν withGmicroν being the Einstein tensor andTmicroν the stress-energy ten-sor Many different formulations of the equations have beenproposed throughout the years starting with the ADM for-mulation in 1962 [14] In our code we have implementedthree different formulations of the field equations includ-ing the ADM formulation a hyperbolic formulation [15] anda more recent conformal-traceless formulation based on theADM construction [2122] (see also Ref [23])

In the ADM formulation [14] the spacetime is foliated witha set of non-intersecting spacelike hypersurfaces Two kine-matic variables relate the surfaces the lapse functionα whichdescribes the rate of advance of time along a timelike unit vec-tor nmicro normal to a surface and the shift three-vectorβi thatrelates the spatial coordinates of two surfaces In this con-struction the line element reads

ds2 = minus(α2 minus βiβi)dt2 + 2βidx

idt+ γijdxidxj (1)

The original ADM formulation casts the Einstein equationsinto a first-order (in time) quasi-linear [24] system of equa-tions The dependent variables are the 3-metricγij and theextrinsic curvatureKij The evolution equations read

parttγij = minus2αKij + nablaiβj + nablajβi (2)

parttKij = minusnablainablajα+ α

Rij +K Kij minus 2KimKmj

minus8π

Sij minus1

2γijS

minus 4πρADM

+βmnablamKij +Kimnablajβm +Kmjnablaiβ

wherenablai denotes the covariant derivative with respect to the3-metricγij Rij is the Ricci curvature of the 3-metric andK equiv γijKij is the trace of the extrinsic curvature In additionto the evolution equations there are four constraint equationsthe Hamiltonian constraint

(3)R+K2 minusKijKij minus 16πρ

ADM= 0 (4)

and the momentum constraints

nablajKij minus γijnablajK minus 8πji = 0 (5)

In equations (2)ndash(5)ρADM

ji Sij S equiv γijSij are the com-ponents of the stress-energy tensor projected onto the 3D hy-persurface (for a more detailed discussion see Ref [25])

As mentioned above in addition to the two formulations de-scribed in paper I we have recently implemented a conformal-traceless reformulation of the ADM system as proposedby [2122] Details of our particular implementation of thisformulation are extensively described in Ref [23] and willnot be repeated here We only mention here that this for-mulation makes use of a conformal decomposition of the 3-metric γij = eminus4φγij and the trace-free part of the extrin-sic curvatureAij = Kij minus γijK3 with the conformal fac-tor φ chosen to satisfye4φ = γ13 equiv det(γij)

13 In this

formulation as shown in Ref [22] in addition to the evo-lution equations for the conformal threendashmetricγij and theconformal-traceless extrinsic curvature variablesAij thereare evolution equations for the conformal factorφ the traceof the extrinsic curvatureK and the ldquoconformal connectionfunctionsrdquo Γi (following the notation of Ref [22]) We notethat the final mixed first and second-order evolution systemfor φK γij Aij Γ

i is not in any immediate sense hyper-bolic [26] In the original formulation of Ref [21] the auxil-iary variablesFi = minussumj γijj were used instead of theΓi

In Refs [2327] the improved properties of this conformal-traceless formulation of the Einstein equations were comparedto the ADM system In particular in Ref [23] a number ofstrongly gravitating systems were analyzed numerically withconvergentHRSC methods withtotal-variation-diminishing(TVD) schemes using the equations described in paper IThese included weak and strong gravitational waves blackholes boson stars and relativistic stars The results showthat our treatment leads to a stable numerical evolution of themany strongly gravitating systems However we have alsofound that the conformal-traceless formulation requires gridresolutions higher than the ones needed in the ADM formula-tion to achieve the same accuracy when the foliation is madeusing the ldquoK-driverrdquo approach discussed in Ref [28] Becausein long-term evolutions a small error growth-rate is the mostdesirable property we have adopted the conformal-tracelessformulation as our standard form for the evolution of the fieldequations

B Hydrodynamic Equations

The GRHydro equations are obtained from the local conser-vation laws of the density current (continuity equation) and ofthe stress-energy tensor which we assume to be that of a per-fect fluid T microν = ρhumicrouν + Pgmicroν with umicro being the fluid4-velocity andP andh the (isotropic) pressure and the spe-cific enthalpy respectively In our code the GRHydro equa-tions are written as a first-order flux-conservative hyperbolicsystem [296]

partt~U + parti

~F i = ~S (6)

where the evolved state vector~U is given in terms of theprimitive variables the rest-mass densityρ the 3-velocityvi = uiW + βiα and the specific internal energyε as

radicγWρ

radicγρhW 2vj

radicγ(ρhW 2 minus P minusWρ)

Hereγ is the determinant of the 3-metricγij andW is the

Lorentz factorW = αu0 = (1 minus γijvivj)

minus12 Further-

more the 3-flux vectors~F i are given by

~F i =

α(vi minus 1αβ

α[(vi minus 1αβ

i)Sj +radicγPδi

α[(vi minus 1αβ

i)τ +radicγviP ]

Finally the source vector~S is given by

αradicγT microνgνσΓσ

microj

αradicγ(T micro0partmicroαminus αT microνΓ0

microν)

whereΓαmicroν are the Christoffel symbols

C Gauge Conditions

The code is designed to handle arbitrary shift and lapse con-ditions which can be chosen as appropriate for a given space-time simulation More information about the possible familiesof spacetime slicings which have been tested and used withthe present code can be found in Refs [623] Here we limitourselves to recall details about the specific foliations used inthe present evolutions In particular we have used algebraicslicing conditions of the form

(partt minus βiparti)α = minusf(α) α2K (10)

with f(α) gt 0 but otherwise arbitrary This choice containsmany well known slicing conditions For example settingf = 1 we recover the ldquoharmonicrdquo slicing condition or bysettingf = qα with q being an integer we recover the gen-eralized ldquo1+logrdquo slicing condition [30] which after integrationbecomes

α = g(xi) +q

2log γ (11)

whereg(xi) is an arbitrary function of space only In partic-ular all of the simulations discussed in this paper are doneusing condition (11) withq = 2 basically due to its compu-tational efficiency (we caution that ldquogauge pathologiesrdquo coulddevelop with the ldquo1+logrdquo slicings see Ref [3132])

The evolutions presented in this paper were carried out withthe shift vector being either zero or constant in time

III NUMERICAL METHODS

We now briefly describe the numerical schemes used in ourcode We will distinguish the schemes implemented in theevolution of the Einstein equations from those implementedin the evolution of the hydrodynamic equations In both casesthe equations are finite-differenced on spacelike hypersurfacescovered with 3D numerical grids using Cartesian coordinates

A Spacetime Evolution

As described in paper I our code supports the use of sev-eral different numerical schemes [623] Currently a Leapfrog(non-staggered in time) and an iterative Crank-Nicholsonscheme have been coupled to the hydrodynamic solver

The Leapfrog method assumes that all variables exist onboth the current time steptn and the previous time steptnminus1Variables are updated fromtnminus1 to tn+1 (future time) evalu-ating all terms in the evolution equations on the current timesteptn The iterative Crank-Nicholson solver on the otherhand first evolves the data from the current time steptn tothe future time steptn+1 using a forward in time centered inspace first-order method The solution at stepstn and tn+1

are then averaged to obtain the solution on the half time steptn+12 This solution at the half time steptn+12 is then usedin a Leapfrog step to re-update the solution at the final timesteptn+1 This process is then iterated The error is defined asthe difference between the current and previous solutions onthe half time steptn+12 This error is summed over all grid-points and all evolved variables Because the smallest numberof iterations for which the iterative Crank-Nicholson evolutionscheme is stable is three and further iterations do not improvethe order of convergence [3323] we do not iterate more thanthree times Unless otherwise noted all simulations reportedin this paper use the iterative Crank-Nicholson scheme for thetime evolution of the spacetime

B Hydrodynamical Evolution

The numerical integration of the GRHydro equationsis based on High-Resolution Shock-Capturing (HRSC)schemes specifically designed to solve nonlinear hyperbolicsystems of conservation laws These conservative schemesrely on the characteristic structure of the equations in order tobuild approximate Riemann solvers In paper I we presenteda spectral decomposition of the GRHydro equations suitablefor a general spacetime metric (see also Ref [34])

Approximate Riemann solvers compute at every cell-interface of the numerical grid the solution of local Riemannproblems (ie the simplest initial value problem with dis-continuous initial data) Hence HRSC schemes automaticallyguarantee that physical discontinuities developing in thesolu-tion (eg shock waves which appear in core-collapse super-novae or in coalescing neutron star binaries) are treated con-sistently HRSC schemes surpass traditional approaches [38]which rely on the use of artificial viscosity to resolve such dis-continuities especially for large Lorentz factor flows HRSCschemes have a high order of accuracy typically second-orderor more except at shocks and extremal points We refer thereader to [3536] for recent reviews on the use HRSC schemesin relativistic hydrodynamics

