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AnILUSmootherfortheIncompressibleNavier-StokesEquationsinGeneralCoordinatesS.ZengP.WesselingAugust28,1992AbstractILUsmoothersaregoodsmoothersforlinearmultigridmethods.Inthispaper,anewILUsmootherfortheincompressibleNavier-Stokesequations,calledCILU(CollectiveILU),isdesigned,basedonr-transformations.ExistingILUdecompositionsfactorizethematrixwithrealelements.InCILUtheelementsofthematrixthatisfactorizedaresubmatrices,cor-respondingtothesetofphysicalvariables.AmultigridalgorithmusingCILUassmootherisinvestigated.Averagereductionfactorsandlimitingreductionfactorsaremeasuredtoexploretheperformanceofthealgorithm.TheresultsshowthatCILUisagoodsmoother.1IntroductionTheoreticalandpracticalinvestigationsforabouttwodecadeshaveshownthatmultigridmethodsareverysuitableforsolvinglargesystemsofalgebraicequationsresultingfromdiscretizationofpartialdi erentialequations.Inthispaper,wewillpresentamultigridmethodfortheincompressibleNavier-Stokesequationsingeneralcoordinatesdiscretizedonastaggeredgrid.AnewsmootherofILUtype,calledCILU(CollectiveILU),isintroduced.Themaincomponentsinamultigridalgorithmaresmoothingandcoarsegridcorrection.Thesmoothershouldpossessthesmoothingproperty,andthecoarsegridapproximationshouldhavetheapproximationproperty([4]).In[16],thesmoothingandapproximationprop-ertiesarestudiedfortheincompressibleNavier-StokesequationsdiscretizedonastaggeredgridinCartesiancoordinates.Ingeneralcoordinatesatheoryisnotavailable.Therefore,theperformanceofCILUistestedinnumericalexperiments.ClassicalJacobiorGau -Seideliterationmaybeusedforsmoothing.Thesemethodsaresimpletoimplement.However,theyarenotrobust.Theyfailwhentheproblemcontainsanisotropies.Examplesofanisotropiesarestrongconvectionandlargeorsmallaspectratioofgridcells,whichoccuroftenindiscretizationsusingboundary- ttedcoordinates.ILUdecompositionforsmoothinginmultigridmethodshasheeninvestigatedbymanyauthors;forasurvey,see[12].ItisfoundthatILUsmoothingisrobustande cient.ThisleadsustoconsiderasmootherbasedonanILUdecomposition.Forreasonsexplainedelsewhere([19]),weuseGalerkincoarsegridapproximation.Thisimpliesthatthenonlinearproblemtobesolvedislinearizedoutsidethemultigridalgorithm.DiscretesystemsapproximatingtheNavier-Stokesequationsareinde nite.SodirectimplementationofILUdecompositionsisproblematic.Thisproblemisovercomebyapplyinganr-transformation,asproposedin[15],[17]and[18].Thispaperisarrangedasfollows.Insection2,thepartialdi erentialequationsandthediscretesystemthataretobesolvedaredescribed.Section3explainsbrie yther-transformation.AnincompleteLUfactorizationcalledCILUisdescribedinsection4.Insection5,alinearmultigridalgorithmispresentedwhichcoverstheV-,W-,F-andA-cycles.Thechoicesforrestrictionandprolongationoperatorsaregiven.UsingskeweddrivencavityproblemsandL-shapeddrivencavityproblemsastestproblems,insection6theperformanceofthelinearmultigridusingCILUassmootherisinvestigated.2PartialDi erentialEquationsandDiscretizationThetensorformulationoftheincompressibleNavier-Stokesequationsingeneralcoordinatesreadsasfollows:U ; =0;(2.1)@@t(U )+(U U ); +(g p); ; =B ;(2.2)where isthedeviatoricstresstensorandisgivenby =Re 1(g U ; +g U ; );(2:3)1withRetheReynoldsnumber,pthepressure,tthetime,U ; =1;2;:::;ndthecontravari-antcomponentsofvelocitywithndthenumberofspacedimensions,andB thecontravariantcomponentofthebodyforce.U andB arederivedfromtheirphysicalcounterpartsuandbthroughthecontravariantbasevectorsa ofthegeneralcoordinatesbyU =a u;B =a b:(2:4)Furthermore,g isthemetrictensorgivenbyg =a a .Forbetteraccuracy,thevariableV =pgU isusedinsteadofU ,wherepgistheJacobianofthemapping;thisismotivatedin[5],[9]and[14].Thediscretesystemoftheaboveequationsdiscretizedingeneralcoordinatesonastag-geredgridintwodimensions(cf. gure2.1)byusingthe nitevolumemethod([5],[6],[14],[9])V1:-pointsV2:-points:-pointspFigure2.1:Astaggeredgridcanbewritten,foragiventimeinterval t,as:1 tVn+1+ Q0(Vn+1)+ Gpn+1=f0v;DVn+1=fc;(2:5)withf0v= Bn+1+(1 )Bn+1 tVn (1 )Q0(Vn) (1 )Gpn:(2:6)HereV=(V1;V2),B=(B1;B2)andpdenotethediscretevelocity,right-handsideandpressuregridfunctions.Thesuperscriptnindicatesthetimelevel.Theparameter isin[0,1],andistakentobe1inthenumericalexperimentshere,whichgivesthebackwardEulermethod.TheunderlyingorderingoftheunknownsisV11;V12; ;V1n1;V21;V22; ;V2n2;p1;p2; ;pn3;(2:7)withsomeordering(forexamplelexicographic)ofthegridpoints.Thiswillbecalledtheblock-wiseordering.2Equation(2.5)givesrisetoasequenceofsystemsofequationsforasequenceoftimelevels.ItislinearizedwiththeNewton’smethod,forexample(U U )n+1=(U )n+1(U )n+(U )n(U )n+1 (U U )n(2:8)ThisgivesQ0(Vn+1)=Q1Vn+1+Q2(Vn)withQ1linear.NotethatbothQ1andQ2areevaluatedbyusingVn.TheresultingsystemisdenotedbyKx=f(2:9)withK=Q GD0!;x=Vn+1pn+1!;f=fvfc!;(2:10)whereQ=1 tI+ Q1;fv=f0v Q2(Vn):(2:11)Ifthereexistsastationarysolution,thenitsatis esKsx=fs(2:12)withKs=Q0GD0!;fs=Bfc!:(2:13)3Ther-Transformation3.1Iterationwithr-TransformationAclassicaliterationmethodsolving(2.9)isgivenbyxi+1=xi M 1(Kxi f)(3:1)withMasplittingofK:K=M N:(3:2)Thismethodconverg
本文标题:An ILU smoother for the incompressible Navier-Stok
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