A Pseudo-Wavelet Scheme for the Two-Dimensional Na

APseudo-WaveletSchemefortheTwo-DimensionalNavier-StokesEquationsPhilippeCharton1;3ValeriePerrier2;3March2,1996AbstractNumericalschemestakingadvantageofthewaveletbasescapabilitiestocom-pressbothfunctionsandoperatorsarepresented.GalerkinmethodsfortheheatandPoissonequationsindimensiontwoaredevelopedandnumericallytested.Thenapseudo-waveletscheme,usingcollocationforthenonlinearterm,isdenedforthetwo-dimensionalNavier-Stokesequations.Numericaltestsproveitsaccuracy.SubmittedtoMatematicaAplicadaeComputacional1IREMIA,UniversitedeLaReunion,15Av.ReneCassin,BP7151,97715SaintDenisMessag.Cedex9,France([e-mail])2Laboratoired’Analyse,GeometrieetApplications,UA742InstitutGalilee,UniversiteParisNord,AvJ.B.Clement,93430Villetaneuse,France3LaboratoiredeMeteorologieDynamique,E.N.S.,24rueLhomond,75231Pariscedex05,France([e-mail]).11IntroductionDespitealargenumberofworksandcomputationsdonesincecomputationaluiddy-namicswasborn25yearsago,thedirectnumericalsimulationofturbulenceisstillagreatchallenge.Itconcernspracticalpurposessuchasindustrialormeteorologicalcom-putations,aswellasfundamentalstudies.DirectnumericalsimulationofturbulencerequirestheintegrationintimeofthenonlinearNavier-Stokesequations.However,atlargeReynoldsnumber,thatiswhennonlinearinteractionsarefardominentuponviscouseects,turbulentowsgenerateincreasinglysmallscales.Inconsequence,toberealistic,thediscretizationinspace(andcorrelativelyintime)oughttohandleahugenumberofdegreesoffreedom.Indimensiontwo,directnumericalsimulationofhomogeneousturbulentowsintheincompressiblecasecanbeperformeduptoquitelargeReynoldsnumbersbywayofspectralFouriertechniques;howevertheseReynoldsnumbersaretoolowtocomparetolargescaleatmosphericdynamicsforexample.ThesituationismuchmoredicultindimensionthreewhereonlymoderateReynoldsnumbersareathandevenwiththepresentlargestsupercomputers.Manytentativeshavebeendoneorareunderwaytoovercomethisproblem;LargeEddySimulations(LES)andsubgrid-scalestechniquestrytoseparatethelargestscales,thatareexplicitlycomputed,fromthesmallscales,thatareparametrizedorstatisticallyrepresented.Vortexmethodsorcontoursurgeryareableindimensiontwo,togenerateverythinscaleshowevertheycan’tadapttoverygeneralproblems.Anotherpropertyofturbulentowsistheirstrongintermittency.Smallscalesmaybelocalizedonlyinaverysmallfractionofspace.Fromthispropertyonemaydreamonnewbasisfunctionsmoresuitabletorepresentthisintermittentspatialstructurewithonlyafewnumberofdegreesoffreedom.Theproblemishowtoconcentratethedegreesoffreedomattheplaceswheresmallscalesarepresent?Thisasksrsttodetectwheresmallscalesareorwillhappen,andthentorepresentthemwithonlyafewparameters.Inthatcontext,waveletsappearedtobeverypromising.Indeedweknow,fromthemathematicalpropertiesofwaveletbases,thatwhereaeldhasasingularityofagivenorder,itswaveletcoecientsatsmallscalesgrowatarelatedpowerofthescale[26,28].Thismeansthatatagivensmallscalethenumberofdegreesoffreedomneedednolongerdependsonthescaleitselfbutonthenumberofactivesingularitiesoftheeldatthisscale.Thispropertywhichcomesfromthedoublelocalization,inspaceandscale,ofthewavelets,hasbeenusedforanalysisandcompressionofturbulentelds[18].Itisalsothebasisofseveralwaveletmethodsdesignedtosolvepartialdierentialequations(PDEs).SchematicallythewaveletbasedmethodsforPDEscanbeseparatedintothreeclasses.Inarstclassofmethods,waveletsareused,intheframeworkofaclassicalgrid-adaptivenumericalcode,todetectwherethegridhastoberenedorcoarsedtooptimallyrepresentthesolution[37,29].Multiresolutionanalysisandtheirassociatedscalefunctionbasesmaybeusedasal-ternativebasesinGalerkinmethods[30,24,42,41].Suchmethodshavethusconvergencepropertiessimilartotheonesofspectralmethods,andsimultaneouslypartialderivativeoperatorsdiscretizesimilarlyasinnite-dierencemethods.However,asthesemethodsdon’tusewaveletsbutratherscalefunctionsasbasisfunctions,theycan’tbeadaptivemethodsandcan’treducesignicantlythenumberofdegreesoffreedominanumericalcode.Thethirdclass,theonlyonewhichusesthecompressionpropertiesofwaveletbases,2containsspecicwaveletmethodsforPDEs.Intheliterature,manytentativeshavebeenperformed,oftenbasedonGalerkinorPetrov-Galerkinmethods.Someofthemtakead-vantageofthewaveletcompressionofthesolution[32,33,3],othersuseinsteadthewaveletcompressionoftheoperator[5,25],theaimofourstudyistotakebenetsimultaneouslyofthesetwoproperties.Thus,ourmethodtakesplaceinthespecic-waveletGalerkinmethodcategoryforsolvingthe2DNavier-Stokesequations.Thealgorithmisbasedonfastmatrix-vectorproducts.Thenonlineartermistreatedbymeansofcollocation.A2Dschemefortheheatequation,alreadyannouncedin[8],isdeveloped.TherstresultsforthewaveletNavier-Stokescodewerepresentedin[10]andpublishedinChartons’PhDthesis[7].Nearlyatthesametimeawavelet-algorithmforthe2DNavier-Stokesequation,basedonthebiorthogonalapproachandaFFT-Poissonsolver,wastestedbyFrohlichandSchneider[22].Thepaperwillbeorganizedasfollows:Section2describesthechosenwaveletapproachforsolvingtheone-dimensionalheatequation,itiscomparedtoclassicalm