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SEMI-LAGRANGIANADVECTION METHODSANDTHEIRAPPLICATIONS INGEOSCIENCE SEMI-LAGRANGIAN ADVECTION METHODSAND THEIR APPLICATIONS INGEOSCIENCE StevenJ.Fletcher
ResearchScientistIII
CooperativeInstituteforResearchintheAtmosphere(CIRA) ColoradoStateUniversity FortCollins,CO,UnitedStates
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1.Introduction
Contents 2.Eulerianmodelingofadvectionproblems
2.1Continuousformoftheadvection equation7
2.2Finitedifferenceapproximationtothe Eulerianformulationoftheadvection equation14
2.3Implicitschemes49
2.4Predictor–correctormethods65
2.5Summary66
3.Stability,consistency,andconvergence ofEulerianadvectionbased numericalmethods
3.1Truncationerror69
3.2Dispersionanddissipationerrors77
3.3Amplitudeandphaseerrors78
3.4Stability81
3.5Quantifyingthepropertiesoftheexplicit finitedifferenceschemes87
3.6Linearmultistepmethods96
3.7Consistencyandstabilityofexplicit Runge–Kuttamethods100
3.8Implicitschemes102
3.9Predictor–correctormethods107
3.10Summary108
4.Historyofsemi-Lagrangianmethods
4.1Fjørtoft(1952)paper111
4.2Welander(1955)paper115
4.3Wiin-Nielsen(1959)paper120
4.4Robert’s(1981)paper124
4.5Summary127
5.Semi-Lagrangianmethodsforlinear advectionproblems
5.1DerivationoftheLagrangianformfor advection129
5.2Derivationofthesemi-Lagrangian approach131
5.3Semi-Lagrangianadvectionofthebell curve135
5.4Semi-Lagrangianadvectionofthestep function140
5.5Summary145
6.Interpolationmethods
6.1Lagrangeinterpolationpolynomials148
6.2Newtondivideddifferenceinterpolation polynomials158
6.3Hermiteinterpolatingpolynomials163
6.4Cubicsplineinterpolation polynomials166
6.5Summary172
7.Stabilityandconsistencyanalysis ofsemi-Lagrangianmethodsforthe linearproblem
7.1Stabilityofsemi-Lagrangian schemes175
7.2StabilityanalysisofLagrangeinterpolation polynomials177
7.3StabilityanalysisofthecubicHermite semi-Lagrangianinterpolation scheme190
7.4Stabilityanalysisofthecubicspline semi-Lagrangianinterpolation scheme195
7.5Consistencyanalysisofsemi-Lagrangian schemes199
7.6Summary203
8.Advectionwithnonconstantvelocities
8.1Semi-Lagrangianapproachesforlinear nonconstantadvectionvelocity205
8.2Twoandthreetimelevelschemes209
8.3Semi-Lagrangianapproximationsto nonlinearadvection219
8.4NonlinearinstabilityI224
8.5NonlinearinstabilityII226
8.6Boundaryconditionsforlimitarea models230
8.7Summary234
9.Nonzeroforcings 9.1Methodsofcharacteristics approach236
9.2Semi-implicitintegration241
9.3Semi-implicitsemi-Lagrangian (SISL)243
9.4Spatialaveraging246
9.5Optimalaccuracyassociatedwith uncenteringtimeaverages247
9.6Semi-Lagrangiantrajectoriesanddiscrete modes252
9.7Time-splitting258
9.8Boundaryconditionsforthe advection-adjustmentequation261
9.9Summary266
10.Semi-Lagrangianmethodsfor two-dimensionalproblems
10.1Bivariateinterpolationmethods269
10.2Gridconfigurations276
10.3Semi-implicitsemi-Lagrangianfinite differencesintwodimensions280
10.4Nonlinearshallowwaterequations284
10.5Finiteelementbasedsemi-Lagrangian method287
10.6Semi-Lagrangianintegrationinflux form294
10.7Semi-Lagrangianintegrationwithfinite volumes302
10.8Semi-Lagrangianadvectioninflowswith rotationanddeformation307
10.9Eliminatingtheinterpolation311
10.10Semi-Lagrangianapproachwithocean circulationmodels318
10.11Transparentboundaryconditions320
10.12Testcasesfortwo-dimensional semi-Lagrangianmethods331
10.13Semi-Lagrangianmethodswiththe2D quasi-geostrophicpotentialvorticity(Eady model)335
10.14Summary346
11.Semi-Lagrangianmethodsfor three-dimensionalproblems
11.1Trivariateinterpolationmethods351
11.2Semi-Lagrangianadvectionintheprimitive equations355
11.33Dfluxformsemi-Lagrangian360
11.4Three-dimensionalfullyelasticEuler equationswithsemi-Lagrangian361
11.5Sensitivitytodeparturepoint calculations368
11.6Consistencyofsemi-Lagrangiantrajectory calculations372
