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CenterforTurbulenceResearchAnnualResearchBriefs200585AhighorderPadeADI


86DYouZhang2004ThehighordercompactADIHOC-ADIschemeretainsthetridiagonalalgorithmofthestandardADIPeacemanRachfordJr1959andatthesametimeachievesfourth-orderaccuracyinspaceAlthoughanumberoftestproblemsat

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1 CenterforTurbulenceResearchAnnualResearc
CenterforTurbulenceResearchAnnualResearchBriefs200585AhighorderPadeADImethodforunsteadyconvection-di usionequationsByD.You1.MotivationandobjectivesTheunsteadyconvection-di usionequationforavariableintwo-dimensionalspacecanbewrittenas@@t=cx@@xcy@@y+x@2@x2+y@2@y2;(x;y)2\n(0;T];(0;x;y)=0(x;y);(x;y)2\n;a(t;x;y)+b(t;x;y)@@n=f(t;x;y);(x;y)2@\n;t2(0;T];(1.1)where\nR2isarectangulardomain,(0;T]isthetimeinterval,and0andfaretheinitialandboundaryconditions.aandbarearbitrarycoecientsdescribingtheboundaryconditionasaDirichlet,Neumann,orRobintypeintheboundarynormaldirectionn.cx(y)andx(y)denotetheconvectionvelocityandviscosityinthex(y)-direction,respectively.Thisequationiscommonlyencounteredinphysicalsciencesgoverningthetransportofaquantitysuchasmass,momentum,heat,andenergy.Finitedi erenceschemeshavebeenwidelyusedtosolvetheequationusingcombina-tionsofvariousspatialandtemporaldiscretizationmethods(Peaceman&RachfordJr.1959;Karaa&Zhang2004;Kalitaetal.2002;Spotz&Carey2001;Rigal1994;Noye&Tan1988;vanderHouwen&deVries1983;deVries1984;vanderHouwen1984).Amongthem,thealternatingdirectionimplicit(ADI)methodproposedbyPeaceman&RachfordJr.(1959)hasbeenpopularduetoitscomputationalcost-e ectiveness.How-ever,likeother rstandsecond-orderaccuracyschemes,thePeaceman-RachfordADI(PR-ADI)scheme,whichissecond-orderaccurate,oftenproducessigni cantdissipationandphaseerrors,especiallyforconvection-dominatedproblems(Karaa&Zhang2004).InvanderHouwen&deVries(1983),deVries(1984),andvanderHouwen(1984),iter-ativeornon-iterativemultistepmethodswerecombinedwiththeADImethodtoachievehigher-ordertemporalaccuracy.Toachievehigherspatialaccuracy,thehighordercompact(HOC)scheme(MacKinnon&Carey1988)hasrecentlybeenutilizedforspatialdiscretizations.Thisschemeleadstofourth-orderaccurateapproximations.Furthermore,theHOCschemeproducesnon-oscillatorysolutionsfor

2 thesteadyhomogeneousconvection-di usione
thesteadyhomogeneousconvection-di usionequation(Spotz&Carey2001).Otherclassesofhighordercompactschemesthathavedi erentweightingparametershavebeenderivedbyRigal(1994).Forsolvingunsteadyconvection-di usionequations,theHOCschemehasbeenutilizedinanumberofdi erentways.Noye&Tan(1988)developedathird-ordernine-pointHOCschemeforunsteadyconvection-di usionequationsandlater,Kalitaetal.(2002)increasedtheorderofaccuracytofourthorderusingthesamebandwidthofstencil.TheADIapproachwasnotemployedinthosemethods,andcomputationalcostswerethereforesigni cantlyhigherthanthatofthePR-ADImethod.ThecomputationaleciencyoftheADIapproachandhigh-orderaccuracyoftheHOCschemewerecombinedinthemethodproposedbyKaraa& 86D.YouZhang(2004).ThehighordercompactADI(HOC-ADI)schemeretainsthetridiagonalalgorithmofthestandardADI(Peaceman&RachfordJr.1959)andatthesametime,achievesfourth-orderaccuracyinspace.AlthoughanumberoftestproblemsatrelativelylowcellReynoldsnumbers(Pe=cx(y)hx(y)=x(y),wherehx(y)isthegridspacing)wereconsideredintheHOCbasedschemes(Karaa&Zhang2004;Kalitaetal.2002;Noye&Tan1988),thecharacteristicsoftheschemesathighcellReynoldsnumbersandtheresultingnumericalerrorsintermsofphaseandamplitudewerenotdiscussed.Manyfully-implicitandsemi-implicitalgorithmsforsolvingNavier-Stokesequationsalsoutilizethecomputationale ectivenessoftheADI-typeapproach(e.g.Choietal.1992;Akselvoll&Moin1995;Youetal.2004;Visbal&Gaitonde1999).Therefore,Navier-StokessolutionswillbeoneofthemostpromisingapplicationsofADImethodswithhigh-orderspatialaccuracy.Itiswellknownthatinturbulent\rowcomputationsusingdirectnumericalsimulationorlargeeddysimulation,numericaldissipationinducedbyarti cialdi usionorbytruncationerrorsofthenumericalschemesigni cantlydegradesthesolutionquality(Mittal&Moin1997).Aswillbeshowninthepresentstudy,HOCbasedschemessu erfromexcessivenumericaldissipationathighcellRey

