Adaptive Refinement Tree - a new high-resolution N-body code

Adaptive Refinement Tree - a new high-resolution N-body code for cosmological simulations

Astrophys.J.Supplement111,73,1997

ADAPTIVEREFINEMENTTREE–ANEWHIGH-RESOLUTION

N-BODYCODEFORCOSMOLOGICALSIMULATIONS

AndreyV.Kravtsov,andAnatolyA.KlypinAstronomyDepartment,NewMexicoStateUniversity,Box30001,Department4500,

LasCruces,NM88003-0001,USA

arXiv:astro-ph/9701195v3 29 Jul 1997andAlexeiM.KhokhlovLaboratoryforComputationalPhysicsandFluidDynamics,Code6404,NavalResearchLaboratory,Washington,DC20375,USAABSTRACTWepresentanewhigh-resolutionN-bodyalgorithmforcosmologicalsimulations.Theal-gorithmemploysatraditionalparticle-meshtechniqueonacubicgridandsuccessivemultilevelrelaxationsonthe nermeshes,introducedrecursivelyinafullyadaptivemannerintheregionswherethedensityexceedsaprede nedthreshold.Themeshisgeneratedtoe ectivelymatchanarbitrarygeometryoftheunderlyingdensity eld–apropertyparticularlyimportantforcos-mologicalsimulations.Inasimulationthemeshstructureisnotcreatedateverytimestepbutisproperlyadjustedtotheevolvingparticledistribution.Thealgorithmisfastande ectivelyparallel:thegravitationalrelaxationsolverisapproximatelyhalfasfastasthefastFouriertrans-formsolveronthesamenumberofmeshcells.TherequiredCPUtimescaleswiththenumberofcells,Nc,as～O(Nc).Thecodeallowsustoimproveconsiderablythespatialresolutionoftheparticle-meshcodewithoutlossinmassresolution.Wepresentadetaileddescriptionofthemethodology,implementation,andtestsofthecode.Wefurtherusethecodetostudythestructureofdarkmatterhalosinhigh-resolution(～2h 1kpc)simulationsofstandardCDM( =1,h=0.5,σ8=0.63)andΛCDM( Λ=1 0=0.7,h=0.7,σ8=1.0)models.We ndthathalodensitypro lesinbothCDMandΛCDMmodelsarewell ttedbytheanalyticalmodelpresentedrecentlybyNavarroetal.,whichpredictsasingular[ρ(r)∝r 1]behaviorofthehalodensitypro lesatsmallradii.Wethereforeconclude

thathalosformedintheΛCDMmodelhavestructuresimilartoCDMhalosandthuscannot

explainthedynamicsofthecentralpartsofdwarfspiralgalaxies,asinferredfromthegalaxies’

rotationcurves.

Subjectheadings:methods:numerical–cosmology:theory–darkmatter

Adaptive Refinement Tree - a new high-resolution N-body code for cosmological simulations

1.Introduction

N-bodytechniquesareusedincosmologicalsimulationstofollowthenonlinearevolutionofasystemofparticles,andtogivetheoreticalpredictionsaboutthematterdistributionthatcanbecomparedwithobservations.ThetraditionalN-bodymethodsaretheparticle-mesh(PM),particle-particle/particle-mesh(P3M),andTREEmethods(Hockney&Eastwood1981;Klypin&Shandarin1983;Efstathiouetal.1985;Bouchet&Hernquist1988,andreferencestherein).Althoughnumerousfundamentalresultshavebeenobtainedusingthesecodes,thecodesoftencannotprovidedesirablespatialormassresolutionwithcurrentlyavailablecomputersbecauseofeithermemoryorCPUlimitations.Thus,forexample,thePMcodecanhandlealargenumberofparticles(thelatestPMsimulationsfollowevolutionofapproximately6×107particles)butislimitedinspatialresolution(toincreaseresolutionbyafactorof2requires8timesasmuchmemory;thelargestPMsimulationshavereacheddynamicrangeof～1500).TREEandP3McodesareCPUlimited1becausecalculationofforcesinthesecodesisconsiderablyslowerthaninthePMcodeand,inthecaseoftheP3Mcode,isalsostronglydependentonthedegreeofparticleclustering.Inanidealcosmologicalsimulationoneneedsaresolution～1 10kpctoresolveagalaxyandasimulationcubeof～100Mpctosampleappropriatelythelongestperturbationwavesortogetsu cientstatistics.Thenumberofparticlesshouldbesu cientlylarge(usuallyafewmillionorlarger)toallowhalopropertiestobereliablydetermined.Therequireddynamicalrangeisthus～104 105,whichishigherthantheabovecodescanprovidefortherequirednumberofparticlesandwithcurrentlyavailablecomputers.Theselimitationshavemotivatedthedevelopmentofnewmethodswithbetterresolutionand/orperformance.

Villumsen(1989)developedacodeinwhichthePMgridwascomplementedby nercubicsubgridstoincreasetheforceresolutioninregionsofinterest.ThelocalpotentialwascalculatedasasumofthepotentialsonthesubgridsandonthePMgrid.AsimilarapproachwasadoptedbyJessop,Duncan,&Chau(1994)intheirparticle-multiple-meshcode.However,insteadofsummingthepotentialsfromsubgrids,thepotentialoneachlevelwasobtainedindependentlybysolvingtheboundaryproblem.Boundaryvaluesofthepotentialwereinterpolatedfromthecoarserparentgrid.Couchman(1991)usedcubicre nementgridstoimprovetheperformanceoftheP3Malgorithm.Here,theresolutionoftheP3Mcodewasretainedwhilethecomputationalspeedwasconsiderablyincreased.IntheLagrangianapproach(Gnedin1995;Pen1995)thecomputationalmeshisnotstaticbutmoveswiththemattersothattheresolutionincreases(smallermeshcells)inthehighdensityregionsanddecreaseselsewhere.Althoughpotentiallypowerful,thisapproachhasitscaveatsanddrawbacks(Gnedin&Bertschinger1996).Themeshdistortions,forexample,mayintroducesevereforceanisotropies.Adi erentapproachwasadoptedbyXu(1995),whodevelopedtheTPMcode,ahybridofthePMandTREEalgorithms.ThegravitationalforcesintheTPMarecalculatedviaaPMschemeonthegridandviamultipoleexpansions(TREEalgorithm)intheregionswherehigherforceresolutionisdesired.Theforcesontheparticlesinlow-densityregionsarecalculatedbythePMscheme,whileforcesontheparticlesinhigh-densityregionsarethesumofexternallarge-scalePMforceandinternalshort-scaleforcefromtheneighboringparticles.AlthoughthiscodemaynotbefasterthanapureTREEcode,itise ectivelyparallelbecauseparticlesindi erentregionscanbeevolvedindependently.AnadaptivemultigridcodeforcosmologicalsimulationswasrecentlypresentedbySuisalu&Saar(1995).Inthiscode, nerrectangularsubgridsareadaptivelyintroducedinregionswherethedensityexceedsaspeci edthreshold.Foreachsubgrid,thepotentialiscalculatedusingboundaryconditionsinterpolatedfromthecoarsergrid.Thesolutiononthe nergridisusedtoimprovethesolutiononthecoarsergrid.Another

Adaptive Refinement Tree - a new high-resolution N-body code for cosmological simulations

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