Almost Optimal Convergence of the Point Vortex Met

MATHEMATICSOFCOMPUTATIONVolume68,Number228,Pages1465{1496S0025-5718(99)01108-4ArticleelectronicallypublishedonMay21,1999ALMOSTOPTIMALCONVERGENCEOFTHEPOINTVORTEXMETHODFORVORTEXSHEETSUSINGNUMERICALFILTERINGRUSSELE.CAFLISCH,THOMASY.HOU,ANDJOHNLOWENGRUBAbstract.StandardnumericalmethodsfortheBirkho-RottequationforavortexsheetareunstableduetotheamplicationofroundoerrorbytheKelvin-Helmholtzinstability.AnonlinearlteringmethodwasusedbyKrasnytoeliminatethisspuriousgrowthofround-oerrorandaccuratelycomputetheBirkho-Rottsolutionessentiallyuptothetimeitbecomessingular.InthispaperconvergenceisprovedforthediscretizedBirkho-RottequationwithKrasnylteringandsimulatedroundoerror.Theconvergenceisprovedforatimealmostuptothesingularitytimeofthecontinuoussolution.TheproofisinananalyticfunctionclassandusesadiscreteformoftheabstractCauchy-Kowalewskitheorem.Inorderfortheprooftoworkalmostuptothesingularitytime,thelinearandnonlinearpartsoftheequation,aswellastheeectsofKrasnyltering,arepreciselyestimated.Thetechniqueofproofappliesdirectlytootherill-posedproblemssuchasRayleigh-Taylorun-stableinterfacesinincompressible,inviscid,andirrotationalfluids,aswellastoSaman-TaylorunstableinterfacesinHele-Shawcells.1.IntroductionStandardnumericalmethodsaregenerallynotconvergentforill-posedproblems.Typically,inanill-posedproblem,thelineargrowthratesincreaseunboundedlywithincreasingwavenumber.SuchproblemsmayhaveshorttimesmoothsolutionsiftheFouriercoecientsoftheinitialdatahaverapidenoughdecay(i.e.,existenceinanalyticfunctionspaces[5,12,23]).However,whenstandardnumericalmethodsareusedtocomputethem,themethodsprovetobehighlyunstable.Thisisbecause,onthenumericallevel,thedecayoftheFouriercoecientsislimitedbythenumericalprecision.Forexample,theFouriercoecientsoftheinitialdatadecayonlyuntiltheroundolevelisreached.Roughlyspeaking,allsubsequentmodesaredominatedbyroundoerroranddonotdecay.Sincethesehighestmodesareampliedthefastestintime,thenumericalsolutionbecomesdominatedReceivedbytheeditorDecember16,1997.1991MathematicsSubjectClassication.Primary65M25;Secondary76C05.Keywordsandphrases.Vortexsheets,pointvortices,numericalltering,discreteCauchy-Kowalewskitheorem.Therstauthor'sresearchwassupportedinpartbytheArmyResearchOceundergrants#DAAL03-91-G-0162and#DAAH04-95-1-0155,thesecondauthor'sbyONRGrantN00014-96-1-0438andNSFGrantDMS-9704976,andthethirdauthor'sbytheMcKnightFoundation,theNationalScienceFoundation,theSloanFoundation,theDepartmentofEnergy,andtheUniversityofMinnesotaSupercomputerInstitute.c1999AmericanMathematicalSociety14651466RUSSELE.CAFLISCH,THOMASY.HOU,ANDJOHNLOWENGRUBbyspuriouserrorandthecomputationbreaksdown,eventhoughthetruesolutionmaystillbeverysmooth.Aprototypicalill-posedproblem,andtheonewewillconsiderinthispaper,istheevolutionofavortexsheetinanincompressible,inviscid,andotherwiseirrotationalfluid.Thisisaclassicalprobleminfluiddynamics,andthesheetundergoestheKelvin-Helmholtzinstability.Inthisproblem,thelineargrowthrateisproportionaltothewavenumberoftheinitialperturbation.Moreover,singularityformationappearstobegeneric,evenforvortexsheetsinitiallynearequilibrium[17,15,6,22,9].Onemotivationforperformingnumericalsimulationsofthevortexsheetproblemistocharacterizethetypesofsingularitiesthatcanformandtodeterminewhetherthereisinfacta\generictype.See[9]foraveryrecentandthoroughstudyofsingularityformationandevolutionforthevortexsheetproblem.Toaccuratelycomputethenumericalevolutionofavortexsheet,onemustover-comethespuriousgrowthofroundoerror.Thiscanbedoneusinganumericallter.However,standardlinearlters,suchasremoving,ordamping,axedbandofmodes,often\over-smooththedetailsofthesolution,makingsingularitycharacterizationdicult.Moreover,throughnonlinearity,thephysicallyrelevantspectrumtypicallyexpandsintimeintotheregionofarticiallyremovedwavenum-bers.Ifthisregionisxedindependentlyofthediscretizationparametersandoftime,thenthistypeoflteringschemewillnolongerconvergeatsuchtimes.Ontheotherhand,anonlinearltering,introducedtothisproblembyKrasny[15],hasprovenverysuccessful.TheKrasnyltersetsequaltozeroallFouriermodeslyingbelowacertainerrortoleranceandleavesthoselyingabovethetoleranceunchanged.Thelterisnonlinear,becausethemodesitremovesdependonthefunctiontowhichthelterisapplied.Importantconsequencesofthislterarethatitallowsnonlinearitytoproducenon-zeromodesanywhereinthespectrum,andthatthelineargrowthrateisdeterminedbythediscretizationandnotthel-ter.Usingthisnonlinearlter,Krasny[15]andsubsequentlyShelley[22]wereabletoaccuratelycomputenumericalsolutionsessentiallyuptothetimetheybecomesingular.InthispaperweprovethatinthepresenceofsimulatedroundoerrorandKrasnyltering,thepointvortexmethod(PVM)andthespectrallyaccuratemodi-edpointvortexmethod(MPVM[22])bothconvergetothesolutionoftheBirkho-Rottequation.TheproofisinananalyticfunctionclassandusesadiscreteformoftheCauchy-Kowalewskitheorem[7,18,19,21].Theproofispresentedforthecaseinwhichthesheetisinitiallynearequilibriumandconvergenceisobtainednearlyuptothesingularitytime.Thisresultisnearlyoptimal