3 noldsnumbers.Inthisreport,aPadescheme-b
noldsnumbers.Inthisreport,aPadescheme-basedADI(PDE-ADI)methodisproposed.Thepresentschemeemploysthestandardfourth-orderaccurateapproximationsfor rstandsecondderivativesintheconvection-di usionequationandsimultaneouslyretainssecond-orderaccuracyintimeutilizingtheeciencyoftridiagonalalgorithms.Furthermore,incon-trasttotheHOCbasedschemesinwhichthephaseandamplitudecharacteristicsofasolutionarealteredbythevariationofcellReynoldsnumber,thepresentmethodretainsthecharacteristicsofthemodi edwavenumbersforspatialderivatives,regardlessofthevariationofcellReynoldsnumber.ThesuperiorityoftheproposedschemecomparedtootherADIschemes(Peaceman&RachfordJr.1959;Karaa&Zhang2004)forsolvingunsteadyconvection-di usionequationsisdiscussedindetail.2.HighorderPadeADIschemeApplyingtheCrank-NicolsonschemeandtheADIfactorizationfortimeintegrationof(1.1)resultsin1+t2cx@@xx@2@x21+t2cy@@yy@2@y2n+1=1t2cx@@xx@2@x21t2cy@@yy@2@y2n+O(t2):(2.1)IntheHOCbasedschemes(Karaa&Zhang2004;Kalitaetal.2002;Spotz&Carey2001;Noye&Tan1988),theconvection-di usiontermsineachdirectionareapproximatedwithfourth-orderaccuracyasfollows:cx@@xx@2@x2=fx;(2.2)cxxix1+Pe2122xi=fxi+h2122xfxiPexhx12xfxi;(2.3) HighorderPadeADImethod87wherexand2xarethesecond-ordercentraldi erenceoperatorsfor rstandsecondderivatives,respectively.HOCapproximationsasgivenin(2.3)areincorporatedintotheADIfactoredequation(2.1)inordertocompletetheHOC-ADIschemeproposedbyKaraa&Zhang(2004).Thismethodisunconditionallystableandretainsfourth-orderaccuracyinspaceandsecond-orderaccuracyintime.Also,theHOC-ADIschemeequippedwithtridiagonalalgorithmsresultsinasigni cantcomputationalcostsavingcomparedtoothernon-ADIbasediterativeHOCschemes(Kalitaetal.2002;Noye&Tan1988).SeeKaraa&Zhang(2004)formoredetails.Twoimportantobservationscanbemadeabout(2.3).First,thecellReynoldsnumb