One of the major advantages of using HRSC schemes isthat we can take advantage of the many different algorithmsthat have been developed and tested in Newtonian hydrody-namics In this spirit our code allows for three alternative

ways of performing the numerical integration of the hydrody-namic equations(i) using a flux-split method [37](ii) usingRoersquos approximate Riemann solver [38] and(iii) using Mar-quinarsquos flux-formula [39] The different methods differ simplyin the way the numerical fluxes at the cell-interfaces are calcu-lated in the corresponding flux-formula The code uses slope-limiter methods to construct second-order TVD schemes [40]by means of monotonic piecewise linear reconstructions of thecell-centered quantities to the left (L) and right (R) sidesof ev-ery cell-interface for the computation of the numerical fluxesMore precisely~UR

i and ~ULi+1 are computed to second-order

accuracy as follows

~URi = ~Ui + σi(xi+ 1

minus xi) (12)

~ULi+1 = ~Ui+1 + σi+1(xi+ 1

minus xi+1) (13)

wherex denotes a generic spatial coordinate We have fo-cused our attention on two different types of slope limitersthe standard ldquominmodrdquo limiter and the ldquomonotonized central-differencerdquo (MC) limiter [41] In the first case the slopeσi iscomputed according to

σi = minmod

~Ui minus ~Uiminus1

∆x~Ui+1 minus ~Ui

where∆x denotes the cell spacing The minmod function oftwo arguments is defined by

minmod(a b) equiv

a if |a| lt |b| andab gt 0

b if |b| lt |a| andab gt 0

0 if ab le 0

On the other hand the MC slope limiter (which was not in-cluded in the previous version of the code discussed in paperI) does not reduce the slope as severely as minmod near a dis-continuity and therefore a sharper resolution can be obtainedIn this case the slope is computed as

σi = MC

~Ui minus ~Uiminus1

∆x~Ui+1 minus ~Ui

where the MC function of two arguments is defined by

MC(a b) equiv

2a if |a| lt |b| and2|a| lt |c| andab gt 0

2b if |b| lt |a| and2|b| lt |c| andab gt 0

c if |c| lt 2|a| and|c| lt 2|b| andab gt 0

0 if ab le 0

and wherec equiv (a + b)2 Both schemes provide the desiredsecond-order accuracy for smooth solutions while still satis-fying the TVD property In sect IV A we will report on acomparison between the two algorithms and justify the use ofthe MC slope limiter as our preferred one

C Equations of State

As mentioned in the Introduction to explore the behaviorof our code in long-term evolutions of equilibrium configura-tions we used two different polytropic equations of state andat various central rest-mass densities In particular we haveimplemented both anadiabatic(or zero temperature) EOS

P = KρΓ = Kρ1+1N (16)

and as a so-calledldquoideal fluidrdquo EOS

P = (Γ minus 1)ρε (17)

whereK is the polytropic constantΓ the polytropic indexandN equiv (Γ minus 1)minus1 the polytropic exponent The ideal fluidEOS (17) depends on both the rest-mass densityρ and on thespecific internal energyε it corresponds to allowing the poly-tropic coefficientK in adiabatic EOS (16) to be a function ofentropy The use of an adiabatic EOS with a constantK iscomputationally less expensive and is physically reasonablewhen modeling configurations that are in near equilibriumsuch as stable stellar models in quasi-equilibrium evolutionsThere are however dynamical processes such as those involv-ing nonlinear oscillations and shocks in which the variationsin the energy entropy cannot be neglected The simulationsdiscussed in section IV C where both equations of state (16)-(17) are used for the same configuration gives direct evidenceof how a more realistic treatment of the internal energy of thesystem can produce qualitatively different results

The increased accuracy in the physical description of thedynamical system comes with a non-negligible additionalcomputational cost It involves the solution of an additionalequation (ie the evolution equation for the specific internalenergyε) increasing the total number of GRHydro equationsfrom four to five and making accurate long-term evolutionsconsiderably harder

D Boundary Conditions

In our general-purpose code a number of different bound-ary conditions can be imposed for either the spacetime vari-ables or for the hydrodynamical variables We refer the readerto [623] for details In all of the runs presented in this paperwe have used static boundary conditions for the hydrodynam-ical variables and radiative outgoing boundary conditionsforthe spacetime variables The only exception to this is the evo-lution of rotating stars (see sect V) for which the spacetimevariables have also been held fixed at the outer boundary

IV SPHERICAL RELATIVISTIC STARS

We turn next to the description of the numerical evolutionsof relativistic star configurations We start by consideringspherical models

A Long-term evolution of stable configurations

Using isotropic coordinates(t r θ φ) the metric describ-ing a static spherically symmetric relativistic star reads

ds2 = minuse2νdt2 + e2λ(dr2 + r2dθ2 + r2 sin2 θdφ2) (18)

whereν andλ are functions of the radial coordinater onlyThe form of the metric componentgrr is much simpler inthese coordinates than in Schwarzschild coordinates whichare often used to describe a Tolman-Oppenheimer-Volkoff(TOV) equilibrium stellar solution In additiongrr is not con-strained to be equal to unity at the center of the stellar config-uration as in Schwarzschild coordinates We have found thatthese two properties of the isotropic coordinates are very ben-eficial to achieve long-term numerical evolutions of relativis-tic stars Therefore all simulations of spherical relativisticstars shown in this paper have been performed adopting theline element (18) expressed in Cartesian coordinates

0 1 2 3 4 5 6t (ms)

ρ cρ

minmodMC

adiabatic

353 Roersquos Solver

FIG 1 Evolution of the central rest-mass densityρc (in units ofthe initial central rest-mass densityρc0) for a nonrotating star withgravitational massM = 165 M⊙ Using Roersquos approximate Rie-mann solver the figure shows a comparison in the use of the minmodand of the MC slope limiters for both the ideal fluid and the adiabaticEOS

Although the initial configurations refer to stellar modelsin stable equilibrium the truncation errors at the center andat the surface of the star excite small radial pulsations thatare damped in time by the numerical viscosity of the codeMoreover these pulsations are accompanied by a secular evo-lution of the values of the central rest-mass density awayfrom its initial value Similar features have been reportedinRefs [4243] These features converge away at the correct ratewith increasing grid resolution and do not influence the long-term evolutions Moreover the secular evolution of the centralrest-mass density varies according to the EOS adopted when

using the ideal fluid EOS we have observed that the seculardrift of the central rest-mass density is towards lower densi-ties However if we enforced the adiabatic condition (whichis justified for the case of a near-equilibrium evolution) wehave observed that the dominant truncation error has oppositesign and the central rest-mass density evolves towards largervalues

0 1 2 3 4 5t (ms)

ρ cρ

minmodMC

Marquina

FIG 2 Evolution of the normalized central rest-mass density ρc

for a nonrotatingM = 165 M⊙ star Different lines show a com-parison between Roersquos Riemann solver and Marquinarsquos flux-formulafor different slope limiters

This is shown in Fig 1 where we plot the evolution of aTOV star with gravitational massM = 165 M⊙ constructedwith a N = 1 polytrope In our units the polytropic con-stant isK = 1235 and the central rest-mass density of thestar isρc = 100 times 10minus3 For these tests a very coarse gridof 353 gridpoints in octant symmetry is sufficient and allowsthe major effects to be revealed with minimal computationalcosts The outer boundary is placed at about17 rs (wherersis the isotropic coordinate radius of the star) We use radiativeboundary conditions with a1r fall-off Irrespective of theslope limiter used the magnitude of the secular drift observedin the central rest-mass density evolution is roughly a factorof two smaller for the adiabatic EOS than for the ideal fluidEOS As a result in all of the evolutions of stable configu-rations which remain close to equilibrium (such as pulsatingstars with no shock developing) the adiabatic EOS is pre-ferred

Fig 1 also gives a comparison of the use of the minmodand the MC slope limiters in the evolution of the normalizedcentral rest-mass density For both the ideal fluid and the adia-batic EOS the MC limiter shows a significantly smaller secu-lar increase in the central rest-mass density as compared to theminmod one The simulations in Fig 1 employed Roersquos ap-proximate Riemann solver in the fluid evolution scheme andthis is then compared to Marquinarsquos flux-formula in Fig 2

for the evolution of the central rest-mass density The secularincrease is significantly smaller when using Marquinarsquos flux-formula than when using Roersquos solver and this is especiallynoticeable for the minmod slope limiter A comparison of theincrease of the maximum error in the Hamiltonian constraintafter several ms of evolution (not shown here) indicates that itis about80 smaller with Marquina than with Roe when us-ing the adiabatic EOS As a result of the above comparisonswe have adopted Marquinarsquos scheme with the MC slope lim-iter as our preferred scheme for evolution of the GRHydroequations Unless otherwise noted all of the simulations pre-sented in this paper have been obtained with such a scheme

