A moving mesh method with variable relaxation time

arXiv:math/0602376v1[math.NA]17Feb2006AmovingmeshmethodwithvariablemeshrelaxationtimeAliRezaSoheiliaandJohnM.Stockieb,1aDepartmentofMathematics,UniversityofSistanandBaluchestan,Zahedan,IranbDepartmentofMathematics,SimonFraserUniversity,Burnaby,BC,CanadaAbstractWeproposeamovingmeshadaptiveapproachforsolvingtime-dependentpartialdifferentialequations.ThemotionofspatialgridpointsisgovernedbyamovingmeshPDE(MMPDE)inwhichameshrelaxationtimeτisemployedasaregulariza-tionparameter.PreviouslyreportedresultsonMMPDEshaveinvariablyemployedaconstantvalueoftheparameterτ.Weextendthisstandardapproachbyincorpo-ratingavariablerelaxationtimethatiscalculatedadaptivelyalongsidethesolutioninordertoregularizethemeshappropriatelythroughoutacomputation.Wefocusonsingularproblemsinvolvingself-similarblow-uptodemonstratetheadvantagesofusingavariablerelaxationtimeoverafixedoneintermsofaccuracy,stabilityandefficiency.AMSClassification:65M50,65M06,35K57Keywords:Movingmeshmethod;Self-similarblow-up;Relaxationtime1IntroductionMovingmeshmethodshavebeenemployedwidelytoapproximatesolutionsofpartialdifferentialequationswhichexhibitlargesolutionvariations,suchasshockwavesandboundaryorinteriorlayers.Severalmovingmeshapproaches1ThisauthorwassupportedbyagrantfromtheNaturalSciencesandEngineeringResearchCouncilofCanada(NSERC).E-mailaddresses:[e-mail](A.R.Soheili),[e-mail](J.M.Stockie).PreprintsubmittedtoElsevierScience2February2008havebeenderivedandmanyauthorshavediscussedthesignificantimprove-mentsinaccuracyandefficiencythatcanbeachievedwithrespecttofixedmeshmethods[9,13,14,17,20,21,22].ThemovingmeshPDE(orMMPDE)approachhasprovenparticularlyef-fectiveinsolvingnonlinearPDEsthatexhibitsolutionshavingsometypeofsingularity,suchasself-similarblow-up[7]ormovingfronts[3,21].Forblow-upproblemsinparticular,movingmeshmethodspermitadetailedstudyofthesingularityformationwithadegreeofaccuracyandefficiencythatissimplynotpossibleusingfixedmeshmethods.Theprimaryadvantageofthemovingmeshapproachstemsfromitsabilitytoexploitspecialfeaturesofthesolution(suchasself-similarity)andbuildthemdirectlyintothenumericalscheme.IntheMMPDEapproach,aseparatePDEisderivedtoevolvethemeshpointsinsuchawaythattheytendtowardsanequidistributedmeshatsteadystate,inthesensethatthemeshpointsarepositionedinspacesoastoequallydistributesomemeasureofthesolutionerror.TheMMPDEiscouplednon-linearlytothephysicalPDEofinterest,andbothPDEsaresolvedsimulta-neously.Akeyparameterinthemovingmeshequationisthemeshrelaxationtime,usuallydenotedasτ;theexactequidistributionequationisnotoriouslyill-conditioned[2,19,16]andsoτactstoregularizethemeshevolutionintime.Thephilosophybehindintroducingtemporalsmoothing,insteadofequidis-tributingexactly,isthatthemeshneednotbesolvedtothesamelevelofac-curacyasthephysicalPDE;infact,solutionaccuracycanstillbesignificantlyimprovedoverfixedmeshmethodsbyonlyapproximatelyequidistributingthemesh.Inpreviousresultsreportedintheliterature,themeshrelaxationtimeisin-variablytakentobeaconstantforanygivensimulation.Furthermore,Huang,RenandRussellobservedin[14]that“whiletheparameterτiscritical,inourexperiencethenumericalmethodsarerelativelyinsensitivetotheactualchoiceofτinapplications,”andsimilarcommentsweremadein[7,13].However,itisessentialtokeepinmindthattheseobservationsweremadeforproblemsinwhichtherangeoftimescalespresentinthesolutionwasfairlylimited.Inpractice,τmustbetunedmanuallytooptimizethebehaviourofthecomputedmesh,andsometimeseventoobtainaconvergentnumericalsolution.Themainpurposeofthispaperistoconsidersituationswheretakingconstantτmaynotbeappropriate.Keepinginmindthatτcanbeinterpretedasatimescaleforthemeshmotion,thenτshouldinfactbetakenasasolution-dependentparameter,becauseassingularitiesform,intensify,propagate,anddissipate,thespeedofsolutionvariations(andhencealsoofthemeshpoints)inagivencomputationmayvaryagreatdeal.Bynomeansarewesuggestingthatavariableτisnecessaryinallmovingmeshcalculations.Nonetheless,thereissomeadvantagetobegainedbyhavinganalgorithmthatiscapa-2bleofdeterminingthevalueofτautomaticallyaspartofthesolutionprocesswithoutrequiringtheusertodetermineitsvaluethroughtrialanderror(sincethecomplicatednonlinearcouplingbetweensolutionandmeshintheMM-PDEapproachmeansthereisnowaytoknowthevalueofτapriori).Themainpurposeofthispaperistodemonstrate,bymeansofspecificexamples,thatvaryingthemeshrelaxationparameterthroughoutacomputationcanbeofsignificantadvantageintermsofbothaccuracyandefficiency.Wewillpresentanapproachforadaptivelyselectingτinsuchawaythatthetemporalevolutionofthemeshisoptimalinanappropriatesense.Thispaperisorganizedasfollows.InSection2,webrieflyreviewmovingmeshmethodsinwhichthemeshequationincorporatesarelaxationtimeτ.Themainmotivatingexampleforintroducinganadaptivestrategyforchoosingatime-dependentmeshsmoothingparametercomesfromaclassofnonlinearparabolicequationsexhibitingself-similarblow-upbehaviour;wethereforein-troduceinSection3theblow-upmodelequation,andmotivateaparticularchoiceofτ(t)whichissuggestedbytheanalysisofblow-upproblems.Numer-icalexperimentsarethenpres