11.7Semi-implicitEulerianLagrangianfinite elements(SELFE)374
11.8Summary380
12.Semi-Lagrangianmethodsonasphere
12.1Vectoroperatorsinspherical coordinates381
12.2Griddevelopmentforasphere383
12.3Grid-pointrepresentationsofthe sphere388
12.4Spectralmodeling394
12.5Semi-Lagrangianandalternatingdirection implicit(SLADI)scheme413
12.6Globalsemi-Lagrangianmodelingofthe shallowwaterequations416
12.7Spectralmodelingoftheshallowwater equations435
12.8Semi-implicitsemi-Lagrangianschemeon thesphere442
12.9RemovingtheHelmholtzequation447
12.10Stableextrapolationtwo-time-levelscheme (SETTLS)449
12.11Fluxformonasphere456
12.12Numericaltestcasesforthesphere464
12.13Summary468
13.Shape-preservingandmass-conserving semi-Lagrangianapproaches
13.1Shape-preservingsemi-Lagrangian advection471
13.2Cascadeinterpolation482
13.3Semi-Lagrangianinherentlyconservingand efficientscheme(SLICE)493
13.4Flux-formsemi-Lagrangianspectralelement approach521
13.5Conservativesemi-LagrangianHWENO methodfortheVlasovequations527
13.6Summary540
14.Tangentlinearmodelingandadjointsof semi-Lagrangianmethods
14.1Derivationofthelinearizedmodel541
14.2Adjoints542
14.3Testofthetangentlinearandadjoint models545
14.4Differentiatingthecodetoderivethe adjoint546
14.5Tangentlinearapproximationsto semi-Lagrangianschemes548
14.6Perturbationforecastmodeling559
14.7Sensitivityofadjointofsemi-Lagrangian integrationtodeparturepoint iterations562
14.8Summary566
15.Applicationsofsemi-Lagrangian methodsinthegeosciences
15.1Atmosphericsciences569
15.2Atmosphericchemistry576
15.3Hydrologicalandocean applications580
15.4Earth’smantleandinterior586
15.5Otherapplications591
15.6Summary592
16.Solutionstoselectexercises
Bibliography 597 Index 605
1 Introduction Advectionplaysavitalpartinmanydifferentformsofgeophysicalmodelingthatcanaffect everybody,everydayinsomeform.Advection,initslinearornonlinearform,affectsthe weatherandoceanforecasts,riversedimentaswellaschemicaltransport,alongwiththe solarwindforecast,modelingofmagmaflows,andhydrothermaltransport.Aninaccurate forecastinanyofthesesituationscouldleadtoquitecatastrophic,andlifethreatening,results.
Moisturetransportforanonshoreflow,ifmiscalculated,couldleadtoamisleadingforecastofanextremerainevent,whichinturncouldleadtoaflashfloodwarningnotbeing issued.TheonsetofthetornadoseasonintheUnitedStates’Midwestispartiallydependent ontheadvectionofthewarmmoistairfromtheGulfofMexicointothecontinent.Anover-, orunder-,predictionofthisadvectioncouldleadtoverydifferentoutcomeswhichcould indicatesevere,ornotsevere,weather.
Aswesawin2010,theeruptionofEyjafjallajökullinIcelandledtotheshutdownofthe NorthAtlanticairspaceforoveraweekbecausethenumericalweatherpredictioncenters didnothavegoodenoughmodelsforthetransportoftheash.Sincethentherehavebeen majordevelopmentsinthisarea,particularlyintransportasaformofadvection,wheresemiLagrangianapproacheshavebeendeveloped.
Thestatementaboveindicatesthatadvectionisalsoreferredtoasatransportproblemin atmosphericchemistry,sedimenttransportinriversandoceanmodeling.Ifwegobelowthe Earth’ssurface,whenweareconcernedwithtemperature,advectionisoftenreferredtoas convectionmodeling.Advectionalsooccursinspacethroughtheionosphereaswellassolar windinteractionswiththemagnetosphere.
ApproximationsofadvectionintheearlynumericalweatherpredictionswereconsideredintheEulerianformwhichisassociatedwithmodelingtheflowasitpassesapoint. Ontheotherhand,theLagrangianformulationdeals withtheflow.Bothapproacheshave advantagesanddisadvantages.Intheearlynumericalweatherprediction,themodelingof advectionintheEulerianformwaseitherdoneusingafinitedifferenceorfiniteelementformulation.Theproblemsthatwereobservedinthoseearlyattemptswereassociatedwiththe Courant–Friedrichs–Lewy,or CFLcondition,whichisassociatedwiththe stability ofthe numericalscheme.Effectively,thisconditionsplacesarestrictiononthesizeofthetimestep, aswellasthegridorelementsize.
