Analysis of the spatial error for a class of finit

AnalysisoftheSpatialErrorforaClassofFiniteDierenceMethodsforViscousIncompressibleFlowBrianR.WettonSeptember19,1994AbstractSeveralrstandsecondordernitedierencemethodsforincompressibleowbasedonprescribedformsofthediscretegradientanddivergenceoperatorsarecon-sidered.Expansionsforthespatialerrorforthesemethodsarepresented.So-calledalternatingexpansionsandnumericalboundarylayersarerequiredtodescribetheerrorsarisingfromschemeswithdecoupledpressureapproximationsandregulariz-ingterms,respectively.Alternatingexpansionsinthediscreteprojectionoperatorcanbeampliedbytheviscoustermandleadtoareductionintheaccuracyofthecomputedpressure.Theseerrorexpansionscanbecombinedwithsimplesta-bilityestimatestoshowtheconvergenceofthediscretesolutionstothenonlineartime-dependentandsteadyproblemswhenadiscreteadjointconditionissatised.However,theanalysisheredoesnotconsiderthesplit-stepnatureofprojectionmethods.Convergenceorderpredictionsareveriedinacarefulnumericalstudy.Keywords:errorexpansions,numericalboundarylayers,gridoscillations.MathematicalSubjectClassications(1991):primary65M06;secondary76D05DepartmentofMathematics,UniversityofBritishColumbia,Vancouver,BC,CanadaV6T1Z2([e-mail]).SupportedbyanNSRECCanadaGrant.11IntroductionWeconsiderthesemi-discretizationinspace,ormethodoflinesapproximation,ofthetime-dependent,incompressibleNavier-StokesEquations.Theclassofmethodswecon-siderarederivedbyassumingaformforthediscretegradientanddiscretedivergenceoperatorsandcalculatingthepressuretoensurethevelocityremainsdiscretedivergencefree.SpatialdiscretizationsthattinthisclassincludetheMarker-and-Cell(MAC)grid[12]andtheregulargriddiscretizationproposedbyChorin[4].Inearlyanalysisofthesemethodsin[4,18]thesolutionofthediscreteproblemwascomparedtoaperturbationoftheexactsolutionthatsatisedthediscretedivergence-freeconditiontohighorder.ThisideawascombinedwiththeasymptoticerroranalysisofStrang[20]bytheauthorandTomHou[15].Inthisworkitwasshownthatperturbationsoftheexactsolutioncouldbeconstructedtosatisfythediscretedivergencefreeconditionandthediscretemomentumequationstoarbitrarilyhighorderaccuracyassumingthesolutionwassuf-cientlysmooth.ThisanalysiswasusedwithsimplediscreteenergyestimatestoshowtheconvergenceoftheMACscheme.Theaccuracyoftheschemewassecondorderinvelocityandpressureandcouldeasilybededucedbytheorderatwhichtheperturba-tionexpansionbegan.Inthispaper,weapplythissamebasicapproachtoothernitedierencemethodsinourclasstodeterminetheiraccuracy.Wewishtoundosomeconfusionfromtheuseoftheword\projectioninourpreviousanalysisfortheMACgrid[15].Forthesemi-discreteversionofthismethod,theactionofthediscretepressureisanorthogonalprojectionofthediscreteaccelerationontothespaceofdiscretedivergencefreevectors.However,ourearlieranalysisandthatpresentedhereisforthespatialdiscretizationonly.Itdoesnotdealwiththeadditionaldelicatestructurearisingfromsplit-steppressureprojectionmethodssuchasthatoriginallyproposedbyChorin[4]andimprovedbymanyothers(see[10]forasurveyofmethods).Theanalysisofthesplit-stepstructurehasbeendonerecentlybyEandLiuinasequenceofpapersbeginningwith[10]forthespacecontinuouscaseandtheMACspatialdiscretization.Inthiswork,theasymptoticcharacteroftheerrorforseveralmethodsinourclassispresented.Forregulargridmethodswherethediscreteequationsforthepressureareuncoupledawayfromboundaries,theerrorexpansionsinvolvetermswhichalternateinsignbetweenadjacentgridpoints.Ifaregularizationmethodisused,suchasthatproposedbyStrikwerde[21],thenthepressureerrorinvolvesanumericalboundarylayer.Adescriptionofstandardor\regularerrorexpansionsaspresentedbyStrang[20],ofnumericalboundarylayersaspresentedbyMichelson[16]andofalternatingerrorexpansionsisgivenforasimplemodelprobleminSection3.Foragivenmethod,wederivetheformoftheexpansionforthediscreteprojectionoperatorinvolvingamixofthetypeoftermsdescribedabove.Whenalternatingtermsarepresentinthisexpansion,theyareampliedthroughthediscreteviscoustermsandcancauseareductionintheorderofaccuracy,althoughonlyforthepressureintheschemesconsideredhere.Acarefulanalysisofthealternatingerrorsatdierentlevelsgivesacompletedescriptionoftheerrorexpansionforthevelocityandpressureandsotheorderofconvergencethemethod.Thepredictionsofthetheoryareveriedincarefulnumericalcomputations.Incertaincases,wherethedivergenceandgradientoperatorsareadjointinsomeinnerproduct(suchastheMACdiscretization)theexistenceoferror2expansionscanbecombinedwithstabilityestimatestoshowconvergencetosmoothsolutionsasin[15].Att=0thesmoothnessofthesolutionandtheerrorexpansioncanbreakdown.Weexamineindetailthespatialerroratt=0andobserveadiscretesmoothingpropertyinallmethodsconsideredfort0.Inthesectionbelowwepresentformallytheclassofmethodsweareconsideringfollowedbythedescriptionofthedierenttypesoferrorexpansionsarisingfromamodelproblem.InSection4,webrieyreviewtheMACanalysisfrom[15]inaslightlycleanerformthatwewilluseasamodelfortheothermethods.Wealsopresenttheanalysisforthecorrespondingtime-independentproblem.InSection5weturntoregulargridmethods.WeexamineChorin’sdiscr