An ILU smoother for the incompressible Navier-Stok

AnILUSmootherfortheIncompressibleNavier-StokesEquationsinGeneralCoordinatesS.ZengP.WesselingAugust28,1992AbstractILUsmoothersaregoodsmoothersforlinearmultigridmethods.Inthispaper,anewILUsmootherfortheincompressibleNavier-Stokesequations,calledCILU(CollectiveILU),isdesigned,basedonr-transformations.ExistingILUdecompositionsfactorizethematrixwithrealelements.InCILUtheelementsofthematrixthatisfactorizedaresubmatrices,cor-respondingtothesetofphysicalvariables.AmultigridalgorithmusingCILUassmootherisinvestigated.Averagereductionfactorsandlimitingreductionfactorsaremeasuredtoexploretheperformanceofthealgorithm.TheresultsshowthatCILUisagoodsmoother.1IntroductionTheoreticalandpracticalinvestigationsforabouttwodecadeshaveshownthatmultigridmethodsareverysuitableforsolvinglargesystemsofalgebraicequationsresultingfromdiscretizationofpartialdierentialequations.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].ItisfoundthatILUsmoothingisrobustandecient.ThisleadsustoconsiderasmootherbasedonanILUdecomposition.Forreasonsexplainedelsewhere([19]),weuseGalerkincoarsegridapproximation.Thisimpliesthatthenonlinearproblemtobesolvedislinearizedoutsidethemultigridalgorithm.DiscretesystemsapproximatingtheNavier-Stokesequationsareindenite.SodirectimplementationofILUdecompositionsisproblematic.Thisproblemisovercomebyapplyinganr-transformation,asproposedin[15],[17]and[18].Thispaperisarrangedasfollows.Insection2,thepartialdierentialequationsandthediscretesystemthataretobesolvedaredescribed.Section3explainsbrieyther-transformation.AnincompleteLUfactorizationcalledCILUisdescribedinsection4.Insection5,alinearmultigridalgorithmispresentedwhichcoverstheV-,W-,F-andA-cycles.Thechoicesforrestrictionandprolongationoperatorsaregiven.UsingskeweddrivencavityproblemsandL-shapeddrivencavityproblemsastestproblems,insection6theperformanceofthelinearmultigridusingCILUassmootherisinvestigated.2PartialDierentialEquationsandDiscretizationThetensorformulationoftheincompressibleNavier-Stokesequationsingeneralcoordinatesreadsasfollows:U;=0;(2.1)@@t(U)+(UU);+(gp);;=B;(2.2)whereisthedeviatoricstresstensorandisgivenby=Re1(gU;+gU;);(2:3)1withRetheReynoldsnumber,pthepressure,tthetime,U;=1;2;:::;ndthecontravari-antcomponentsofvelocitywithndthenumberofspacedimensions,andBthecontravariantcomponentofthebodyforce.UandBarederivedfromtheirphysicalcounterpartsuandbthroughthecontravariantbasevectorsaofthegeneralcoordinatesbyU=au;B=ab:(2:4)Furthermore,gisthemetrictensorgivenbyg=aa.Forbetteraccuracy,thevariableV=pgUisusedinsteadofU,wherepgistheJacobianofthemapping;thisismotivatedin[5],[9]and[14].Thediscretesystemoftheaboveequationsdiscretizedingeneralcoordinatesonastag-geredgridintwodimensions(cf.gure2.1)byusingthenitevolumemethod([5],[6],[14],[9])V1:-pointsV2:-points:-pointspFigure2.1:Astaggeredgridcanbewritten,foragiventimeintervalt,as:1tVn+1+Q0(Vn+1)+Gpn+1=f0v;DVn+1=fc;(2:5)withf0v=Bn+1+(1)Bn+1tVn(1)Q0(Vn)(1)Gpn:(2:6)HereV=(V1;V2),B=(B1;B2)andpdenotethediscretevelocity,right-handsideandpressuregridfunctions.Thesuperscriptnindicatesthetimelevel.Theparameterisin[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(UU)n+1=(U)n+1(U)n+(U)n(U)n+1(UU)n(2:8)ThisgivesQ0(Vn+1)=Q1Vn+1+Q2(Vn)withQ1linear.NotethatbothQ1andQ2areevaluatedbyusingVn.TheresultingsystemisdenotedbyKx=f(2:9)withK=QGD0!;x=Vn+1pn+1!;f=fvfc!;(2:10)whereQ=1tI+Q1;fv=f0vQ2(Vn):(2:11)Ifthereexistsastationarysolution,thenitsatisesKsx=fs(2:12)withKs=Q0GD0!;fs=Bfc!:(2:13)3Ther-Transformation3.1Iterationwithr-TransformationAclassicaliterationmethodsolving(2.9)isgivenbyxi+1=xiM1(Kxif)(3:1)withMasplittingofK:K=MN:(3:2)Thismethodconverg