4 erispresentinthestencilcoecients.Thisfe
erispresentinthestencilcoecients.Thisfeatureresultsinanon-oscillatorysolutionbyaddingnumericaldissipationdependingonthecellReynoldsnumberforthesteadyhomogeneousconvection-di usionequation(Spotz&Carey2001).However,thenon-oscillatoryfeatureisnotguaranteedinunsteadyproblems(Spotz&Carey2001),andaswillbeshowninthisnote,theschemeproducessigni cantlyenhanceddissipationincasesofhighcellReynoldsnumbers.Theotherpointisthattheapproximationbecomessingularforpureconvectionproblems(x=0),whereasforpuredi usionproblems(cx=0),theHOCapproximationbecomesthestandardfourth-orderPadescheme.ToexaminethecharacteristicsoftheHOCschemeinmoredetail,amodi edwavenum-beranalysisof(2.2)isperformed.Themodi edwavenumberanalysisallowsonetoassesshowwelldi erentfrequencycomponentsofaharmonicfunctioninaperiodicdomainarerepresentedbya nite-di erencescheme(Moin2001).Replacingthedi erenceoperatorsin(2.3)withmodi edwavenumbersforsecond-ordercentraldi erencesresultsinHOC=icxk1+x1+Pe212k2,iPexhx12k1+1h212k2:(2.4)Forcomparison,thestandardsecond-ordercentraldi erence(CD)andfourth-orderPade(PDE)approximationsof(2.2)arealsoconsidered,forwhichCD=icxk1+xk2;(2.5)PDE=icx^k1+x^k2;(2.6)wherek1=sinkhx=hx,k2=(22coskhx)=h2,^k1=3sinkhx=hx(2+coskhx)and^k2=12(1coskhx)=h2x(5+coskhx).Figure1showstherealandimaginarypartsofsasfunctionsofkhxfortwodi erentcellReynoldsnumbers.InthecaseofPe=0:1,showninFig.1(a),therealpartsforboththeHOCandPadeschemesarealmostindistinguishable,whilethePadeschemeshowsbetterresolutionthantheHOCschemeintheimaginarypartsof.Bothschemesshowbetterresolutionpropertiesthandoesthesecond-ordercentraldi erencescheme.However,ifthecellReynoldsnumberisincreasedto10,theHOCschemedramaticallyincreasesdissipationerror(Real())andproducesasigni cantovershootintheimagi-narypartof(seeFig.1(b)).ThissuggeststhatthesolutionqualityoftheHOCschemeishighl

5 ydependentonthecellReynoldsnumber.Especi
ydependentonthecellReynoldsnumber.Especially,atahighReynoldsnum-ber,signi cantnumericaldissipationisexpected,whichisundesirableinturbulent\rowcomputationsusingdirectnumericalsimulationorlarge-eddysimulation(Mittal&Moin1997).IncontrasttotheHOCscheme,thePadescheme-basedalgorithmdoesnotalteritsnon-dissipativecharacteristicswiththecellReynoldsnumber.However,thetridiagonalmatrixalgorithmsarenotretainedifweapplythePadeapproximationsdirectlyto(2.1). 88D.You(a)(b)Figure1.Realandimaginarypartsofforthreenumericalschemes:(a)Pe=0.1and(b)Pe=10.,PDE;,HOC;,CD;,exact.Toovercomethisde ciency,furtherfactorizationsaremadewithoutlossoftemporalaccuracy:1+t2cx@@x1t2x@2@x21+t2cy@@y1t2y@2@y2n+1=1t2cx@@x1+t2x@2@x21t2cy@@y1+t2y@2@y2n+O(t2);(2.7)wherethespatialderivativetermsareapproximatedwiththestandardfourth-orderPadeschemes.Forinstance,the rstandsecondderivativesinx-directionareapproximatedas01;j+40i;j+0i+1;j=3h(i1;ji+1;j);(2.8)001;j+1000i;j+00i+1;j=12h2(i1;j2i;j+i+1;j);(2.9)orinmatrix-vectornotations,Lx;x=Ax;Lxx;xx=Bxx:(2.10)Thesameapproximationsarealsoappliedtothespatialderivativetermsiny-direction.Inmatrix-vectornotation,(2.7)becomesLx1T+xLxx1TxxLy1T+yLyy1Tyyn+1=Lx1TxLxx1T+xxLy1TyLyy1T+yyn;(2.11) HighorderPadeADImethod89whereTx=Lxt2cxAx;Txx=Lxxt2xBxx;Ty=Lyt2cyAy;Tyy=Lyyt2yByy:(2.12)Thesolutionprocedureonlyconsistsofanumberofmultiplicationsandinversionsoftridiagonalmatrices,whichmakesthecomputationmoreecientthanothernon-ADIbasedschemes(Kalitaetal.2002;Noye&Tan1988)thatrequireiterativemethodsforsolvingbandedsparsematrices(alsoseeRef.Karaa&Zhang(2004)forthecomparisonofcomputationalcostsforADIandnon-ADIbasedschemes).From(2.8)-(2.10)and(2.12),itcaneasilybeshownthatT+x(y)andTxx(yy)arenonsingulartridiagonalmatrices.Thepresentfactorizationretain