Itwasshownearlyoninnumericalweatherpredictionthatthisconditioncouldleadto veryunstableforecastsifitwasnotmet,butthisrestrictedthesizeofthetimestep,whichin turnpreventedrunninglongerforecastsasthecomputationalresourceswerenotavailableto
runatsuchafinetemporalresolution.Therewasalsotheproblemofdampingwithcertain Eulerian-basedfinitedifferenceschemes.
IfweweretoconsidertheLagrangianframework,wewouldstartwithasetofparticles, followthemintime,andthenapplythenumericalapproximationtothedifferentialoperators there.Theproblemwiththisapproachisthattheparticlescouldbecometoofarapart,which isnotpracticaltoachieveviableapproximations.WeshowacopyofFig.2fromWelander [198]inFig. 1.1,whichillustratesthisproblem.
FIGURE1.1 CopyoftheLagrangiandeformationplot,Fig.2fromWelander,1955:Studiesonthegeneraldevelopmentofmotioninatwo-dimensionalidealfluid, Tellus, 17,141–156. https://www.tandfonline.com/doi/abs/10. 3402/tellusa.v7i2.8797 https://creativecommons.org/licenses/by/4.0/
TheadvantageoftheLagrangianapproachisthatthevalueoftheparticlefollowingthe trajectoryisconstant.However,wecannotkeeptrackofalltheseparticles,butwewould likesomeformofatechniquethatutilizesthis.Herecomesemi-Lagrangianapproaches.The basisofsemi-LagrangianapproachesisthatwehaveafixedEuleriangridwithknownvalues ofthefieldatthesegridpointsattime t n andweknowthelocationofthegridpointsat t n+1 , butwedonotknowwhatisthevalueofthefieldthere.
TheLagrangianformoftheadvectionequation,whichyouwillseethroughoutthebook, isgivenby
Dψ Dt = 0, subjecttothekinematicequation
D x Dt = u, where u isreferredtoastheadvectionvelocity.
Thesemi-Lagrangianapproachamountstosayingthatthesecondequationabovetellsus howfartheparticlehastraveledinoneortwotimestepswithoutforcing.Whereasthefirst equation,whenintegratedwithrespecttotime,tellsusthatthevalueofthefieldat t n 1 , or t n ,dependingonwhetheroneusesatwoorthreetimeleveldiscretizationforthetime derivative,isthevalueofthetraceratthe arrivalpoint at t n+1 .Theproblemwehaveisthat wearenotguaranteedthatthe departurepoint willbeatagridpointat t n or t n 1 .Thevalue of ψ canbefoundthroughinterpolationbutwehavetobecarefulwhichorder,andwhat properties,wewishtheinterpolationpolynomialtosatisfy.Thisisthefinitedifferencepoint ofview;thereisalsoasimilarviewwiththefiniteelementapproachwhereitnotapointbut anelementtodealwith.Finally,therearethefluxformandfinitevolumeapproachesthat conservepropertiesofthefieldbetweenthetimelevels.
Giventhismotivation,wenowmoveontobrieflysummarizewhatwehaveineachchaptertoaddressthesituationdescribedabove.Wefinishwithapplicationsofsemi-Lagrangian approachesinthegeosciences,followedbysolutionstoselectexercises.
InChapter 2 westartbyshowingderivationsoftheadvection/transportequationsand showthecharacteristicsbasedapproachwhichwillenableustomoveontothesemiLagrangianapproaches.WewillpresentdifferentexplicitandimplicitEulerian-basedfinite differenceschemesanddemostratetheperformanceoftheseschemeswithasmoothGaussianbelladvection,aswellaswithadiscontinuousstepfunction.
Chapter 3 coversthedifferentpropertiesoftheEulerian-basedfinitedifferencescheme, whereweshallintroducedifferentformsofstability,convergence,andconsistency.Wewill lookatdispersionanddiffusionpropertiesofthedifferentschemessothatweknowtheir likelyperformance.
GiventhebehavioroftheEulerianscheme,wemoveontothehistoryoftheideastowardsadoptingasemi-LagrangianapproachinChapter 4;weshouldnoteherethatthe semi-Lagrangianapproachwasreferredtoinitiallyasquasi-Lagrangian,buttheschemesare alsoreferredtoas Eulerian–Lagrangian approach,andinsomedisciplinesasa particletrajectorytracking.
InChapter 5 wewillintroducethesemi-Lagrangianapproachforthelinearonedimensionalscalaradvectionandshowitsperformancewithdifferentorderofinterpolation fortheadvectionoftheGaussianbellcurveandthestepfunction.
