High accuracy iterative solution of convection dif

HighAccuracyIterativeSolutionofConvectionDiusionEquationwithBoundaryLayersonNonuniformGridsLixinGeyandJunZhangzDepartmentofComputerScience,UniversityofKentucky,773AndersonHall,Lexington,KY40506-0046,USAMay10,1999AbstractAfourthordercompactnitedierenceschemeandamultigridmethodareemployedtosolvethetwodimensionalconvectiondiusionequationswithboundarylayers.Thecompu-tationaldomainisrstdiscretizedonanonuniform(stretched)gridtoresolvetheboundarylayers.Agridtransformationtechniqueisusedtomapthenonuniformgridtoauniformone.Thefourthordercompactschemeisappliedtothetransformeduniformgrid.Amultigridmethodisusedtosolvetheresultinglinearsystem.Weshowhowthegridstretchingaectsthecomputedaccuracyofthesolutionsfromthefourthordercompactschemeandhowthegridstretchinginuencestheconvergencerateofthemultigridmethod.Numericalexperimentsareusedtoshowthatagradedmeshandagridtransformationarenecessarytocomputehighaccuracysolutionsfortheconvectiondiusionproblemswithboundarylayersanddiscretizedbythefourthordercompactscheme.Keywords:convectiondiusionequation,boundarylayer,gridstretching,gridtransformation,multigridmethod1IntroductionNumericalsimulationoftheconvectiondiusionequationplaysaveryimportantroleincomputa-tionaluiddynamicstosimulateowproblems.AtwodimensionalconvectiondiusionequationsatisfyingDirichletboundaryconditionscanbewrittenintheformofuxx+uyy+p(x;y)ux+q(x;y)uy=f(x;y);(x;y)2;u(x;y)=g(x;y);(x;y)2@:(1)Theconvectioncoecientsp(x;y)andq(x;y)arefunctionsoftheindependentvariablesxandyandareassumedtobesucientlysmooth.HereisaconvexdomainconsistingofaunionofrectanglesTechnicalReportNo.288-99,DepartmentofComputerScience,UniversityofKentucky,Lexington,KY,1999.ThisresearchwassupportedinpartbytheUniversityofKentuckyCenterforComputationalSciencesandinpartbytheUniversityofKentuckyCollegeofEngineering.yE-mail:[e-mail]-mail:[e-mail],URL:[e-mail]/~[e-mail](x;y)andq(x;y)maybereferredtoastheReynoldsnumber(Re)anditdeterminestheratiooftheconvectiontodiusion.Inmanyproblemsofpracticalinterest,theconvectivetermsdominatethediusion.ManynumericalsimulationsoftheEquation(1)basedoniterativesolutionmethodsbecomeincreasinglydicult(convergeslowlyorevendiverge)astheratiooftheconvectiontodiusionincreases[32].Traditionalnitedierencediscretizationschemessuchasthesecondordercentraldierenceschemeandtherstorderupwindschemehavethedrawbacksofeitherlackofstability(centraldierence)orlackofaccuracy(upwind).Thereisconsiderableliteratureindevelopingimprovednitedierencediscretizationschemesfortheconvectiondiusionequationsinone,two,andthreedimensions[1,16,18,19].Recently,theclassofhigherordercompactdiscretizationschemeswithsuperconvergentpropertieshaveattractedconsiderableattentionsandhavebeenappliedtotheone,two,andthreedimensionalconvectiondiusionequations[6,8,13,14,17,29,39,40].InthevariouswaysofdierencingtheEquation(1),themostfamiliarschemesarethecentraldierenceschemeandtheupwinddierencescheme.ThesetwoschemesyieldalinearsystemwithavepointsparsematrixoftheformAu=f:(2)Inthecaseofthecentraldierencescheme,classicaliterativemethodsforsolvingtheresultinglin-earsystem(2)donotconvergewhentheconvectivetermsdominateandthecellReynoldsnumberisgreaterthanacertainconstant.Conventionalupwinddierenceapproximationiscomputa-tionallystable,butisonlyrstorderaccurate;andtheresultingsolutionexhibitstheeectofarticialviscosity[23,27].Thesecondorderupwindmethodssuerfromsimilarproblems.Thehigherordernitedierencemethodsofconventionaltypesarecomplicatedtoimplementandarecomputationallyinecient.Afteracontinuousproblemisdiscretized,itisimportantthattheresultingdiscreteproblembesolvedeciently.Thelastfewdecadeshaveseenatremendousamountofworkindevisingcomputationalmethodsforsolvingsparsesystemsoflinearequations.Existingsolutionmethodsfallintotwolargecategories:directmethodsanditerativemethods.Directmethods,ofwhichGaussianeliminationistheprototype,computeasolutionexactly(uptothemachineaccuracy)inanitenumberofarithmeticsteps.However,theyareratherspecializedandcanbeappliedprimarilytosystemswhicharisefromseparableselfadjointboundaryvalueproblems.Relaxationmethods,suchastheJacobiandGauss-Seideliterations,areeasytoimplementandmaybesuc-cessfullyappliedtomoregenerallinearsystems.However,theserelaxationmethodsalsosuerfromsomeinherentlimitationsandarenotrobustformanyproblemsofpracticalinterest.Multi-gridmethodsweredevelopedfromtheattemptstocorrecttheselimitations.Usedinamultigridmethod,relaxationmethodsarecompetitivewiththefastdirectmethodswhenappliedtothemodelproblems.Inaddition,themultigrid/relaxationmethodshavemoregeneralityandawiderrangeofapplicationsthanthespecializeddirectmethodsdo.Forthetwodimensionalconvectiondiusionequation,Guptaetal.[12,13]proposedafourthorderninepointcompactnitedierenceformula,whichwasshowntobecomputationallyecientandstableandtoyieldhighlyaccuratenumericalsolutions.TheresultinglinearsystemcanbesolvedbyclassicaliterativemethodsforlargevaluesoftheReynoldsnumber[13].Zhang[33