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ApproximationSchemesforconvectivetermstructuredgridsCommonFromCFDWiki

Contents1DiscretisationSchemesforconvectivetermsinGeneralTransportEquation.FiniteVolumeFormulation,structuredgrids2Introduction3BasicEquationsofCFD4ConvectionSchemes5BasicDiscretisationschemes

5.1CentralDifferencingScheme(CDS)5.2UpwindDifferencingScheme(UDS)also(FirstOrderUpwindFOU)5.3HybridDifferencingScheme(HDSalsoHYBRID)5.4PowerLawScheme(alsoExponentialschemeorPLDS)

6HighResolutionSchemes(HRS)6.1ClassificationofHighResolutionSchemes

6.1.1Linearschemes6.1.2Kappaformulation,KappaSchemesandOtherschemes6.1.3NonLinearschemes

6.2NumericalImplementationofHRS(Defferedcorrectionprocedure)7Diagonaldominancecriterion8NormalisedVariablesFormulation(NVF)9NormalisedVariablesDiagram(NVD)10NormalisedVariableandSpaceFormulation(NVSF)11ConvectionBoundednessCriterion(CBC)12S.K.Godunovtheorem13MonotonicityCriterion14TotalVariationDiminishing(TVD)SimplifiedDescription

14.1Generalissues15TotalVariationDiminishingDiagram(Swebydiagram)

15.1Fluxlimitingformulation16DiscretizationschemesQualityCriterions

DiscretisationSchemesforconvectivetermsinGeneralTransportEquation.



FiniteVolumeFormulation,structuredgrids

Introduction

Thissectiondescribesthediscretizationschemesofconvectivetermsinfinitevolumeequations.Theaccuracy,numericalstability,andboundnessofthesolutiondependonthenumericalschemeusedfortheseterms.Thecentralissueisthespecificationofanappropriaterelationshipbetweentheconvectedvariable,storedatthecellcenter,anditsvalueateachofthecellfaces.

BasicEquationsofCFD

Alltheconservationequationscanbewritteninthesamegenericdifferentialform:

(1)

Equation(1)isintegratedoveracontrolvolumeandthefollowingdiscretizedequationfor isproduced:

(2)

where isthesourcetermforthecontrolvolume ,and and represent,respectively,theconvectiveanddiffusivefluxesof acrossthecontrolvolumeface

Theconvectivefluxesthroughthecellfacesarecalculatedas:

(1)

where isthemassflowrateacrossthecellface .Theconvectedvariable associatedwiththismassflowrateisusuallystoredatthecellcenters,andthussomeformofinterpolationassumptionmustbemadeinordertodetermineitsvalueateachcellface.Theinterpolationprocedureemployedforthisoperationisthesubjectofthevariousschemesproposedintheliterature,andtheaccuracy,stability,andboundednessofthesolutiondependontheprocedureused.

Ingeneral,thevalueof canbeexplicityformulatedintermsofitsneighbouringnodalvaluesbyafunctionalrelationshipoftheform:



(1)

where denotestheneighbouringnode values.Combiningequations(\ref{eq3})through(\ref{eq4a}),thediscretizedequationbecomes:

(1)

ConvectionSchemes

Alltheconvectionschemesinvolveastencilofcellsinwhichthevaluesof willbeusedtoconstructthefacevalue

Whereflowisfromlefttoright,and isthefaceinquestion.

meanUpstreamnode

meanCentralnode

meanDownstreamnode

Inthefirstplot,itisnotsonatraltothinkthecentralnode"C"notasthepresentnode"P".Itmaybethoughtasthefirstnodetotheupstreamdirectionofthesurfaceinquestion"f".

BasicDiscretisationschemes

CentralDifferencingScheme(CDS)

Italsocanbeconsideredaslinearinterpolation.

Themostnaturalassumptionforthecellfacevalueoftheconvectedvariable wouldappeartobetheCDS,whichcalculatesthecellfacevaluefrom:

(1)

orformorecommoncase:



(1)

wherethelinearinterpolationfactorisdefiniedas:

(1)

normalizedvariablesuniformgrids

(1)

normalizedvariablesnonuniformgrids

(1)

Thisschemeis2ndorderaccurate,butisunboundedsothatnonphysicaloscillationsappearinregionsofstrongconvection,andalsointhepresenceofdiscontinuitiessuchasshocks.TheCDSmaybeuseddirectlyinverylowReynoldsnumberflowswherediffusiveeffectsdominateoverconvection.

UpwindDifferencingScheme(UDS)also(FirstOrderUpwindFOU)

TheUDSassumesthattheconvectedvariableatthecellface isthesameastheupwindcellcentrevalue:



(1)

normalisedvariables

(1)

TheUDSisunconditionallyboundedandhighlystable,butasnotedearlieritisonly1storderaccurateintermsoftruncationerrorandmayproduceseverenumericaldiffusion.Theschemeisthereforehighlydiffusivewhentheflowdirectionisskewedrelativetothegridlines.