0 1 2 3 4 5 6 7t (ms)

ρ cρ

FIG 3 Time evolution of the normalized central rest-mass den-sity at three different grid resolutions (323 643 and963 gridpointsrespectively) for aM = 14 M⊙ N = 1 relativistic sphericalpolytrope The evolution of the central rest-mass density is mainlymodulated by the fundamental radial mode of oscillation of the starThe initial amplitude of the oscillation converges to zero at sec-ond-order while the secular increase in the central rest-mass densityconverges away at almost second-order

Next we show in Fig 3 the long-term evolution of thecentral rest-mass density for three different grid resolutionsFor this we consider a nonrotatingN = 1 polytropic starwith gravitational massM = 14 M⊙ circumferential radiusR = 1415 km central rest-mass densityρc = 128 times 10minus3

andK = 100 The different simulations used323 643

and963 gridpoints with octant symmetry and with the outerboundary placed at17 rs These grid resolutions correspondto about 19 38 and 56 gridpoints per star radius respectivelyFig 3 shows the oscillations in the central rest-mass densityand the secular evolution away from the initial value men-tioned above The oscillations are produced by the first-ordertruncation error at the center and the surface of the star (ourhydrodynamical evolution schemes are globally second orderbut only first-order at local extrema see related discussions inRef [23] where long-term convergence tests are presented)

but both the amplitude of the initial oscillation and the rateof the secular change converge to zero at nearly second-orderwith increasing grid resolution

Note that the evolutions shown in Figs 3-5 extend to 7 mscorresponding to about 10 dynamical times (taking the fun-damental radial mode period of pulsation as a measure of thedynamical timescale) significantly longer than say the onesreported by other authors [844] Our evolutions are limitedby the time available (a simulation with963 gridpoints and upto 7 ms takes about 40 hours on a 128 processor Cray-T3E su-percomputer) We have found that for a resolution of963 thecentral density at the end of the 7ms evolution is just 025larger than the initial central density

For the same configuration we show in Fig 4 the timeevolution of the L2-norm of the violation of the Hamiltonianconstraint at the three different grid resolutions Also inthiscase the violation of the Hamiltonian constraint converges tozero at nearly second-order with increasing grid resolution

0 1 2 3 4 5 6 7t (ms)

FIG 4 Convergence of the L2-norm of the Hamiltonian con-straint at three different grid resolutions (323 643 and 963 grid-points respectively) for aM = 14 M⊙N = 1 polytropic spheri-cal relativistic star The rate of convergence is close to second-orderwith increasing grid resolution

In Fig 5 we show other aspects of the accuracy of the sim-ulation with963 gridpoints by comparing the initial profilesof the rest-mass densityρ and of the lapse functionα of theTOV star with those obtained after 7 ms of evolution Thesmall deviations from the original profiles are worth empha-sizing The small inset shows a magnification of the rapidchange in the gradient of the rest-mass density profile at thesurface of the star

0 2 4 6 8 10 12 14 16 18x (km)

8 10 12 14minus01

t=0t=7 ms

FIG 5 Variation of the original profiles along thex-axis of therest-mass density (left vertical axis) and lapse function (right verticalaxis) for aM = 14 M⊙ N = 1 polytropic spherical relativisticstar after 7 ms of evolution A963 grid in octant symmetry was usedin the simulation The small inset shows a magnification of the rapidchange in the gradient of the rest-mass density profile at thesurfaceof the star

B Radial pulsations

As mentioned in the previous section the truncation errorsof the hydrodynamical schemes used in our code trigger radialpulsations of the initially static relativistic star (see Ref [45]for a review) These pulsations are initiated at the surfaceofthe star where the gradients of the rest-mass density are thelargest (cf Fig 5) Because gravitational waves cannot beemitted through the excitation of radial pulsations of nonrotat-ing relativistic stars these pulsations are damped only bythenumerical viscosity of the code in numerical simulations ofinviscid stars In treatments more dissipative than the HRSCschemes used in our code such as those using artificial vis-cosity or particle methods (eg Smoothed Particle Hydrody-namics) these oscillations will be damped significantly faster

In order to test the properties of the long-term hydrodynam-ical evolution separately from those of the spacetime evolu-tion we have first examined the long-term hydrodynamicalevolution separately from those of the spacetime evolution wehave first examined the small-amplitude radial pulsations in afixed spacetimeof an initially static relativistic star As initialdata we use theM = 14 M⊙ polytropic star of the previoussection We show in Fig 6 the evolution up to 7 ms of thenormalized starrsquos central rest-mass density with a numericalgrid of 963 gridpoints The amplitude of the excited pulsa-tions in this purely hydrodynamical evolution is minute (lessthan 1 part in 200) and is significantly smaller than the corre-sponding amplitude in a coupled hydrodynamical and space-time evolution (compare the vertical axes of Figs 3 and 6)

0 1 2 3 4 5 6 7t (ms)

FIG 6 Time evolution of the central rest-mass density of aM = 14 M⊙ N = 1 polytropic spherical relativistic star Inthis the simulation the spacetime is heldfixedand the hydrodynamicvariables have been evolved on a numerical grid of963 gridpointsThe evolution is a superposition of radial normal modes of pulsationexcited by truncation errors of the hydrodynamical schemeHigherovertones are damped faster by the small but non-zero numerical vis-cosity

A closer look at Figure 6 reveals that the evolution of thecentral rest-mass density is a superposition of different radialnormal modes of pulsation The higher-frequency modes aredamped faster so that after a certain time the evolution pro-ceeds mainly in the fundamental mode of pulsation Note alsothe small damping rate of the fundamental pulsation mode in-dicating the small effective numerical viscosity of our HRSChydrodynamical scheme The evolution towards larger valuesof the central rest-mass density is similar to that discussed inSection IV A but less pronounced in this case At a resolutionof 963 gridpoints the secular change in the average centralrest-mass density is less than 002 for the total evolutiontime shown

The use of truncation error as an initial perturbation de-serves commenting on The oscillations caused by truncationerror will converge away with increasing resolution hencetheoverall oscillation amplitude can carry no physical informa-tion about the system However the frequencies and normal-ized eigenfuntions of particular normal-modes of oscillationof the star are physical (in the sense that they match the eigen-frequencies and eigenfunctions calculated through perturba-tive analyses) and can be extracted from these simulationsby carrying out a Fourier transform of the time evolution ofthe radial velocity or of the rest-mass density As the small-amplitude pulsations are in the linear regime the eigenfunc-tions can be normalized arbitrarily (eg to 10 at the surface ofthe star) At increasing resolution the solution converges tothe mode-frequencies and to the normalized eigenfunctions

even though the overall oscillation amplitude converges tozero Such evolutions are useful for extracting the propertiesof linear normal-modes of oscillation as long as the resolu-tion is fine enough that the pulsations excited by truncationerrors are in the linear regime and as long as the resolution iscoarse enough that the various local 1st and 2nd order trunca-tion errors of the numerical scheme result in a time evolutionthat is dominated by a sum of normal modes (at very fine res-olutions the Fourier transform of the time evolution would bevery small and thus have a very noisy power spectrum due toroundoff errors in which case the physical normal-mode fre-quencies would be difficult to extract - this has not been thecase for the resolutions used in this paper) We also note thatdifferent variants of our hydrodynamical evolution schemesexcite the various physical normal-modes at different ampli-tudes For example 2nd order schemes employing the min-mod limiter tend to clearly excite a large number of high-frequency overtones whereas the use of the MC limiter resultsin the clear excitation of only a few low-frequency overtonesand a more noisy FFT power spectrum at higher frequencies(for the resolutions used in this paper) This difference inbe-haviour is due to the differences in the local truncation errorsinherent in these numerical schemes

The radial pulsations are a sum of eigen modes of pulsa-tion Since the radial pulsations triggered by truncation errorshave a small amplitude one can compare the frequencies withthat computed by linear perturbation theory [43] or with hy-drodynamical evolutions of similar models in 2D [4243] Inthis way we can validate that the ldquoartificialrdquo perturbationspro-duced by the truncation errors do excite ldquophysicalrdquo modes ofoscillation for a relativistic star However before discussingthe results of this comparison it is important to emphasizethat the identification of the frequency peaks in the Fouriertransform of the time evolution of a given variable with phys-ical frequencies must be done with care A real pulsation fre-quency must be global (the same at every point in the star atleast for discrete normal mode frequencies) and it should ap-pear in the time evolution of different physical quantitiesde-scribing the starrsquos structure and dynamics To eliminate possi-ble ambiguities we have carried out our frequency identifica-tion procedure for different variables and at different positionsin the star