Giventhatwerequireinterpolationpolynomials,weshallintroducedifferentformsof interpolationpolynomialsinChapter 6,andshowtheirperformancesinreconstructingthe bellcurveandstepfunctionwithdifferentnumberofinterpolationpointstoillustratehow welltheapproximationsimprovewiththenumberofpoints.
Thenextstageoftheintroductionofthesemi-Lagrangianapproachesistoconsiderhow todetermineiftherearerestrictionsonthetimeandspacestepsofthedifferentschemes;this isaccomplishedinChapter 7.
Thepropertiesandtechniquestoperformthesemi-Lagrangianintegrationsofarhave beenintendedfortheconstantvelocityandzeroforcingone-dimensionalscalarproblem. InChapter 8 wewillintroducethetechniquesthatarerequiredtobeabletosolvethecase wherethetrajectoryvelocityisnolongerconstant,whichimpliesthatwehavetoestimate thevelocityalongthetrajectory,butalsotoevaluatethevelocityatthedeparturepointinan implicititerativesolution.Wewillconsidereithertwoorthreetimelevelschemesandlook atthepropertiesofbothoftheseapproaches.
InChapter 9 wewillintroducethesituationwherewehaveaforcingtermontherighthandsideoftheadvectionequations.Therearemanydifferentformsofadvectionproblems thathavenonzeroforcingterms,andgiventhedynamicalsituationthatisbeingconsidered, theapproachtodealwiththesesituationscouldleadtononphysicalmodesbeingexcitedin thenumericalsolution.Awidelyusedapproachtodealwiththissituationisreferredtoas thesemi-implicitsemi-Lagrangian(SISL)method,wherenonlineartermsaretreatedasexplicit,whilethelineartermsaretreatedimplicitly.Weshallintroducethistechniquealong withothertechniquesthathavebeendevelopedovertheyearstostabilizeandincreasethe efficientlyofthesemi-Lagrangianintegrationwhenaforcingtermispresent.
Wemoveontothetwo-dimensionalprobleminChapter 10.Inthischapterweshallintroducethetechniquesofbivariateinterpolationtoestimatethevalueofthefieldatthe departurepoint,alongwiththestabilityanalysisintwodimensions.Weshallalsointroducedifferentstaggeredhorizontalandverticalgridsthatarequiteoftenusedfornumerical modeling.Weshallalsolookattwo-dimensionalapproximationsfornonconstantvelocities, aswellasfornonzeroforcingtermswherethedeparturepointnowisinanarea,element,or cellforthefinitedifference,finiteelements,orfinitevolume/fluxformapproaches.Weshall alsopresentsometestcasesthatarefrequentlyusedtoanalyzetheperformanceofnewdevelopmentsinsemi-Lagrangiantheory.Finally,weshallshowanexampleofsemi-Lagrangian advectionwiththeEadymodelwhichdescribesbaroclinicinstabilityinan x –z planeapproximationoftheatmosphere.
Thenextstepistointroducethethirddimensionfortheadvectionproblem,whichwe doinChapter 11.Inthischapterweshallextendtheideasfromthepreviouschaptersto thethree-dimensionalproblem.Weconsiderthecasethatweneedtointerpolatetoadeparturepointinthreedimensions.Weshallintroducesomedifferentverticalcoordinatesystems thatareusedinoperationalnumericalweatherpredictionsystems,aswellasinoceanmodeling.Weshallconsiderfinitedifference,finiteelement,andfinitevolumeapproaches,all asanextensionoftheflux-formfromthelastchapteronchemicaltransport.Werevisitthe assumptionsofonlyneedingtwoiterationstofindthedeparturepoint,aswellasbeinginconsistentintheorderofthevelocityinterpolation,comparedtothefields’interpolationto thedeparturepoint.
Aswedonotliveonacube,weneedtolookathowwecanextendthetechniqueto sphericalcoordinates,andwedothisinChapter 12.Hereweshallintroducesemi-Lagrangian developmentonthesphere.Thischaptercomprisesoftwoparts:thefirstintroduceshow thevectorcalculustransferstosphericalcoordinates,projections,numericalgrids,aswellas introducesthetheoryofspectralmethods.Thesecondpartdescribeshowsemi-Lagrangian
theoryisappliedtodifferentmodelsinsphericalcoordinatesandmultipledimensions,where wewillpresentfinitedifferenceandfinitevolumeapproaches,aswellassemi-Lagrangian methodswithspectralmethods.