6 sthewavenumbercharacteristicsdiscussedin
sthewavenumbercharacteristicsdiscussedinFig.1withO(t2)accuracy.Considersubsetsofthefactoredequations(2.1)and(2.7):1+t2cx@@xx@2@x2=g(2.13)and1+t2cx@@x1t2x@2@x2=g:(2.14)Itcaneasilybeshownthatmodi edwavenumberspectrumsfor(2.13)and(2.14)areequivalentbysecond-ordertemporalaccuracy:1+t2icx^k1+t2x^k2=1+t2icx^k11+t2x^k2+O(t2):(2.15)ThevonNeumannstabilityanalysisisperformedtoexaminethestabilityofthescheme.Byassumingn=neikxhxeikyhy,weobtaintheampli cationfactorinthefollowingform=(1i x)(1 x)(1+i x)(1+ x)(1i y)(1 y)(1+i y)(1+ y);(2.16)where x(y)=t2cx(y)3sin(kx(y)hx(y))hx(y)(2+cos(kx(y)hx(y))); x(y)=t2x(y)12(1cos(kx(y)hx(y)))h2(y)(cos(kx(y)hx(y))+5):(2.17)Itcaneasilybeshownthatthecriterionjj1issatis edforall x(y), x(y).Therefore,thenewschemeisunconditionallystable.ExtensionofthepresentADImethodtoathree-dimensionalproblemandasystemofequationscanbemadeinthesamewayforPR-ADI(Peaceman&RachfordJr.1959)orforHOC-ADI(Karaa&Zhang2004)methods,exceptfortheadditionalfactoriza-tionintheadvectionanddi usionsteps.StabilityandconvergenceofthepresentADImethodforlinearandnonlinearsystemsofequationscanbeprovedbyapplyingthesameapproachusedforthePR-ADImethod(Hundsdorfer&Verwer1989).TodemonstratethepotentialofthepresentPade-ADImethodforthree-dimensionalnonlinearsystems,aformulationfortheincompressibleNavier-StokesequationsispresentedintheAppendix.Forwardorbackwarddi erencescanbeemployedtoimposeDirichlet,Neumann,or 90D.YouRobintypeboundaryconditionswhileretainingthebandwidthandthestrictdiagonaldominanceofthematrices(Peaceman&RachfordJr.1959;Karaa&Zhang2004;Spotz&Carey2001).Mattsson&Nordstrom(2004)showedthattwoandoneorderslessaccurateboundaryclosuresforthesecondand rstderivativesterms,respectively,canmaintaintheinternalorderofaccuracy.ItisalsoworthnotingthatastrongstabilityofthepresentADIschemefornon-period