Fig 7 shows the Fourier transform of the time evolution ofthe central rest-mass density of the same initial model as inFig 6 but using theminmodlimiter (which gives a clearer ex-citation of the higher overtones) We indicate withF the fun-damental normal mode frequency and withH1minusH6 the nextsix higher frequency modes (overtones) We have also com-pared the frequency peaks in the Fourier spectrum to both thenormal mode frequencies expected by linear perturbation the-ory in the Cowling approximation (see Ref [46]) and to thefrequencies computed with an independent 2D axisymmetricnonlinear code [43] which uses the same HRSC schemes butin spherical polar coordinates (shown as dashed vertical linesin Fig 7)

As can be seen from Table I the agreement is extremelygood The relative difference between the 3D and 2D resultsat this grid resolution is better than1 up to (H4) and slightly

0 2 4 6 8 10 12 14f (kHz)

H1 H2 H3 H4 H5 H6F

FIG 7 Fourier transform of the central rest-mass density evolu-tion of aM = 14 M⊙N = 1 polytropic spherical relativistic starin a fixed spacetimeevolution HereF represents the fundamentalnormal mode frequency whileH1 minusH6 indicate the first six over-tones The frequency peaks in the power spectrum are in excellentagreement with the radial normal mode frequencies (shown here asdashed vertical lines) computed with an independent 2D codeusingspherical polar coordinates The solid and dotted lines were com-puted with963 and 643 gridpoints respectively The units of thevertical axis are arbitrary

larger for higher frequencies (H5 andH6) which becomeunder-resolved at this grid resolution This excellent agree-ment is a significant test for the correct implementation of thehydrodynamicalevolution schemes in our code and is an indi-cation of the level of accuracy we can achieve resolving andfollowing these small deviations away from the equilibriumconfiguration As one would expect lower or higher resolu-tion runs (eg with643 or1443 gridpoints) which have intrin-sically larger or smaller perturbation amplitudes respectivelyreproduce the peaks in the power spectrum shown in Fig 7(see dotted line in Fig 7 which corresponds to an evolutionwith 643 grid-points

After establishing the accuracy of the long-term evolutionof the GRHydro equations we have examined the eigenfre-quencies of the radial pulsations of spherical stars incoupledhydrodynamical and spacetime evolutions A Fourier trans-form of the evolution of the radial velocity (for the same equi-librium model as the one discussed before) is shown in Fig 8Again in this case we have been able to identify several fre-quency peaks in the Fourier spectrum with the normal modefrequencies obtained with linear perturbation techniques[47]A detailed comparison of these frequencies is shown in Ta-ble II The agreement is again excellent Note the rather largedifferences between the frequencies shown in Tables I and IIThe Cowling approximation is rather inaccurate for the lowestradial mode-frequencies [48] but is increasingly more accu-

0 1 2 3 4 5 6 7 8 9f (kHz)

F H1 H2 H3

FIG 8 Fourier transform of the evolution of the radial velocityfor aM = 14 M⊙ N = 1 polytropic spherical relativistic star ina coupledspacetime and hydrodynamical evolution The frequencypeaks in the spectrum are in excellent agreement with the radial nor-mal mode frequencies computed by perturbation theory (shown hereas dashed vertical lines) As in Fig 7 hereF represents the funda-mental normal mode frequency whileH1 minus H3 are the next threehigher frequency modes The units of the vertical axis are arbitrary

rate for nonradial pulsations or for higher frequencies [48]All of the results discussed so far refer to simulations in-

volving stable relativistic configurations In the followingsection we consider numerical evolutions of relativistic starswhich are initially in an unstable equilibrium

C Migration of unstable configurations to the stable branch

The numerical evolution of a nonrotating relativistic starin an equilibrium unstable to the fundamental radial mode ofpulsation is mainly determined by the numerical truncationerrors that cause it to evolve away from its initial configura-tion Depending on the type of perturbation the star can eithercollapse to a black hole or expand and migrate to the stablebranch of the sequence of equilibrium models reaching a newstable equilibrium configuration with approximately the samerest-mass of the perturbed star We have therefore constructeda model of aN = 1K = 100 polytropic star with rest-massM0 = 1535 M⊙ (M = 1447 M⊙) and a central rest-massdensityρc = 80times 10minus3 which is larger than the central rest-mass density of the maximum-mass stable model The star istherefore initially in an unstable equilibrium (see the inset ofFig 9) and under the perturbation introduced by the truncationerror it expands evolving rapidly to smaller central rest-massdensities until it reaches the stable branch of equilibrium con-figurations An analogous behavior has been observed in nu-merical simulations of relativistic boson stars [18] (see also

TABLE I Comparison of small-amplitude radial pulsation fre-quencies obtained with the present 3D nonlinear evolution code withfrequencies obtained with an independent 2D code Both codesevolve the GRHydro equations in afixed spacetimeand for an equi-librium model of aN = 1 relativistic polytrope withMR = 015

Mode Present 3D code 2D code Relative Difference(kHz) (kHz) ()

F 2696 2701 02H1 4534 4563 06H2 6346 6352 01H3 8161 8129 04H4 9971 9875 10H5 11806 11657 13H6 13605 13421 17

TABLE II Comparison of small-amplitude radial pulsation fre-quencies obtained with the present 3D nonlinear evolution code withlinear perturbation mode frequencies in fullycoupledevolutionsThe equilibrium model is a nonrotatingN = 1 relativistic polytropewithMR = 015

Mode Present 3D code Perturbation code Relative Difference(kHz) (kHz) ()

F 1450 1442 06H1 3958 3955 00H2 5935 5916 03H3 7812 7776 04

Ref [49] for recent numerical simulations of expanding un-stable boson stars)

In a realistic astrophysical scenario a stable neutron starcan accrete matter eg from a companion star in a binary sys-tem or from infalling matter after its formation in a supernovacore-collapse The star would then secularly move towardslarger central densities along the stable branch of equilibriumconfigurations exceed the maximum-mass limit and collapseto a black hole No secular mechanism could evolve the starto the unstable branch In this respect the migration mech-anism discussed here cannot occur in practice Neverthelessit provides a consistent solution of the initial value problemand represents an important test of the accuracy of the codein a highly dynamical and non-adiabatic evolution We usesuch an initial data set to study large amplitude oscillationsof relativistic stars which cannot be treated accurately by lin-ear perturbation theory Large amplitude oscillations about aconfiguration on the stable branch could occur after a super-nova core-collapse [20] or after an accretion-induced collapseof a white dwarf While the actual set of quasi-normal modesexcited will depend on the excitation process the ability tosimulate large amplitude oscillations is important

Fig 9 shows the evolution of the central rest-mass densityρc normalized to its initial value and up to a final time of 426ms On a very short dynamical timescale of 05 ms the starhas expanded and has its central density dropped to about 3 of its initial central rest-mass density Note that this isless

0 1 2 3 4t (ms)

12ρ c

0 0004 0008ρc

FIG 9 Evolution of the (normalized) central rest-mass densityρc

during the migration of an unstable relativistic star to a stable modelwith the same rest-mass When an adiabatic EOS is used (dottedline) the difference in gravitational binding energy between the un-stable and stable models is periodically converted in bulk kinetic en-ergy through highly nonlinear nearly constant amplitude pulsationsIn contrast when an ideal fluid EOS is used (solid line) the grav-itational binding energy is gradually converted into internal energyvia shock heating As a result the oscillations are damped and theheated stable equilibrium model approaches a central density slightlysmaller than the rest-mass density of a zero temperature star of thesame rest-mass (indicated by an asterisk on the left vertical axis)

than the central rest-mass densityρc = 135 times 10minus3 of thestable model of same rest-mass which is indicated with an as-terisk on the vertical axis of Fig 9 During the rapid decreaseof the central rest-mass density the star acquires a large radialmomentum The star then enters a phase of large amplituderadial oscillations about the stable equilibrium model with thesame rest-mass Because the unstable and stable models haverather different degrees of compactness the migration to thestable branch will be accompanied by the release of a signif-icant amount of gravitational binding energy which could ei-ther be converted to bulk kinetic energy or to internal energydepending on the choice of EOS

In order to investigate both responses we have performedtwo different evolutions of the same initial model In the firstcase (the ldquoadiabatic EOSrdquo in Fig 9) we have enforced the adi-abatic condition during the evolution ie we have assumedthat the star remains at zero temperature following an adia-batic EOS As shown in Fig 9 with a dotted line in this casethe star behaves like a compressed spring which is allowedto expand oscillating with a nearly constant amplitude Thisindicates that the star periodically converts all of the excessgravitational binding energy into the kinetic energy and viceversa As the oscillations are highly nonlinear the restoringforce is weaker at higher densities than at lower densities and