Oneoftheproblemsthathasbeennoticedinthestandardfinitedifferenceapproachto thesemi-Lagrangianmethodsisthattheymaynotpreservetheshapeoftheobjectbeing advected;itmayalsoberequiredthatthe mass isconserved.InChapter 13 wewillintroduce differenttechniquesthathavebeendevelopedtoensurethatthesemi-Lagrangianscheme canpreservetheshapeofthebodybeingadvected,toavoidtheGibbsphenomenaofunderandovershoots.WeshallintroducethecascadeinterpolationmethodsbetweenEulerianand Lagrangiangrids,alongwithdifferentfinitevolumeapproachesthatenabletheconservation ofmasseitherlocallyofglobally.WewillalsointroduceSLICEwhichisafinitevolumebased schemeutilizingthecascadeinterpolation.ThechapterfinishesbyintroducingtheVlasov familyofnonlinearPDEsandtheHWENOapproachtosolvethem.
Inthelasttheoreticalchapter,Chapter 14,weshallintroducetheconceptofthetangent linearmodel,alongwithadjoints.Weshallalsointroducethenotionofdifferentiatingthe codetoobtaintheadjointratherthanderivingitanalytically.Wewillalsointroducetheperturbationforecastmodelwhichisseenasanalternativetothetangentlinearapproximation. Afterwardswewillapplyallthesetechniquestosemi-Lagrangianmethods,wherewewill seethatthereisanimportantpropertyinthederivationthatputsalimitonthesizeofthe variationinthedeparturepointandtheinterpolation.
Thepenultimatechapterofthebook,Chapter 15,willpresetdifferentapplicationsofsemiLagrangian,Eulerian–Lagrangian,andquasi-Lagrangianapproachesforsolvingadvection, convection,transport,Navier–Stokes,andVlasov–Maxwellequations.Wewillshowapplicationsinnumericalweatherandoceanprediction,hydrologyglaciermovement,volcanicash transport,airpollution,oceanridgehydrothermalmodels,tonameafew,toshowhowfar fetchingsemi-Lagrangianmethodsareused.Thefinalchapeterissolutionstoselectexercises.
Thatbeingsaid,wenowmoveontolearnaboutsemi-Lagrangianadvectionmethodsand theirapplicationsingeosciences.
Eulerianmodelingofadvection problems Asweshallmentionagainsoon,therearedifferentwaysofdescribingthemovementofa particle.Itcanbewithrespecttotheparticleitself,oritcanbewithrespecttotheparticles goingpastacertainfixedpoint.Itisthelatterdescriptionthatthenexttwochaptersare concernedwith.Inthischapterweshallpresentdifferentnumericalapproximationstothe one-dimensionalconstant-velocityadvectionequation.Forsomeofthenumericalschemes, weshallpresentplotsoftheirperformancewith advecting aGaussianbellcurveandadiscontinuousstepfunctionaroundaperiodicdomain.Weshallseesomebehaviorsoftheschemes thatweshallquantifyandverifyinthenextchapter.Westartthischapternowwithabrief summaryoftheone-,two-,andthree-dimensionalgeneralEulerianformsoftheadvection equation,followedbyacoupleofbriefderivationsoftheadvectionequation.
2.1Continuousformoftheadvectionequation Therearetwoapproachesthatcanbeusedtodescribetheprocessofadvection:Eulerian, wherewehaveavolumeelementthatisfixedinspaceinasetframeofreference,orLagrangian,wherethesurfaceofthevolumeelementis co-moving withthefluid,inthefluid’s frameofreference.ThusinEulerianmodelingofadvectionweareconsideringtheproblem oftheflowpastapoint,andassuchthecontinuousone-dimensionalgenerallinearadvection modelisgivenbythepartialdifferentialequation
(x,t )
+ u (x,t )
where ψ (x,t ) isthedependentvariable, t istime, x isthespatialcoordinate, u (x,t ) isthe advectionvelocity,whichcanbeafunctionofbothspaceandtime,andfinally, f (x,t ) is referredtoasthe forcingterm.Notethattheforcingtermcanbezero,linear,ornonlinear. Theadvectionvelocitycanalsobeconstant,linear,ornonlinear.
Ifwenowthinkthatthereisafieldthatistwo-dimensional,meaningthatitcanmovein bothhorizontaldirections,orinonehorizontalandtheverticaldirections,thenthereisan associatedversionoftheadvectionequationintwodimensions.Forexample,ifweconsider thehorizontalcase,thentheassociatedtwo-dimensionalpartialdifferentialequationforthis
situationisgivenby ∂ψ (x,y,t )
+ u (x,y,t ) ∂ψ (x,y,t )
(x,y,t )
)
)
where v (x,y,t ) istheadvectionvelocityinthe y direction,butitcanbeafunctionofthe otherspatialcoordinate.Finally,ifweareconsideringathree-dimensionalfieldthenitis possiblethatitcanbeadvectedwithrespecttoallthreespatialcoordinatesinaCartesian formulation,oranyothercoordinatesystem,andtheassociatedpartialdifferentialequation forthissituationisgivenby ∂ψ (x,y,z,t )
+ u (x,y,z,t )
(x,y,z,t )
w (x,y,z,t )
(x,y,z,t )
+ v (x,y,z,t )
(x,y,z,t )
where w (x,y,z,t ) istheadvectionvelocityintheverticaldirection.Notethatallthreeadvectingvelocitiescanbefunctionsofallthreespatialdimensions,alongwithtime.