(1)

(1)

UDSmaybewrittenas

(1)

orinmoregeneralform

(1)

where

(1)

(1)



HybridDifferencingScheme(HDSalsoHYBRID)

TheHDSofSpalding[1972]switchesthediscretizationoftheconvectiontermsbetweenCDSandUDSaccordingtothelocalcellPecletnumberasfollows:

(1)

(1)

ThecellPecletnumberisdefinedas:

(1)

inwhich and arerespectively,thecellfaceareaandphysicaldiffusioncoefficient.When ,CDScalculationstendstobecomeunstablesothattheHDSrevertstotheUDS.Physicaldiffusionisignoredwhen .

TheHDSschemeismarginallymoreaccuratethantheUDS,becausethe2ndorderCDSwillbeusedinregionsoflowPecletnumber.

D.B.Spalding(1972),"Anovelfinitedifferenceformulationfordifferentexpressionsinvolvingbothfirstandsecondderivatives",Int.J.Numer.Meth.Engng.,4:551559,1972.

PowerLawScheme(alsoExponentialschemeorPLDS)

Patankar,S.V.(1980),NumericalHeatTransferandFluidFlow,ISBN0070487405,McGrawHill,NewYork.



HighResolutionSchemes(HRS)

ClassificationofHighResolutionSchemes

HRScanbeclassifiedaslinearornonlinear,wherelinearmeanstheircoefficientsarenotdirectfunctionsoftheconvectedvariablewhenappliedtoalinearconvectionequation.Itisimportanttorecognisethatlinearconvectionschemesof2ndorderaccuracyorhighermaysufferfromunboundedness,andarenotunconditionallystable.

Nonlinearschemesanalysethesolutionwithinthestencilandadaptthediscretisationtoavoidanyunwantedbehavior,suchasunboundedness(seeWaterson[1994]).ThesetwotypesofschemesmaybepresentedinaunifiedwaybyuseoftheFluxLimiterformulation(WatersonandDeconinck[1995]),whichcalculatesthecellfacevalueoftheconvectedvariablefrom:

(1)

where istermedalimiterfunctionandthegradientration isdefinedas:

(1)

ThegeneralisationofthisapproachtohandlenonuniformmesheshasbeengivenbyWaterson[1994]

Fromtheequation(\ref{eq9})itcanbeseenthat givestheUDSand givestheCDS.

Pleasenotethatlineardoesnotmeanfirstorder

Linearschemes

Linearschemesarethoseforwhich islinearfunctionof

isupwinddifferencing(firstorderaccurate)

iscentraldifferencing(secondorderaccurate)

Kappaformulation,KappaSchemesandOtherschemes

kappaformulation

B.vanLeer(1985),"UpwinddifferencemethodsforaerodynamicsproblemsgovernedbytheEulerequations",LecturesinAppl.Math.,22:327336.

Higherorderschemesareusuallymembersofthe class,forwhich

(1)

Usingthisequationfacevariablescanbeexpressed:

inusualvariables



(1)

innormalisedvariables

(1)

Themainschemesare

CDS(centraldifferencingscheme)QUICK(quadraticupwindscheme)LUS(linearupwindscheme)FrommCUS(cubicupwindscheme)

NonLinearschemes

Nonlinearschemesarethoseforwhich isnotalinearfunctionof .Theyfallintothreecategories,dependingonthelinearschemesonwhichtheyarebased.

QUICKbased:

SMART(piecewiselinear,bounded)

(1)

HQUICK(smooth)

(1)

UMIST(piecewiselinear,bounded)

(1)

CHARM(smooth,bounded)

(1)

(1)



Frommbased:

MUSCL(piecewiselinear)

(1)

vanLeer(smooth)

(1)

OSPRE(smooth)

(1)

vanAlbada(smooth)

(1)

other:

Superbee(piecewiselinear)

(1)

MinMod(piecewiselinear)

(1)

Waterson,N.PandDeconinck,H(1995),"Aunifiedapproachtothedesignandapplicationofboundedhighordercovectionschemes",9thInt.Conf.onNumericalMethodsinLaminarandTurbulentFlow,Atlanta,USA,July1995,TaylorandDurbetakieds.,PineridgePress.

Waterson,N.P.(1994),"Developmentofboundedhighorderconvectionschemeforgeneralindustrialapplications",ProjectReport199433,vonKarmanInstituteforFluidDynamics,SintGenesiusRode,Belgium.

NumericalImplementationofHRS(Defferedcorrectionprocedure)

TheHRSschemescanbeintroducedintoequation(\ref{eq4b})byusingthedeferredcorrectionprocedureofRubinandKhosla[1982].Thisprocedureexpressesthecellfacevalue by:

(1)



where isahigherordercorrectionwhichrepresentsthedifferencebetweentheUDSfacevalue andthehigherorderschemevalue ,i.e.

(1)

Ifequation(\ref{eq10a})issubstitutedintoequation(\ref{eq4b}),theresultingdiscretisedequationis:

(1)

where isthedeferredcorrectionsourceterms,givenby:

(1)

ThistreatmentleadstoadiagonallydominantcoefficientmatrixsinceitisformedusingtheUDS.