0 5 10 15 20 25 30x (km)

minus04

minus02

084 ms098 ms113 ms

supersonic infallhomologous infall

bounce

shock wave

FIG 10 Shock formation in the outer core mantle during thefirstbounce at equilibrium densities of an unstable star evolved with anideal fluid EOS The top and bottom panels show the internal energyǫ and radial velocityvx respectively at three different times thehomologous infall phase the inner core bounce and the outwardsshock propagation The oscillations of the inner core are damped byshock heating

the oscillations are therefore far from being sinusoidalIn the second case (the ldquoideal fluid EOSrdquo in Fig 9) we do

not enforce the abiabatic condition but allow all of thermody-namic variables to evolve in time As a result the oscillationsare gradually damped in time while the star oscillates arounda central density close to that of a stable star with the samerest-mass

The rapid decrease in the oscillation amplitude is due to thedissipation of kinetic energy via shock heating At the end ofthe first expansion (ie at the first minimum in Fig 9) thestar has expanded almost to the edge of the numerical grid Atthis point the outer parts of the initial star have formed a low-density outer-core mantle around the high-density inner coreand the star then starts to contract Fig 10 shows with solidlines the supersonic infall of the outer core mantle att = 084ms while the inner core is contracting homologously Af-ter this ldquopoint of last good homologyrdquo the high-density innercore reaches its maximum infall velocity and then starts slow-ing down The infalling low-density mantle forms a shock atthe inner coremantle boundary (dotted lines att = 098 ms inFig 10) After the inner core bounces it expands and pressurewaves at the inner core-mantle boundary feed the shock wavewith kinetic energy (dashed lines att = 113 ms in Fig 10)In this way the shock wave is dissipating the initial bindingenergy of the star so that the amplitude of the central densityoscillations decreases with time The above process is verysimilar to the core bounce in neutron star formation (see forinstance the description in [50]) except for the fact thatherethe outer mantle is created during the first rapid expansion

from material of the initial unstable starAs a result of the damping of the radial oscillations the star

settles down on a secular timescale to a stable equilibriumconfiguration with central density somewhat smaller than thecentral density of a stable star with same rest-mass as the ini-tial unstable star This is because part of the matter of theinitial star forms a heated mantle around the inner core

The evolution shown in Fig 9 was obtained using a resolu-tion of 963 gridpoints Since the initial unstable configurationis much more compact than the final configuration the bound-aries of the computational grid were placed at about45 rs Asa result the grid resolution of the initial configuration isratherlow causing an additional non-negligible deviation of the av-erage central rest-mass density of the pulsating star away fromthe expected central rest-mass density of the zero-temperaturestar of the same rest-mass

The evolution of the highly nonlinear and nonadiabatic pul-sations of a star when it settles down on the stable branch un-derlines the importance of evolving all of the thermodynamicvariables (including the specific internal energy) and the im-portance of using HRSC methods in order to resolve the for-mation and evolution of shocks correctly These capabilitiesof the numerical code will be important in the correct simu-lation of general relativistic astrophysical events such as themerging of a neutron star binary system or the formation of aneutron star in an accretion-induced collapse of a white dwarf

D Gravitational collapse of unstable configurations

As mentioned in the previous section the numerical schemeused in the hydrodynamical evolution is such that it causes anonrotating relativistic star in an unstable equilibrium to ex-pand and migrate to the configuration of same rest-mass lo-cated on the stable branch of equilibrium configurations Inorder to study the gravitational collapse to a black hole of anunstable model we need to add to the initial model a small ra-dial perturbation in the rest-mass density distribution Averysmall perturbation of the order ofsim 1 is sufficient and its ra-dial dependence can be simply given bycos(πr2rs) wherer is coordinate distance from the center andrs its value atthe surface of the star The addition of this small perturba-tion dominates over the truncation error and causes the starto collapse to a black hole Note that after the perturbationisadded to the initial equilibrium configuration the constraintequations are solved to provide initial data which is a solutionto the field equations [25]

The (forced) collapse to a black hole of an unstable spher-ical relativistic star is shown in Fig 11 for a simulation with1283 gridpoints in octant symmetry using Roersquos solver and anideal fluid EOS The figure shows the profiles along thex-axisof the lapse function (top panel) of thegxx metric compo-nent (middle panel) and of the normalized rest-mass density(bottom panel) Different lines refer to different times oftheevolution with the thick solid line in each panel indicating theinitial profile and with the thick dashed line correspondingtothe final timeslice att = 029 ms intermediate times (shown

0 1 2 3 4 5 6 7 8 9 10x (km)

FIG 11 Profiles along thex-axis of representative metric andfluid quantities during the gravitational collapse to a black hole ofan unstableN = 1 ρc = 80 times 10minus3 relativistic polytrope show-ing different snapshots of the time evolution The top mediumand bottom panels show the evolution of the lapse function of thegxx metric component and of normalized rest-mass density respec-tively The thick solid and dashed lines indicate the initial and final(t = 029 ms) profiles Intermediate profiles indicated by thin dot-ted ashed lines are shown every 0049 ms

every 0049 ms) are indicated with dotted lines The evolu-tion of the lapse function shows the characteristic ldquocollapseof the lapserdquo a distinctive feature of black hole formationThe evolution of thegxx metric component and of the rest-mass density also clearly exhibit features typical of blackholeformation such as the large peak developing ingxx or thecontinuous increase in the central rest-mass density

While the collapse of the lapse is a good indication of theformation of a black hole the formation of an apparent hori-zon (the outermost of the trapped surfaces) in the course of thesimulation is an unambiguous signature of black hole forma-tion An apparent horizon finder based on the fast-flow algo-rithm [51] was used to detect the appearance of horizons andto calculate the horizon mass This apparent horizon finderand its validation is described in Ref [52]

Fig 12 shows the evolution of the horizon mass as a func-tion of time Initially there is no horizon At a timet = 021ms a black hole forms and an apparent horizon appears As theremaining stellar material continues to accrete onto the newlyformed black hole its horizon mass increases finally levellingoff until aboutt = 027 ms The subsequent growth of thehorizon mass is the result of the increasing error due to grid

02 022 024 026 028 03t (ms)

FIG 12 Horizon Mass as a function of time A black hole isformed att = 021 ms and the horizon mass then starts to increaseas a result of accretion

stretching - the radial metric function develops a sharp peakwhich cannot be resolved adequately

V RAPIDLY ROTATING RELATIVISTIC STARS

A Stationary equilibrium models

The long-term evolution of rapidly rotating stable equilib-rium relativistic stars represents a much more demanding testfor a numerical code In this case in fact the use of a non-zeroshift vector is strictly necessary and this in turn involves thetesting of parts of the code that are not involved in the evolu-tion of a non-rotating stellar model The initial data used hereare numerical solutions describing general relativistic station-ary and axisymmetric equilibrium models rotating uniformlywith angular velocityΩ The models are constructed with therns code [5354] (see Ref [55] for a recent review of rotatingstars in relativity) which provides the four metric potentialsνB micro andω needed to describe the spacetime with line ele-ment

ds2 = minuse2νdt2 +B2eminus2νr2 sin2 θ(dφ minus ωdt)2

+e2micro(dr2 + r2dθ2) (19)

In the nonrotating limit the above metric reduces to the metricof a static spherically symmetric spacetime in isotropic coor-dinates A rotating model is uniquely determined upon spec-ification of the EOS and two parameters such as the centralrest-mass density and the ratio of the polar to the equatorialcoordinate radii (axes ratio)

Using the standard Jacobian transformations between thespherical polar coordinates(r θ φ) and the Cartesian coor-

0 2 4 6 8 10 12 14 16x (km)

ρρ c

t=0t=378 ms (3P)

xminusaxiszminusaxis

FIG 13 Profiles of the (normalized) rest-mass density along thex-axis andz-axis at two coordinate timest = 0 (solid lines) andt = 378 ms (dashed lines) corresponding to three rotational peri-ods (P ) The star is aN = 1 ρc = 128 times 10minus3 polytrope rotatingat 92 of the mass-shedding limit The simulation has been per-formed only in the volume above the(x y) plane which is coveredwith 129 times 129 times 66 gridpoints

dinates(x y z) the initial data for a rotating star are trans-formed to Cartesian coordinates Convergence tests of theinitial data on the Cartesian grid at various resolutions showthat the Hamiltonian and momentum constraints converge atsecond-order everywhere except at the surface of the starwhere some high-frequency noise is present This noise isdue to Gibbs phenomena at the surface of the star which areinherent to the method [56] used in the construction of the2D initial data (see the relevant discussion in Ref [54]) Toour knowledge all currently available methods for construct-ing initial data describing rotating relativistic stars suffer fromsome kind of Gibbs phenomena at the surface of the starwith the only exception being a recent multi-domain spec-tral method that uses surface-adapted coordinates [57] Thehigh-frequency noise does not appear to affect the long-termevolution of the initial data at the grid resolutions employedin our simulations The evolution is carried out up to severalrotational periods using the shift 3-vector obtained fromthesolution of the stationary problem which we do not evolve intime

