An-introduction-to-SPH

ComputerPhysicsCommunications48(1988)89—9689North-Holland,AmsterdamANINTRODUCTIONTOSPHJ.J.MONAGHANInstituteofAstronomy,Cambridge,EnglandandDepartmentofMathematics,MonashUniversity,Clayton,Vict.3168,AustraliaThispapergivesthederivationoftheequationsforSPH(smoothedparticlehydrodynamics)anddescribestheirapplicationtoawidevarietyofproblemsincompressiblegasflow.1.Preliminaryremarkssarytoconstructtheforceswhichanelementoffluidwouldexperience.Theseforcesmustbecon-Smoothedparticlehydrodynamics(SPH)wasstructedfromtheinformationcarriedbythepar-inventedtodealwithproblemsinastrophysicstides.involvingfluidmassesmovingarbitrarilyinthreeWebeginthetaskofconstructingtheseforcesdimensionsintheabsenceofboundaries.Atypi-byconsideringintegralinterpolants.Startingwithcalexample[1,2]isthenumericalsimulationofthetrivialidentitythefissionofarapidlyrotatingstar.ThewidevarietyofapplicationsofSPHwhichA(r)=fA(r’)6(r_r’)dr’,(2.1)rangefrompost-Newtonianfluiddynamicstohighlysupersonicmetalandrockcollisions(otherforanyfieldA(r),weconsideranapproximationexamplesaredescribedbyBenzintheseproceed-ings[3]),showthatSPHisaversatiletool.ThereA(r)=JA(r’)w(r_r’,h)dr’,(2.2)arearguments(seesection9)tosuggestthatSPHismosteffectiveinthreedimensionalcalculationswherew(u,h)isaninterpolatingkernelwiththeandleastefficientinonedimension,butthefullpropertieseffectivenessofSPHisyettobedetermined.Afterall,finitedifferencemethodshavebeenvigorouslyI~~h)du=1(2.3)studiedforfortyyearsbyanarmyofresearchJworkers,whileSPHhasbeenwithusfortenyearsandanditsseriousstudyisonlybeginning.w(u,h)h_9O(U~2.InterpolationfromdisorderedpointsForexample,wecanchoose,inthreedimensionsSPHinvolvesthemotionofasetofpoints.Atw(u,h)=3/23exp(—u2/h2),(2.4)anytimethevelocityandthermalenergyare~hknownatthesepoints.Amassisalsoassignedtoalthoughinpracticewewouldusuallychooseaeachpointand,forthisreason,thepointsarekernelwithcompactsupport.Thereareinfinitelyreferredtoasparticles.Inordertomovethemanypossiblekernels.Adiscussionofkernelsisparticlescorrectlyduringatimestepitisneces-giveninrefs.[4—6].OO1O-4655/88/$03.50©ElsevierSciencePublishersB.V.(North-HollandPhysicsPublishingDivision)90J.J.Monaghan/AnintroductiontoSPHSupposenowwehaveafluidwithdensityp(r).followthemotionofparticleiweneedanesti-WecanwritetheRHSof(2.2)asmateofVP/pwherePisthepressure.Wecouldcalculatethisdirectly,butthesymmetrizedformJ[A(r’)/~(r’)]w(r—r’,h)p(r’)dr’.(2.5)obtainedfromtheidentity1/p\PToevaluatetheintegralwecanimaginethe—VP=Vt—)+—jVp,(3.1)P~pJPmatterdividedintoNsmallvolumeelementswithmassesm1,m2,...,mAT.Thecontributiontotheleadstoexactlinearandangularmomentumcon-integralfromthevolumeelementkwithmassmkservation.From(2.7)wefindandcentreofmass~kisIP\/P\=~mk~Vw(r—,~,A(rk)w(r—rk,h)mk,(2.6)P~\p/k~1Pkp(rk)(3.2)andanapproximationtoA(r))isgivenbyand~mk—w(r-rk,h)(2.7)_Nk=1PkVp—V(p=~mkVw(r—rk,h).(3.3)k=iwhere,forexample,Ak:=A(rk).Whenthepar-Themomentumequationforparticleicanthere-tidesareequi-separated,andtheirmassesareforebewrittenequal,(2.7)isasimpleRiemannsumwhichisequivalenttothetrapezoidalruleifAk/pkvanishesdv(Pk‘~~whenPkvanishes.Inothercasesitismoredif-=—~mk~+jJ~(3.4)kPkP~/ficulttoestimatetheaccuracy.IntheoriginalpapersonSPH(2.7)wasconsideredtobeaMontewhere~meanstakethegradientwithrespecttoCarloestimateof(2.2),butitisnowclearthatthisthecoordinatesofparticleiandWik:=w(i~—givesmuchtoopessimisticanestimateoftherk,h).accuracy.Amorereasonableestimateisbased°“Itisworthspendingalittletimestudying(3.4).thebehaviourofquasi-orderednumbers[5]whichIfWi~istheGaussian(2.4)thentheforceperunithaveanerror~(log~N)d/Nwheredisthenum-massonparticleiduetoparticlek,~k’~berofdimensions.Using(2.7)wecanapproximateanyfieldAbyF:=+i,\2(i—‘k)(3.5)ananalyticalfunctionA(r)(ifwisntimesPkh2W1k.differentiablethensoisA(r)).ThedensityestimateisThisisacentralforceandlinearandangularNmomentumarethereforeconservedexactly.Al-=~~h),(2.8)thoughF,kisapairforceitisreallyacamou-k—iflagedmany-bodyforcebecausethecoefficientwhichissometimesinterpretedasthesmoothingdependsonthepressureanddensitywhichareoftheparticle’spointmassbythekernelsoastodeterminedbytheneighbours(seeforexampleobtainacontinuousdensityfieldfromasetof(2.8)).particles.Intheabsenceofheatsourcesorsinkstherateofchangeofthethermalenergyperunitmassuisgivenby3.Equationsofmotiondu—P(3.6)dtpForthepresentweconfineourattentiontoinviscidfluiddynamicswithoutbodyforces.ToThereareseveralwaysthiscanbewritteninaJ.J.Monaghan/AnintroductiontoSPH91formsuitableforcomputation.Forexample,startwouldpreferitnotto!Ifweuse(3.11)wecanfromlegislatethatforallparticles,p=p0att=0,andthedensitywillonlychangeifparticlesmovePP1—V•V=—,V(pv)—VVP],(3.7)towardseachotheri.e.intheimpact.p2Ifthefluidisbarytropic,u=u(p),andtheanduse(2.7)toestimatepvandp.Then(3.6)formomentumequationcanbeobtainedfromtheLagrangianparticleibecomes=~m,jv,—Vk)~‘,W,k.(3.8)L=fp(~v2+u(p))dr,(3.12)dtPik=1whichcanbeapproximatedbyTheinterpretationof(3.8)isthatthethermalenergyincreaseswhenparticlesmovetowardseachL—~mk(~v~+U(pk))=:L’(3.13)other(assumingP0,whichmaybeviolatedf