Thefinalformofthediscretizedequation:

(1)

Subscrit representsthecurrentcomputationalcell representthesixneighbouringcellsandrepresentstheprevioustimestep(transistentcasesonly)

Thecoefficientscontaintheappropriatecontributionsfromthetransient,convectiveanddiffusivetermsin(\ref{eq1})

P.K.KhoslaandS.G.Rubin(1974),"Adiagonallydominantsecondorderaccurateimplicitscheme",Comput.Fluids,2207209.

S.G.RubinandP.K.Khoshla(1982),"Polynomialinterpolationmethodforviscousflowcalculations",J.Comp.Phys.,Vol.27,pp.153.

Diagonaldominancecriterion

NormalisedVariablesFormulation(NVF)

B.P.Leonard(1988),"Simplehighaccuracyresolutionprogramforconvectivemodellingofdiscontinuities",InternationalJ.NumericalMethodsFluids,8:12911318.



NormalisedVariablesDiagram(NVD)

AccordingtoLeonard[1988],forany(ingeneralnonlinear)characteristicsinthenormalizedvariablediagram(seefigurebelow):

Passingthrough isnecessaryandsufficientforsecondorderaccuracyPassingthrough withaslopeof0.75(forauniformgrid)isnecessaryandsufficientforthirdorderaccuracy

Thehorizontalandverticalcoordinatesofpoint inthenormalizedvariablediagram,andtheslopeofthecharacteristicsatthepoint forpreservingthethirdorderaccuracyforanonuniformgrid,canbeobtainedbysimplealgebrausingeqs.[.....]

(1)

(1)

(1)

where



(1)

(1)

(1)

Forauniformqrid, and

Normalisedvariablediagramforvariouswellknownschemes

NormalisedVariableandSpaceFormulation(NVSF)

DarwishM.S.andMoukalledF.(1994),"NormalizedVariableandSpaceFormulationMethodologyforHighResolutionSchemes",Num.HeatTrans.,partB,vol.26,pp.7996.

AlvesM.A.,CruzP.MendesA.MagahaesF.D.PinhoF.T.,OliveiraP.J.(2002),"AdaptivemultiresolutionapproachforsolutionofhyperbolicPDEs",ComputationalMethodsinAppliedMechanicsandEngineering,191,39093928.



ConvectionBoundednessCriterion(CBC)

ChoiS.K.,NamH.Y.andChoM.(1995),"Acomparisonofhighorderboundedconvectionschemes",ComputationalMethodsinAppliedMechanicsandengineering,Vol.121,pp.281301.

GaskellP.H.andLauA.K.C.(1988),"Curvativecompensatedconvectivetransport:SMART,anewboundednesspreservingtrasportalgorithm",InternationalJournalforNumericalMethodsinFluids,Vol.8,No.6,pp.617641.

GaskelandLauhaveformulatedtheCBCasfollows.Anumericalapproximationto isboundedif:

for , isboundedbelowbythefunction andabovebyunityandpassesthroughthepoints(0,0)and(1,1)

for or , isequalto

TheCBCisclearlyillustratedinfigurebelow,wheretheline andtheshadedareaaretheregionoverwhichtheCBCisvalid.TheimportanceoftheCBCistoprovideasufficientandnecessaryconditionforguaranteeingtheboundedsolutionifatmostthreeneighbouringnodalvaluesareusedtoapproximatefacevalues.Itiswellknownthatthepositivityoffinitedifferencecoefficientsisalsoasufficientconditionforboundedness,butthisisoverlystringent,fortheexistenseofnegativecoefficientsdoesnotneccesarilyleadtooverorundershoots.



S.K.Godunovtheorem

MonotonicityCriterion

TotalVariationDiminishing(TVD)SimplifiedDescription

Generalissues

A.Harten(1984),"Onaclassofhighresolutiontotalvariationstablefinitedifferenceschemes",SIAMJ.Num.Analysis,21,p1.

A.Harten(1983),"Highresolutionschemesforhyperbolicconservationlaws",J.Comput.Phys.,49:357393,1983.

P.K.Sweby(1984),"Highresolutionschemesusingfluxlimitersforhyperbolicconservationlaws",SIAMJ.Num.Analysis,21,p995.

TVDcriterion

nonewlocalextremamustbecreatedthevalueofanexistinglocalminimummustbenondecreasingandthatofthelocalmaximummustbenonincreasing

TotalVariation(TV)ofafunction isdefinedby

(1)

TotalVariation(TV)ofanumericalsolutionisdefinedby



(1)

where gridpointindex

forasetofdiscretedata

theTVisdefinedby

(1)

(1)

Formonotonicitytobesatisfied,thisTVmustnotbeincreased!

FinallyanumericalschemeissaidtobeTVDif:

(1)

where timesteporiterationindex

Usingnormalisedvaribles,TVDconditioncabbewritten:

(1)

(1)

Toobtaindifferencingscheme,satisfyingTVDcondition,fluxlimiter isincluded,whichdependsuponfunction'sgradients.

Inordertoprovidemonotonicityofthesolution,itisnecessarytoimplementcondition[K.Fletcher]

(1)

where

(1)



TotalVariationDiminishingDiagram(Swebydiagram)

Fluxlimitingformulation

DiscretizationschemesQualityCriterions

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