Forthemultidimensionalequationsabove,(2.2)and(2.3),itispossibletosimplifythese equationsusingvector-differentialnotationas
isthespatial
gradientoperatorgivenby
Wenowconsiderhowtoderivetheone-dimensionaladvectionequationthatwepresented above.Weshallpresenttwodifferentapproaches:onebasedonadynamicaldescriptionof thesituation,andonethatusesamoremathematicalargument.
2.1.1Derivationoftheone-dimensionalEulerianadvectionequation Asjustmentioned,inthissectionweshallpresenttwodifferentapproachestoderivethe one-dimensionaladvectionequation.Weshouldnoteherethattheadvectionequationhas manydifferentnames,itisquiteoftenreferredtoasthetransportequation,butitisalso referredtoastheconvectionequation,alongwithbeingassociatedwiththewaveequation. Thefirstderivationthatweconsiderhereisadynamics-basedapproachemployingtheideas ofmassconservation,continuity,andincompressibility.
Massconservationderivationoftheadvectionequation Atthebeginningofthischapterwereferredtoavolumeelement,whichwewillnow denoteas dV asshowninFig. 2.1,where P (x,y,z) isthecentroidofthevolumeelement.As weareconsideringtheEulerianformulation,wehavethatthesidesofthevolumeelement
arefixedinspace.Therefore,wewishtoconsiderthesituationwherewehaveaflowinand outofthevolumeelementthroughthesides.
FIGURE2.1 Schematicofagenericvolumeelement, dV
Ifwedenotethemassdensityatthepoint P (x,y,z) as ρ (x,y,z),whichrepresentsthe massdividedbythevolume,thenweassumethatitisanaverage,andnearlyuniformmass densityinallof dV .Thusthetotalmass, M ,thatiswithin dv isgivenbytheintegralquantity
Thenextassumptionthatwemakeisthattherearenosourcesorsinksofmasswithin dV .As suchwehavethat dM dt representstherateatwhichmassentersorexitsthevolumethrough thesurface dS .
Whenweareconsideringthethree-dimensionalvolume,therearemultiplesurfacesofthe volume,assuchwehaveasideofthevolumedenotedby d S,andtheunitnormalvectordenotedby n,whereitisassumedthatthenormalvectorpoints“out”ofthevolumeelement. Thenextimportantquantitythatwehavetointroduceisthe flux ofthemass,whichrepresentsmassperunitarea,perunittime,thatwehavepassingthroughasurface,andwhichis givenby ρ v,where v isthefluidvelocity.
Wenowrequirethatthemassperunittimethatisflowingthroughthesideofthevolume, d S,begivenby ρ v d S ≡ ρ v ndS ,andthusthetotalrateofflowofmassoutofthe volumeelement dV isgivenbythesumofthemassperunittimeflowingthrougheachof thesurfaces/facesofthevolumeelement.Thusincontinuousformthisisgivenbythesurface integrals
Asmentionedbefore,wehave dM dt representingtherateatwhichmassisenteringor exitingthevolumeelementthroughthesurface,andassuchwerequire(2.6)tobeequalto
2.Eulerianmodelingofadvectionproblems
dM dt ,whichimpliesthat
AsweareconsideringafixedEuleriansurface,weareabletobringthetotaltimederivative d dt insidethevolumeintegralasapartialderivative,whichthengivesus
Whiletheexpressionin(2.8)maylookquiteintimidating,wenowintroduceatheoremthat willhelpushere.Thistheoremisreferredtoas Gauss’Theorem andisstatedbelow.
Theorem2.1 (Gauss’Theorem). Let V bearegioninspacewithboundary ∂V .Thenthevolume integralofthedivergence ∇· F of F over V andthesurfaceintegralof F overtheboundary ∂V of V arerelatedby
Gauss’Theoremisquiteoftenreferredtoasthe divergencetheorem,wherethetheoremis themathematicalexpressionforthephysicalsituationthat,intheabsenceofthecreationor destructionofmatter,thedensitywithinaregionofspacecanchangeonlybyhavingitflow inoroutfromtheregionthroughitsboundary,whichisthesituationthatwehavehere.