We have evolved models at various rotation rates and forseveral polytropic EOS all showing similar long-term be-haviour and convergence Hereafter we will focus on aN =1 polytropic model rotating at92 of the allowed mass-shedding limit for a uniformly rotating star with the same cen-tral rest-mass density In particular we have chosen a stellarmodel with the same central rest-mass density as the nonrotat-ing model of Section IV A and which is significantly flattenedby the rapid rotation (the polar coordinate radius is only 70

0 5 10 15 20 25x (km)

t=0t=378 ms (3P)

FIG 14 Profile of the metric componentgxx along thex-axisandz-axis at two different coordinate times for the same evolutionshown in Fig 13

of the equatorial coordinate radius)Similarly to what is observed in the numerical evolution of

nonrotating stars the truncation errors trigger in a rapidly ro-tating star oscillations that are quasi-radial As a result therotating star pulsates mainly in its fundamental quasi-radialmode and during the long-term evolution its central rest-mass density drifts towards higher values Also in this caseboth the amplitude of the pulsations and the central densitygrowth rate converge to zero at nearly second-order with in-creasing grid resolution

Our simulations have been performed only in the volumeabove the(x y) plane which is covered with129 times 129 times 66gridpoints At such grid resolutions we have been able toevolve a stationary rapidly rotating relativistic star forthreecomplete rotational periods before the numerical solutiondeparts noticeably from the initial configuration Note thatmuch longer evolution times (more than an order of magni-tude longer and essentially limited by the time available) canbe achieved if the spacetime is held fixed and only the hy-drodynamical equations in a curved background are evolvedThis has been demonstrated recently in Ref [17] with a codebased on the one used in the present paper and in which athird-order Piecewise Parabolic Method (PPM) [58] was usedfor the hydrodynamical evolution and applied to the study ofnonlinearr-modes in rapidly rotating relativistic stars and theoccurrence of differential of a kinematical differential rota-tion [59] (see Ref [6061] for a recent review on ther-modeinstability) While our current second-order TVD methodwith the MC limiter is not as accurate (for the same grid reso-lution) as the third-order PPM method it has nevertheless avery good accuracy significantly better than that of the min-mod limiter

Results of our simulations of rapidly-rotating stars are plot-

ted in Figs 13-15 In particular Fig 13 shows the (normal-ized) rest-mass density along thex andz axes at two coor-dinate timest = 0 (solid lines) andt = 378 ms (dashedlines) with the latter corresponding to three rotational peri-ods The outer boundary of the grid is placed at about twicethe equatorial radius After three rotational periods therest-mass density profile is still very close to the initial one Sim-ilarly Fig 14 shows the metric componentgxx along thexandz axes at the same coordinate times of Fig 13 Againthe change ingxx is minimal and only near the stellar surfacecan one observe a noticeable difference (the error there growsfaster due to the Gibbs phenomenon in the initial data)

0 2 4 6 8 10 12 14 16x (km)

t=0t=378 ms (3P)

FIG 15 The velocity componentvy along thex-axis at twodifferent coordinate times for the same evolution as in Fig 13

Besides triggering the appearance of quasi-radial pulsationsand the secular increase in the central rest-mass density thetruncation errors also induce the formation of a local maxi-mum at the stellar surface for the evolved ldquomomentumrdquo vari-able Sj [cf Eq (7)] The existence of this local extremumreduces at the surface of the rotating star the order of ourTVD schemes to first-order only As a result the angular mo-mentum profile at the surface gradually drifts away from theinitial uniformly rotating one with the rate of convergence ofthis drift being only first-order with increasing grid resolutionWe emphasize however that this is only a local effect every-where else inside the star the angular momentum evolution issecond-order accurate Fig 15 shows the velocity componentvy along thex-axis at the same coordinate times of Fig 13 and14 Alternative evolution schemes based on third-order meth-ods have been shown to have a smaller truncation error at thesurface of the star both for 2D and 3D evolutions of the sameinitial data [4317] at least in the Cowling approximation

Note that plotting the velocity profile as in Fig 15 allowsone to ascertain the accuracy in the preservation of the veloc-ity field Isocontours or vector plots of the velocity field canin fact easily mask the secular evolution shown in Fig 15 We

also note that the variable evolved in the code is not the ro-tational velocity but a corresponding momentum componentwhich depends on the local rest-mass [cf Eq( 7)] The er-ror in the rotational velocity near the surface is thereforealsoinfluenced by the small value of the rest-mass density in thatregion

B Quasi-radial modes of rapidly rotating relativistic stars

The quasi-radial pulsations of rotating neutron stars are apotential source of detectable gravitational waves and couldbe excited in various astrophysical scenarios such as a ro-tating core-collapse a core-quake in a rotating neutron star(due to a large phase-transition in the equation of state) orthe formation of a high-mass neutron star in a binary neu-tron star merger An observational detection of such pulsa-tions would yield valuable information about the equation ofstate of relativistic stars [62] So far however the quasi-radialmodes of rotating relativistic stars have been studied onlyun-der simplifying assumptions such as in the slow-rotation ap-proximation [6364] or in the relativistic Cowling approxima-tion [4865] The spectrum of quasi-radial pulsations in fullGeneral Relativity has not been solved to date with perturba-tion techniques (see Ref [55] for a recent review of the sub-ject)

In this section we take a step forward in the solution of thislong standing problem in the physics of relativistics starsandobtain the first mode-frequencies of rotating stars in full Gen-eral Relativity and rapid rotation As done in Section IV B forthe radial pulsation of nonrotating stars we take advantageof the very small numerical viscosity of our code to extractphysically relevant information from the quasi-radial pertur-bations induced by truncation errors The ability to do sodemonstrates that our current numerical methods are matureenough to obtain answers to new problems in the physics ofrelativistics stars

TABLE III Comparison of small-amplitude quasi-radial pulsa-tion frequencies obtained with the present 3D code infixed space-time with frequencies obtained with an independent 2D code Theequilibrium model is aN = 1 relativistic polytrope rotating at 92of the mass-shedding limit

F 2468 2456 05H1 4344 4357 03H2 6250 6270 03

Following the approach outlined in Section IV B we havefirst computed the quasi-radial mode frequencies from numer-ical evolutions of the GRHydro equations in afixed space-timeevolution in order to compare with recent results comingfrom an independent 2D nonlinear evolution code [65] Ta-ble III shows the comparison of between the eigenfrequenciescomputed in the Cowling approximation with the 2D code for

TABLE IV Quasi-radial pulsation frequencies for a sequence ofrotatingN = 1 polytropes with rotation rates up to 97 of themass-shedding limit The frequencies of the fundamental modeFand of the first overtoneH1 are computed fromcoupledhydrody-namical and spacetime evolutions The ratio of polarrp to equatorialre coordinate radii of the rotating models is also shown

rpre ΩΩK F (kHz) H1 (kHz)1000 0000 1450 39580950 0407 1411 38520850 0692 1350 38670825 0789 1329 38940775 0830 1287 39530750 0867 1265 40310725 0899 1245 39740700 0929 1247 38870675 0953 1209 38740650 0974 1195 3717

the equilibrium model of the previous Section Note that thenewly obtained frequencies differ by less than 05 verifyingthat our code can accurately reproduce them

Next we have computed the quasi-radial frequencies incoupledhydrodynamical and spacetime evolutions for rapidlyrotating stars As mentioned before this is a novel study andthe results obtained cannot be compared with correspondingresults in the literature To study this we have carried outtwo types of analysis Firstly we have followed the sameprocedure used in the case of a nonrotating star case and ob-tained the normalized frequency spectrum of oscillations in-duced by the truncation errors Secondly we have computedthe frequency spectrum of oscillations triggered by a smallbut specified perturbation More precisely we have intro-duced the same radial perturbation in the rest-mass densityused in Sect IV D to induce collapse ieA cos(πr2rp)whereA = 002 r is coordinate distance from the centerandrp is the radial coordinate position of the poles Whencompared the results of the two treatments indicate that thefundamental mode frequency agrees to within 2 while theH1 mode near the mass-shedding limit is probably accurate toseveral percent only (at this resolution)