7 icboundaryconditionscanbeachievedbyemplo
icboundaryconditionscanbeachievedbyemployingthesimultaneousapproximationterm(SAT)procedure(Carpenteretal.1994;Mattsson2005).TheSATprocedureisapenaltymethodinwhichthepenaltyparametersaredeterminedbystabilityconsiderations(seeRefs.Mattsson&Nordstrom(2004);Carpenteretal.(1994);Mattsson(2005);Strand(1994);Carpenteretal.(1999)fordetails).NumericalexamplesToexaminethevalidityande ectivenessofthepresenthighorderPadeADImethod,anunsteadyproblemconcerningtheconvection-di usionofaGaussianpulseinthesquaredomain[0;2][0;2]isconsideredwiththefollowinginitialcondition(Karaa&Zhang2004):(0;x;y)=exp(x0:5)2x(y0:5)2y:(3.1)Ananalyticalsolutiontothisproblemis(t;x;y)=14t+1exp(xcxt0:5)2x(4t+1)(ycyt0:5)2y(4t+1):(3.2)TheDirichletboundaryconditionsaretakenfromtheanalyticalsolution.Auniformgridofhx=hy=0:025isemployedtocomparetheaccuracyofthecomputedsolutionsfromthepresentPade-ADI,theHOC-ADI,andthePR-ADIschemes.Theviscosityvaluesare xedatx=y=0:01.TwocellReynoldsnumbersofPe=2and200areconsideredbysettingconvectionvelocitiescx=cy=0:8andcx=cy=80.Constanttimestepsizesof2:5103and2:5105areusedforPe=2and200,respectively.Figure2showsL2-normerrorsofthecomputedsolutionsatPe=2withrespecttotheexactsolutionateachtimestep.ThepresentADIschemeproducesasigni cantlymoreaccuratesolutionthandotheotherschemeswithwhichitiscompared.Lessac-curatepredictionsoftheHOC-ADIschemearemainlyduetophaseerrorsasexpectedthroughthemodi edwavenumberstudy(seeFig.1(a)).ThePR-ADImethodshowssig-ni cantlyhighermagnitudesofL2-normerrors.InKaraa&Zhang(2004),inthiscase,itwasshownthatthePR-ADImethodcannotachievethesameorderofL2-normerrorsoftheHOC-ADIscheme,evenwithtwicethegridresolution.Numericalsolutionsarecomparedwiththeexactsolutionatthe naltimestep(t=1:25)inFig.3.SolutionsobtainedfromthepresentADI(Fig.3(b))andtheHOC-ADI(Fig.3(c))schemesarevisuallyindistinguis

8 hablefromtheexactsolution(Fig.3(a)).Howe
hablefromtheexactsolution(Fig.3(a)).However,noticeablephasedi erencesareobservedbetweenthePR-ADIsolution(Fig.3(d))andtheexactsolution.ThesuperiorityofthepresentADIschemeismoreclearlyobservedinthehighcellReynoldsnumbercase(Pe=200).Figure4showscontourplotsofthenumericalandexactsolutionsatt=0:0125.ThepresentADIschemeproducesasolutionthatisingoodagreementwiththeexactsolutionintermsofamplitudeandphase(Fig.4(a)and(b)).However,theHOC-ADIschemeandthePR-ADIschemeleadtosigni cantlydissipated HighorderPadeADImethod91Figure2.L2-normerrorsproducedbythreenumericalschemesateachtimestep:,presentADIscheme;,HOC-ADIscheme(Karaa&Zhang2004);,PR-ADIscheme(Peaceman&RachfordJr.1959).t=0:0025,x=y=0:025,cx=cy=0:8andx=y=0:01.SchemeL2-normerrorCPUtimeCPUtimeforPR-ADIPR-ADI(Peaceman&RachfordJr.1959)3:291041:0HOC-ADI(Karaa&Zhang2004)1:821041:4PresentADI7:681062:2Table1.L2-normerrorsatt=0:0125andCPUtimesusedfornumericalintegrations.t=2:5105,x=y=0:025,cx=cy=80andx=y=0:01.solutionsthatarealsohighlydistortedandoscillatory(Fig.4(c)and(d)).Inparticular,theenhancednumericaldissipationmakestheHOC-ADIschemeunattractivefordirectnumericalsimulationsorlargeeddysimulationsofturbulent\rows.Thedistortionsandoscillationsareinopposite-directioninthesolutionofHOC-ADIandPR-ADIschemes,andthisfeatureisexplainedbythecharacteristicsoftheimaginarypartsofthesforahighcellReynoldsnumbercase(seeFig.1(b)).AtPe=200,thethreenumericalschemesarecomparedquantitativelyinTable1.ThepresentADIschemeshowsamuchsmallerL2-normerrorthandothoseofotherADIschemes.However,incontrasttothecaseofPe=2,theL2-normerroroftheHOC-ADIschemeisnotsigni cantlylowerthanthatofthePR-ADImethod.AdisadvantageofthepresentADIschemeisthehighercomputationalcostduetotheincreasednumberoffactorizationsofthegoverningequation.However,itisworthnotingthatatthishighcellReynoldsnumber,thepresentschemepr