UponapplyingGauss’theoremtothesituationabove,weobtain
Therefore,wenowhave
Werequiretheexpressionin(2.11)toholdforallarbitraryshapedvolumes,andthustheonly waythattheintegralequationin(2.11)canbesatisfiedisiftheintegrandisequaltozero.This thenimpliesthat
Eq.(2.12)isreferredtoasthe continuityequation,whereitisdescribingthe conservation ofmass withintheEulerianframeofreference.Wecanseethatthisisalmosttheadvection equationthatwepresentedearlier,butwehavetomakeonemoreassumption,namelythat thefluidisincompressible,whichimpliesthatthedensitywithinthevolumeisconstant. Anotherwayofinterpretingthisassumptionisthatthedivergenceoftheflowvelocityis
zero.Mathematicallythenon-divergenceofthevelocityfieldisequivalentto ∇· v = 0.Ifwe nowexpandtheterm ∇· (ρ v) wehave
(ρ
ρ, wherewecanseethatwerequirethedivergenceofthevelocity.Usingtheincompressibility assumptionresultsinthecontinuityequationforanincompressiblefluidas
whichisthevectorialformofthethree-dimensionaladvectionequationthatwepresented in(2.4),butwherethereisnoexternalforcingterm.
Taylorseriesexpansionderivationoftheone-dimensionaladvectionequation
Thisderivationoftheadvectionequationisfortheone-dimensionsituation,whichstarts byconsideringtheone-dimensionaldriftofanincompressiblefluid.Therefore,let ψ (x,t ) representaparticledensitythatonlychangesduetoadvectionprocessessothat
Ifwenowassumethat t issufficientlysmallsothatwecanapplyaTaylorseriesexpansion toeithersideof(2.14),thenweobtain,tofirstorder,
whichsimplifiesto
where u isanonzeroconstantvelocity.
Asanasidehere,weshouldnotethataswearedealingwithapartialdifferentialequation.Therearethreedifferenttypesthatapartialdifferentialequationcanbe:hyperbolic, parabolic,orelliptical,whichrepresentmanydifferentphysicalandgeophysicalprocesses. Thepartialdifferentialequationin(2.16)ishyperbolic.Thustheuniquesolutionof(2.16)is determinedbyaninitialcondition,denotedby
,suchthat
and
Oneapproachtosolvetheproblemaboveisbythe MethodofCharacteristics.
2.1.2Methodsofcharacteristics
Thefirstthingtonoticehereisthattheadvectionequationin(2.16)canbefactorizedas
Thenextstepistotransform x , t tonewvariables r , s suchthatthederivativetransformsas
Ifitispossibletointroducethischangeofvariable,thenwewouldhavethat ∂ψ ∂r = 0,where thisequationiseasiertosolve.Therefore,ifwelet x = x (r,s ) and t = t (r,s ),then,applying thechainrule,weobtain
Ifwenowcompare(2.20)with(2.18),weseethat ∂x ∂r = u and ∂t ∂r = 1.Integratingthese twoequationsresultsin x = ur + q (s ) and t = r + p (s ).Wenowusethe s dependenceby applyingthepropertythattheinitialconditionsspecifythesolutionalongthe x -axis.Tomake theapplicationoftheinitialconditionsinthetransformedspace,weintroducethecondition thatthe x -axisat t = 0 transformsontothe s -axis (r = 0);specifically,weneed r = 0,implying that t = 0 and x = s .Thusbysetting r = 0 and t = 0,wehave q (s ) = s and p (s ) = 0,implying that
(2.21)
Thenextstepistoinvertthetransformationabove,whichresultsin r = t and s = x ut , whichthenenableustowrite(2.18)intheform ∂ψ
= 0,implyingthat ψ = ψ (s ) = ψ (x ut ).
Giventheinitialconditions,wethereforecanconcludethatthesolutionoftheadvection problemis
Therefore,theexpressionin(2.22)definesatravelingwavethatmoveswithspeed u,andit isalsoseenthatthesolutionisconstantalongthelinesoftheform x ut = β where β is aconstant.Theselinesarereferredtoasthe characteristics forthepartialdifferentialequation,andthemethodthatwasappliedheretofindthesolutionisknownasthe methodof characteristics
InFig. 2.2 wehaveanillustrationoftheappearanceofthecharacteristicsfortheadvection equation,wereweseethatthecharacteristicsaredefinedbytheirvalueastheycrossthe x -axis,butthenhavethatvaluealongthewholepath.