To study quasi-radial modes of rapidly rotating relativisticstars we have built a sequence of models having the same gridresolution the same equation of state and central rest-massdensity used in the previous section varying only the rota-tion rateΩ The sequence starts with a nonrotating star andterminates with a star at 97 of the maximum allowed rota-tional frequencyΩK = 05363times 104 sminus1 for uniformly rotat-ing stars (mass-shedding limit) The results of these simula-tions are reported in Table IV and shown in Fig 16 where thefrequencies of the lowest two quasi-radial modes are shownInterestingly the fundamental mode-frequencies (solid lines)and their first overtones (dashed lines) show a dependence onthe increased rotation which is similar to the one observedfor the corresponding frequencies in the Cowling approxima-tion [65]

In particular theF -mode frequency decreases monotoni-

0 02 04 06 08 1ΩΩΚ

full GRH

relativ Cowling approx

FIG 16 Quasi-radial pulsation frequencies for a sequenceof ro-tatingN = 1 polytropes and a number of different rotation ratesThe frequencies of the fundamental modeF (filled squares) and ofthe first overtoneH1 (filled circles) are computed fromcoupledhy-drodynamical and spacetime evolutions (solid lines) The sequencesare also compared with the corresponding results obtained from com-putations in the relativistic Cowling approximation

cally as the maximum rotation rate is approached Near themass-shedding limit the frequency is 18 smaller than thefrequency of the nonrotating star The difference between theF -mode frequency computed here and the corresponding re-sult in the Cowling approximation is nearly constant Thusone can construct an approximate empirical relation for thefundamental quasi-radial frequency of rapidly rotating starsusing only the corresponding frequency in the Cowling ap-proximationFCowling and the frequency of the fundamentalradial mode in the nonrotating limitFΩ=0 For the particularsequence shown above the empirical relation reads

F = (FCowling minus 1246) kHz (20)

and yields the correct frequencies with an accuracy of betterthan 2 for the most rapidly rotating model More gener-ally if FCowlingΩ=0 is the frequency of the fundamental ra-dial mode in the Cowling approximation then the empiricalrelation can be written as

F = FΩ=0 + FCowling minus FCowlingΩ=0 (21)

Such an empirical relation is very useful as it allows one toobtain a good estimate of the fundamental quasi-radial modefrequency of rapidly rotating stars by solving the hydrody-namical problem in a fixed spacetime rather than solving themuch more expensive evolution problem in which the space-time and the hydrodynamics are coupled

The frequency of theH1 mode shows a non-monotonicdecrease as the mass-shedding limit is approached depart-

ing from the behavior obtained in the Cowling approxima-tion The oscillations in the frequency at larger rotationrates could be due to ldquoavoided crossingsrdquo with frequenciesof other modes of oscillation (We recall that is referred to asldquoavoided crossingrdquo the typical behaviour shown by two eigen-frequency curves which approach smoothly but then departfrom each other without crossing At the point of closest ap-proach the properties of the modes on each sequence are ex-changed [66]) Similar avoided crossings have been observedalso in the Cowling approximation for higher overtones andnear the mass-shedding limit (see Refs [4865]) Our resultsindicate therefore that the avoided crossings in a sequenceof relativistic rotating stars occur for smaller rotation ratesthan predicted by the Cowling approximation This increasesthe importance of avoided crossings and makes the frequencyspectrum in rapidly rotating stars more complex than previ-ously thought

VI GRAVITATIONAL WAVES FROM A PULSATING STAR

The ability to extract gravitational wave information fromsimulations of relativistic compact objects is an importantfeature of any 3D General Relativistic hydrodynamics codeTo assess the ability of our code to extract self-consistentand accurate gravitational waveforms we have excited simplequadrupolar perturbations in our standard sphericalN = 1polytrope In particular on the basis of the angular behaviorof the ℓ = 2 f -mode in linear perturbation theory we haveintroduced in the initial model a perturbation in the velocityof the form

uθ(t = 0) = A sin (πrrs) sin θ cos θ (22)

whereA = 002 is the amplitude of the perturbation andrs isthe coordinate radius of the star

Following York [25] we have then constructed the initialdata for the perturbed model by solving the constraint equa-tions for the unperturbed model with added perturbations andthen proceeded to evolve this solution in time As a responseto the initial perturbations the star has started a series ofperiodic oscillations mainly in the fundamental quadrupolarmode of oscillation Other higher-order modes are also ex-cited (and observed) but these are several orders of magnitudesmaller and play no dynamical role

As a consequence of the time-varying mass quadrupolartriggered by the oscillations the perturbed star emits gravita-tional waves which are extracted through a perturbative tech-nique discussed in detail in Refs [67ndash69] and in which theZerilli function is expanded in terms of spherical harmonicswith each component being the solution of an ordinary differ-ential equation

We plot in Fig 17 theℓ = 2m = 0 component of theZerilli function ψ20 The upper panel in particular showsthe waverforms as extracted atr

E= 177 km (dotted line)

and atrE

= 236 km (solid line) respectively with the firsthaving been rescaled asrminus32 to allow a comparison The

0 1 2 3 4 5t (ms)

minus005

010ψ2

minus005

rE=177 KmrE=236 km

FIG 17 Gravitation-wave extraction (ψ(20)) of a perturbednonrotating relativistic star pulsating mainly in the fundamentalquadrupolar mode The top panel shows the rescaled waverformsas extracted atr

E= 177 km (dotted line) and atr

E= 236 km

(solid line) The lower panel on the other hand shows the amplitudeof ψ20 extracted atr

E= 236 km for two different resolutions of

643 (dashed line) and963 gridpoints (solid line) respectively

very good agreement between the two waveforms is an indi-cation that the gravitational waves have reached their asymp-totic waveform The lower panel on the other hand shows theamplitude ofψ20 extracted atr

E= 236 km for two differ-

ent resolutions of643 (dashed line) and963 gridpoints (solidline) respectively Note that the gravitational wave signalconverges to a constant amplitude as the true gravitational-wave damping timescale for this mode is several orders ofmagnitude larger than the total evolution time shown Thesmall decrease in the amplitude observed in our numericalevolution is thus entirely due to the effective numerical vis-cosity of our scheme At a resolution of963 gridpoints theeffective numerical viscosity is sufficiently low to allow for aquantitative study of gravitational waves from pulsating starsover a timescale of many dynamical times (the largest rela-tive numerical error estimated on the basis of the simulationspresented in Fig 17 is 33)

As done in the previous sections we have compared the fre-quencies derived from our numerical simulations with thoseobtained from perturbative techniques forf -mode oscillationsof N = 1 polytropes Again in this case the comparison hasrevealed a very good agreement between the two approachesAs one would expect the dominant frequency of the gravita-tional waves we extract (156 kHz) agrees with the fundamen-tal quadrupolef -mode frequency of the star (158 kHz) [47]to within 13 (at a resolution of963 grid-points)

VII CONCLUSIONS

We have presented results obtained with a 3D general rela-tivistic code GRAstro in a comprehensive study of the long-term dynamics of relativistic stars The code has been builtbythe Washington UniversityAlbert Einstein Institute collabora-tion for the NASA Neutron Star Grand Challenge Project [12]and is based on the Cactus Computational Toolkit [13] Thesimulations reported here have benefited from several new nu-merical strategies that have been implemented in the code andthat concern both the evolution of the field equations and thesolution of the hydrodynamical equations In addition to thefeatures of the code discussed in paper I the present version ofthe code can construct various type of initial data representingspherical and rapidly rotating relativistic stars extract gravi-tational waves produced during the simulations and track thepresence of an apparent horizon when formed

All of these improvements have allowed tests and perfor-mances well superior to those reported in the companion pa-per I With this improved setup we have shown that our codeis able to succesfully pass stringent long-term evolution testssuch as the evolution of both static and rapidly rotating sta-tionary configurations We have also considered the evolu-tion of relativistic stars unstable to either gravitational col-lapse or expansion In particular we have shown that unsta-ble relativistic stars can in the course of a numerical evolu-tion expand and migrate to the stable branch of equilibriumconfigurations As an application of this property we havestudied the large-amplitude nonlinear pulsations produced bythe migration Nonlinear oscillations are expected to accom-pany the formation of a proto-neutron star after a supernovacore-collapse or after an accretion-induced collapse of a whitedwarf

Particularly significant for their astrophysical applicationwe have investigated the pulsations of both rapidly rotatingand nonrotating relativistic stars and compared the computedfrequencies of radial quasi-radial and quadrupolar oscilla-tions with the frequencies obtained from perturbative methodsor from axisymmetric nonlinear evolutions We have shownthat our code reproduces these results with excellent accu-racy As a particularly relevant result we have obtained thefirst mode-frequencies of rotating stars in full general relativ-ity and rapid rotation A long standing problem such frequen-cies had not been obtained so far by other methods

In our view the results discussed in this paper have a dou-ble significance Firstly they establish the accuracy and reli-ability of the numerical techniques employed in our code andwhich to the best of our knowledge represent the most accu-rate long-term 3D evolutions of relativistic stars available todate Secondly they show that our current numerical methodsare mature enough for obtaining answers to new problems inthe physics of relativistic stars