9 oducesbetterresultsthandoestheHOC-ADIsch
oducesbetterresultsthandoestheHOC-ADIschemethatemploysthedoublemeshsize. 92D.You0.050.080.100.130.16xy1.21.41.61.81.21.41.61.8xy1.21.41.61.81.21.41.61.80.050.080.100.130.16xy1.21.41.61.81.21.41.61.8(a)(b)xy1.21.41.61.81.21.41.61.80.050.080.100.130.16xy1.21.41.61.81.21.41.61.8xy1.21.41.61.81.21.41.61.80.050.080.100.130.16xy1.21.41.61.81.21.41.61.8(c)(d)Figure3.Contourplotsofthepulseintheregion1:2x;y1:8att=1:25:(a)exact,(b)presentADI,(c)HOC-ADI(Karaa&Zhang2004),and(d)PR-ADI(Peaceman&RachfordJr.1959).t=2:5103,x=y=0:025,cx=cy=0:8andx=y=0:01.Dottedcontourlinesin(b)-(d)correspondtotheexactsolution.4.ConcludingremarksAhighorderalternatingdirectionimplicit(ADI)methodforcomputationofunsteadyconvection-di usionequationshasbeenproposed.ThePadeapproximationsofspatialderivativesleadtofourth-orderaccuracywithhighresolutionpropertiesinspace,whilesecond-orderaccuracyismaintainedintime.Thesolutionprocedureconsistsofanum-berofmultiplicationsandinversionsoftridiagonalmatricesthatarecomputationallycoste ective.Thepresentmethodisunconditionallystableandproducesmoreaccu-ratesolutionsintermsofphaseandamplitudeerrorsthandothestandardsecond-orderADImethod(Peaceman&RachfordJr.1959)andthefourthorderHOC-ADIscheme(Karaa&Zhang2004).Inparticular,itdoesnotintroducenumericaldissipation,whichissigni cantintheconvectiondominatedcasewhenotherpreviousADIschemesareemployed. HighorderPadeADImethod930.050.260.470.690.90xy1.21.41.61.81.21.41.61.80.050.260.470.690.90xy1.21.41.61.81.21.41.61.80.050.260.470.690.90xy1.21.41.61.81.21.41.61.8(a)(b)0.020.150.320.500.670.840.02xy1.21.41.61.81.21.41.61.8-0.07-0.070.100.270.450.620.79xy1.21.41.61.81.21.41.61.8(c)(d)Figure4.Contourplotsofthepulseintheregion1:2x;y1:8att=0:0125:(a)exact,(b)presentADI,(c)HOC-ADI(Karaa&Zhang2004),and(d)PR-ADI(Peaceman&RachfordJr.1959).t=2:5105,x=y=0:025,cx=cy=80andx=

