Stability Analysis of Finite-difference Schemes fo

StabilityAnalysisofFinite-dierenceSchemesfortheViscoelasticWaveEquationJoakimO.BlanchWilliamW.SymesyOctober5,1994AbstractItisdiculttopredictstabilitypropertiesofanitedierencescheme.IthastobeinvestigatedthroughtherootsoftheZ-trans-formedandFouriertransformeddierencescheme(modalequation).Tosimultaneouslyinvestigateseveralschemesfortheviscoelasticwaveequation,itispossibletoderivethemodalequationwithparame-terizedcoecients.Severalconditionallystableschemeswerefound,wherethemostecientisastaggeredschemewithastabilitycondi-tioncloselyresemblingthatofanelasticscheme.1IntroductionFinitedierenceschemesareoftenusedtonumericallysolveordinaryandpartialdierentialequations,[7].Itiseasytocreatesuchascheme,sincederivativesaresimplyexchangedfordierenceratios,whichapproximatethederivativestosomeaccuracy.TheaccuracyisfoundthroughTaylorexpan-sion.ThereisnostandardruleforhowtheseschemesarecreatedandthusTheRiceInversionProject,DepartmentofGeologyandGeophysics,RiceUniversity,HoustonTX77251-1892yTheRiceInversionProject,DepartmentofComputationalandAppliedMathematics,RiceUniversity,HoustonTX77251-18921leavesquitesomeroomforexperimentation.Theaccuracyofthenitedier-enceschemeiseasytondanditiscommonlyagoodmeasureforhowwelltheschemewillsolvethedierentialequation,i.e.,thehighertheaccuracy,thecloserthenumericalsolutionwillbetothetruesolution.Fortimedepen-dentproblemsthesolutionatearliertimeisusedtoupdatethesolutiontothenexttimelevel.Therearetwodistinctivetechniquestoupdatethesolu-tionintime,explicitorimplicitupdating(timestepping).Forimplicittimesteppingitisnecessarytosolveanequationsystemtoupdatethesolutionintime.Explicittimesteppingisthusnaturallyfaster.Implicittimesteppingtendstoattenuatehigherfrequencymodesinthesolutionaswell.Explicittimesteppingisalwaysunstable(solutiongrowsuncontrollably)foratleastsomechoicesofparameters,e.g.,timestep,whereasimplicittimesteppingisstable(forwell-posedproblems).Explicittimesteppingispreferredforhyperbolicproblems,sinceitdoesnotnecessarilyattenuatethesolution(hy-perbolicsystemsareenergypreserving)andisfast.Hence,itisnecessarytondtheparameterchoicesforwhichtheexplicitschemeisnotunstable,i.e.thestabilitycondition.Thiscanprovedicultsincesomeschemesareunconditionallyunstable(nosuchchoicesexist)andonlysomeconditionallystable.Theproblemisthustondaconditionallystableschemeandthestabilitycondition.ItispossibletoinvestigatethestabilityofaparticularnitedierenceschemebyFouriertransformingtheschemeinspaceandZ-transformingtheschemeintime.Theresultisanequationwhichdependsonwavenumber.Theschemeisstableiftheabsolutevalueoftheroots(polesofthescheme)oftheequationareallequaltoorlessthanoneforallpossiblewavenumbers[7].Themorestrictformulationstates:ItisannecessaryconditionforstabilityorItisasucientconditionforinstabilityIFtheabsolutevalueofoneormoreofthepolesisgreaterthanone.Themethodimplicitlyimpliesthatparametersforthedierentialequationareindependentofspace(constants).Someinitialworktosolvetheviscoelasticwaveequationwithanexplicitnitedierenceschemeencounteredstabilityproblems,[1],[2].Tondareliableschemeitwasclearlynecessarytothoroughlyinvestigatethestabilitypropertiesofseveraldierentrealizations.Alltherealizationsarerstorsecondorderaccurateintimeandfourthorderaccurateinspaceandbasedonastress-velocityformulationwithonememoryvariable,[1],[9].Thedierentialequationandtheseveraldierentnite-dierenceschemerealizationswillbecoveredintherstsectionaswellastheinvestigation2method.Thefollowingsectionpresentstheinvestigationandstabilityresultsfortheschemes.Finally,themostecientconditionallystableschemewillbethoroughlyanalyzedforitsstabilitycondition.2ViscoelasticWavePropagation,DiscreteSchemesandMethodforInvestigationThe1-Dviscoelasticwaveequationforaconstitutiverelationcorrespondingtoastandardlinearsolidis,8:p;t=K(1+)v;xrr;t=1(r+Kv;x)v;t=1p;x;(1)[5].Itisaninitialvalueproblem,i.e.,p=r=v=0,t0.Kisthebulkmodulus,isthedensity,andisroughlyproportionaltothereciprocalofQandameasureofattenuation,[5].isarelaxationtimeanddeterminesthefrequencyrangewherethereismostattenuation.pisthepressure(stress)andvistheparticlevelocity.risthe,socalled,memoryvariable,whichreectsthe’memory’oftheviscoelasticmedium.Severaldierenttypesofschemesandgridscanbeusedtodiscretizethecontinuoussystem,Equation(1).Thegeneralschemecreatedherecanberstorsecondorderaccurateintimeandfourthorderaccurateinspace.Itispossibletottheschemeforsomeinstancesofparametervaluesonastaggeredgrid.Acoustic/elasticproblemshavegenerallybeensolvedwithleap-frogschemesforupdatingintime[8].Hence,leap-frogstencilswillbeusedfortoupdatepandv.TheequationforthememoryvariablermightforsmallvaluesofbestianditisnaturaltosolvetheequationwithanEuler-BackwardorCrank-Nicolsonscheme,cf.,[6].WiththesespecicationafairlygeneralnitedierenceschemeforEquation(1)canbeconstructed.38:pn+1m=pn1mtxK(1+)D0;xvnm2t(c1c2rn+1m+(1c1)rnm+c1(1c2)rn1m)rn+1m=rnm(1+)t(arn+1m+(1a)rnm)+K2xD0;x(