ACKNOWLEDGMENTS

It is a pleasure to thank KD Kokkotas for many discus-sions and for providing us with the linear perturbation fre-quencies We have also benefited from many discussions withM Alcubierre S Bonazzola B Brugman JM Ibanez JMiller M Shibata K Uryu and Shin YoshidaThe simulations in this paper have made use of code com-ponents developed by several authors In what followswe report the names of the different components their useand their main author BAM (elliptic equation solver) BBrugman AH-FINDER (apparent horizon finder) M Alcu-bierre CONF-ADM (evolution scheme for the field equa-tions) and MAHC (evolution scheme for the GRHydro equa-tions) M Miller PRIM-SOL (solver for the hydrodynamicalprimitive variables) P Gressman RNS-ID (initial data solverfor rotating and perturbed relativistic stars) N StergioulasEXTRACT (gravitational wave analysis) G Allen The ap-plication code is built on the CACTUS Computational Toolkitwritten by P Walkeret al (version 3) and T Goodaleet al(version 4)Financial support for this research has been provided bythe NSF KDI Astrophysics Simulation Collaboratory (ASC)project (Phy 99-79985) NASA Earth and Space ScienceNeutron Star Grand Challenge Project (NCCS-153) NSFNRAC Project Computational General Relativistic Astro-physics (93S025) and the EU Programme rdquoImproving the Hu-man Research Potential and the Socio-Economic KnowledgeBaserdquo (Research Training Network Contract HPRN-CT-2000-00137)

[1] For a summary of the present and planned high energy mis-sions seeheasarcgsfcnasagovdocsheasarcmissionshtml

[2] K Thorne in Proceedings of theldquoEight Nishinomiya-YukawaSymposium on Relativistic Cosmologyrdquo M Sasaki ed Univer-sal Academy Press Japan 1994

[3] J R Wilson G J Mathews and P Marronetti Phys RevD 541317 1996

[4] T Nakamura and K Oohara invited talk atNumerical Astro-physics 1998 preprintgr-qc9812054

[5] T W Baumgarte S A Hughes and S L Shapiro Phys RevD 60 087501 1999

[6] J A Font M Miller W M Suen and M Tobias Phys RevD61044011 2000

[7] M Miller W-M Suen and M Tobias Phys Rev D631215012001

[8] M Shibata Phys Rev D60104052 1999[9] M Shibata and K Uryu Phys Rev D61064001 2000

[10] M Shibata T W Baumgarte and S L Shapiro Phys Rev D61044012 2000

[11] The NASA Grand Challenge Project is described athttpwugravwustleduRelativnsgchtml

[12] The code and its documentation can be found atwu-gravwustleduCodesGR3D

[13] For information seewwwcactuscodeorg[14] R Arnowitt S Deser and C W Misner inldquoGravitation An

Introduction to Current Researchrdquo p 227 L Witten ed JohnWiley New York 1962

[15] C Bona J Masso E Seidel and J Stela Phys Rev D563405 1997

[16] M Alcubierre W Benger B Brugmann G LanfermannL Nerger E Seidel and R Takahashi 2000preprintgr-qc0012079

[17] N Stergioulas and JA Font Phys Rev Lett86 1148 2001[18] E Seidel and W-M Suen Phys Rev D42 384 1990[19] T Zwerger and E Muller Astron and Astrophys320 209

1997[20] H Dimmelmeier J A Font and E Mueller Astrophys J Lett

in press 2001preprintastro-ph0103088[21] M Shibata and T Nakamura Phys Rev D525428 1995[22] T W Baumgarte and S L Shapiro Phys Rev D59 024007

1999[23] M Alcubierre B Brugmann T Dramlitsch JA FontP Pa-

padopoulos E Seidel N Stergioulas W-M Suen andR Takahashi Phys Rev D62044034 2000

[24] R D Richtmyer and KW MortonDifference Methods for Ini-tial Value Problems Interscience Publishers New York 1967

[25] J York in Sources of Gravitational Radiation L Smarr edCambridge University Press Cambridge England 1979

[26] H Friedrichprivate communication 2000[27] M Alcubierre G Allen B Brugmann E Seidel and W-M

Suen Phys Rev D62 124011 2000[28] J Balakrishna G Daues E Seidel W-M Suen M Tobias

and E Wang Class Quan Grav Lett13 L135 (1996)[29] F Banyuls J A Font J M Ibanez J M Martı and JA

Miralles Astrophys J476 1997[30] C Bona J Masso E Seidel and J Stela Phys Rev Lett 75

600 1995[31] M Alcubierre Phys Rev D555981 1997[32] M Alcubierre and J Masso Phys Rev D574511 1998[33] S Teukolsky Phys Rev D61 087501 2000[34] JM Ibanez MA Aloy JA Font JM Martı JA Miralles

and JA Pons In EF Toro editorGodunov methods theoryand applications Kluwer Academic Plenum Publishers 2001

[35] JM Ibanez and JM Martı J of Comput and Appl Math 109173 1999

[36] J A Font Living Rev Relativity 3 2 2000httpwwwlivingreviewsorgArticlesVolume22000-2font

[37] C Hirsch Numerical Computation of Internal and ExternalFlows Wiley-Interscience 1992

[38] P L Roe J of Comput Phys43357 1981[39] R Donat and A Marquina J of Comput Phys12542 1996[40] A Harten SIAM J Numer Anal21 1 1984[41] B J van Leer J of Comput Phys23276 1977[42] N Stergioulas J A Font and K D Kokkotas inProceed-

ings of the 19th Texas Symposium on Relativistic AstrophysicsE Aubourg et al eds p 233 Nucl Phys B (Proc Suppl)802000

[43] J A Font N Stergioulas and K D Kokkotas Mon NotRAstron Soc313678 2000

[44] T W Baumgarte S A Hughes L Rezzolla S L Shapiroand M Shibata inldquoGeneral Relativity and Relativistic Astro-physics ndash Eighth Canadian Conferencerdquo C P Burgess andR C Myers eds p 53 AIP Conference Proceedings 493

Melville New York 1999[45] K DKokkotas and B G Schmidt Living Rev Relativity 2

2 1999 httpwwwlivingreviewsorgArticlesVolume21999-2kokkotas

[46] P N McDermottH M Van Horn and J F Sholl Astrophys J268837 2001

[47] K D Kokkotasprivate communication 2001[48] S Yoshida and Y Eriguchi Mon Not R Astron Soc322389

2001[49] S H Hawley and M W Choptuik Phys Rev D62 104024

2000[50] R Moenchmeyer G Schaefer E Mueller and R E KatesAs-

tron and Astrophys246417 1991[51] C Gundlach Phys Rev D57863 1998[52] M Alcubierre S Brandt B Brugmann C Gundlach JMasso

and P Walker Class Quant Grav172159 2000[53] N Stergioulas and J L Friedman Astrophys J444306 1995[54] T Nozawa N Stergioulas E Gourgoulhon and Y Eriguchi

Astron Astrophys Suppl132431 1998[55] N Stergioulas Living Rev Relativity 1

8 1998 httpwwwlivingreviewsorgArticlesVolume11998-8stergio

[56] H Komatsu Y Eriguchi and I Hachisu Mon Not R AstronSoc237355 1989

[57] S Bonazzola E Gourgoulhon and J-A Marck Phys Rev D58 104020 1998

[58] P Collela and P R Woodward J Comput Phys54174 1984[59] L Rezzolla F K Lamb and S L Shapiro Astroph Journ

531 L141 2000 L Rezzolla F K Lamb D Markovic andS L Shapiro Phys Rev Din press 2001

[60] N Andersson and K DKokkotas Int J Mod Phys D103812001

[61] J L Friedman and K H Lockitch In V Gurzadyan R Jantzenand R Ruffini edtsProceedings of the 9th Marcel GrossmanMeeting World Scientific 2001gr-qc0102114

[62] K D Kokkotas N Andersson and T A Apostolatos MonNot R Astron Soc320307 2001

[63] J B Hartle and J L Friedman Astrophys J196653 1975[64] B Datta S S Hasan P K Sahu and A R Prasanna Int J Mod

Phys D7 49 1998[65] J A Font H Dimmelmeier A Gupta and N Stergioulas

Mon Not R Astron Soc3251463 2001[66] W Unno Y Osaki H Ando H Saio and H ShibahashiNon-

radial Oscillation of Stars2nd Ed University of Tokyo PressTokyo 1989

[67] K Camarda and E Seidel Phys Rev D59 064026 1999[68] M E Rupright A M Abrahams and L Rezzolla Phys Rev

D 58 044005 1998[69] L Rezzolla A M Abrahams R Matzner M Rupright and

S Shapiro Phys Rev D59 064001 1999