10 y=0:01.AcknowledgmentsThehelpfulcomment
y=0:01.AcknowledgmentsThehelpfulcommentsofDr.MengWangonadraftofthispaperaregreatlyappreci-ated.Appendix.PadeADImethodforincompressibleNavier-StokesequationsConsidertheincompressibleNavier-Stokesequationsintheconservativeform:@ui@t=@@xjuiuj+1Re@2@xj@xjui@p@xi;@ui@xi=0:(5.1) 94D.YouApplyingtheCrank-Nicolsonschemeandthefractional-stepmethodKim&Moin(1985)to(5.1)leadsto^uiunt=2=xj^ui^uj1Re2xjxj^uixjunun1Re2xjxjun;(5.2)un+1i^uit=n+1xi;(5.3)un+1ixi=0;(5.4)whereisreferredtoasthepseudo-pressureandisdi erentfromtheoriginalpressurepbypn+1xi=n+1xi+O(t):(5.5)(5.2)canberecastas^ui+t2xj^ui^uj1Re2xjxj^ui=unt2xjunun1Re2xjxjunRni(5.6)orFi=^ui+t2xj^ui^uj1Re2xjxj^uiRni=0(5.7)ApplyingaNewton-iterationmethodto(5.7)leadsto@Fi@^ujr^ur+1j=Fri;(5.8)where^ur+1j=^ur+1j^ur,ristheiterationindex,andj=1;2;3.Thenij+t2@@^ujxj^ui^uj1Re2xjxj^uir^ur+1j=Fri;(5.9)andweintroduceamatrixoftheformMij=@@^ujxj^ui^uj1Re2xjxj^ui:(5.10)NowwesplitMij=(M1ij+M2ij+M3ij)intothreeparts,eachcontainingx1;x2,andx3-derivatives,respectively.UsinganADIfactorizationtechnique,(5.9)becomes1+t2M1iir1+t2M2iir1+t2M3iir^ur+1i=Frit2(Mij)r^u;(5.11)where=1forj6=iand=0forj=iwithj=1;2;3.Nosummationruleisrepeatedoni(=1;2;3).^uisupdatedduringtheiterationstep.Thefactoredtermsintheleft-handsideof(5.11)becometridiagonalmatriceswhenthespatialderivativesareapproximatedbythesecond-ordercentraldi erences.Inversionsofthetridiagonalmatricesresultinasigni cantreductionincomputingcostandmemory.Thisformulationhasbeensuccessfullyemployedinanumberofdirect-andlarge-eddysimulationsofturbulent\rows(Choietal.1992;Akselvoll&Moin1995;Youetal.2004). HighorderPadeADImethod95Interestingly,Visbal&Gaitonde(1999)employedsecond-ordercentraldi erencesandfourth-orderPadeschemesforthespatialderiva

11 tivetermsintheleft-andright-handsidesof(
tivetermsintheleft-andright-handsidesof(5.11)toobtainglobalhigh-orderspatialaccuracy.Second-ordercentraldi erencesforthefactoredtermswerenecessarytoretaintheeciencyoftridiagonalmatrixoperations.ThepresentADImethodallowsfortheuseoffourth-orderPadeschemesfortheleft-handsidefactoredterms,therebymaintainingtheeciencyoftridiagonalmatrixoperations.ApplicabilityofthepresentADImethodisconsideredforthecaseofi=1as:1+t2x12^u1r1t21Re2x1x1r1+t2x2^u2r1t21Re2x2x2r1+t2x3^u3r1t21Re2x3x3r^ur+11=Fr1t2x2(^u1^u)t2x3(^u1^u);(5.12)whereM111^u1=x1(2^u1^u1)1Re2x1x1^u1;M211^u1=x2(^u2^u1)1Re2x2x2^u1;M311^u1=x3(^u3^u1)1Re2x3x3^u1;M12^u2=x2(^u1^u2);M13^u3=x3(^u1^u3):(5.13)REFERENCESAkselvoll,K.&Moin,P.1995Largeeddysimulationofturbulentcon nedcoannularjetsandturbulent\rowoverabackwardfacingstep.ReportTF-63.DepartmentofMechanicalEngineering,StanfordUniversity,Stanford,California.Carpenter,M.H.,Gottlieb,D.&Abarbanel,S.1994Time-stableboundaryconditionsfor nite-di erenceschemessolvinghyperbolicsystems:methodologyandapplicationtohigh-ordercompactschemes.J.Comp.Phys.111,220{236.Carpenter,M.H.,Nordstrom,J.&Gottlieb,D.1999Astableconservativeinterfacetreatmentofarbitraryspatialaccuracy.J.Comp.Phys.148,341{365.Choi,H.,Moin,P.&Kim,J.1992Turbulentdragreduction:studiesoffeedbackcontroland\rowoverriblets.ReportTF-55.DepartmentofMechanicalEngineering,StanfordUniversity,Stanford,California.deVries,H.B.1984ComparativestudyofADIsplittingmethodsforparabolicequa-tionsintwospacedimensions.J.Comp.App.Math.10,179{193.Hundsdorfer,W.H.&Verwer,J.G.1989StabilityandconvergenceofthePeaceman-RachfordADImethodforinitial-boundaryvalueproblems.Math.Comp.53,81{101.Kalita,J.C.,Dalal,D.C.&Dass,A.K.2002Aclassofhigherordercompact 96D.Youschemesfortheunsteadytwo-dimensionalconvection-di usion

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