Tohelpillustratethisapproach,weshallconsidertheexampleofastepfunction,which weshallrevisitmanytimesthroughoutthebookindifferentforms.Theinitialconditionsthat definethestepfunctionshowninFig. 2.3 aregivenby
Thusfrom(2.22)thesolutionis
FIGURE2.2 Exampleoftheconstantcharacteristicsfortheone-dimensionallinear,constantvelocity,advection partialdifferentialequation,withinitialconditions ψ (x, 0) = (x )
FIGURE2.3 Schematicofadvectingastepfunctionstartingatinitialtimeinthetopplot,andthenafteratimeof 1inthelowerplot.
2.Eulerianmodelingofadvectionproblems orequivalently,
0 (x ) = 1,ut ≤ x ≤ 1 + ut, 0,x<ut,x> 1 + ut. (2.25)
InFig. 2.3 wehavepresentedboththeinitialstepfunction,asmentionedabove,andthe stepfunctionatafuturetime t ,wherethesolutionistheoriginalstepfunctionthathasbeen movedtotheinterval ut ≤ x ≤ 1 + ut
Thereareafewpropertiesofthelinearconstantvelocityequationthatareusefulwhen assessingtheperformanceofthenumericalapproximationtoit:
1. Theinitialprofileispreserved;thisistrueforjumpsanddiscontinuities.
2. Theinformationoftheinitialprofiletravelsatspeed u,andtheassociateddirectionofthe travelisone-wayonly.
3. Thesolutionatagivenspatiallocation x andattime t isentirelydeterminedbythevalues oftheinitialconditionsat x = x ut .Wehaveprovidedaschematicofthissituationin Fig. 2.4.
FIGURE2.4 Schematicofback-tracingofthecharacteristicstotheinitialconditionfromthepoint x, t .
GiventhecontinuousEulerianapproachforadvection,wenowconsiderseveraldifferent numericalapproximationtotheone-dimensionallinearadvectionequationwithconstant velocity.Theapproachesthatweshallshowinthenextsectionarereferredtoa finitedifferencing scheme,wherewewillpresentexplicitandimplicitschemes,andacombinationof thetwo,whichcanbemultistepalgorithmsreferredtoaspredictor–correctormethods.
2.2FinitedifferenceapproximationstotheEulerianformulationofthe advectionequation
Inthissectionweshallintroducemanydifferentapproximationstothelinear,zero-forcing versionoftheone-dimensionaladvectionequation.Thefirstthingthatwehavetointroduce hereistheconceptofthenumericalgridforbothtimeandspace.
InFig. 2.5 wehavepresentedauniformdistributedsetofgridpoints,whereforthissetup wehaveequalspacingintimeandspace,althoughweshouldnotethatspacinginthetemporaldirectionisnotalwaysthesamesizeasthatinthespatialdirection.Wecanseethat
thereisadistancebetweenthepointsinbothtimeandspace.Thedistancebetweenthegrid pointsinthe x -directionisdenotedby x ,whereasthedistancebetweenthetimegridpoints isdenotedby t ,andtheyarereferredtoasthegridspacingandtimestep,respectively.
Illustrationofanumericalgrid.
Ifwestartwiththespatialdirection,thenweintroducetheindex i forthe x direction,while forthetemporaldirectionwithhavetheindex n.Thusatpoint (i,n) wehaveanumerical approximationtothecontinuoustracer, ψ (x,t ),as ψ (xi ,t n ),whichisusuallydenotedby ψ n i ,for i = 1, 2,...,I and n = 0, 1,...,N .Pleasenotethatthesuperscript n heredoesnot representapowerofthefunction,itissimplythetemporalindex.Therefore,ifweconsider the x directionderivative,thenwecouldusethefollowingapproximation:
wheretheapproximationin(2.26)isreferredtoasan upwind scheme. Itisalsopossibletodefineanupwindschemeforthetemporalderivativeforthenumerical approximationtothetimecomponentaswell,whichisgivenby
Theapproachespresentedhereareexamplesofwhatisreferredtoas finitedifference approximationstothegradientsofthefieldinthepartialdifferentialequation.Giventhe conceptofthegridpointsintroducedabove,weshallnowpresentaseriesofdifferentexplicitnumericalandimplicitnumericalschemes,andtheschemesthatcombineexplicitand implicitfeaturesfortheEulerianformulationoftheone-dimensionaladvectionequation.
2.2.1UpwindforwardEuler
Dependingonthesignoftheadvectivevelocity,wecandefinetwodifferentversionsof thisscheme.If u> 0,thentheversionofthe forwardEuler approximationisbaseduponthe upwindapproximationtothespatialderivativepresentedabovecombinedwiththetemporal
FIGURE2.5
2.Eulerianmodelingofadvectionproblems approximationaboveaswell,whichresultsin
Toimplementthisnumericalschemepresentedin(2.28),werearrange(2.28)as
where C ≡ u x t isreferredtoasthe Courantnumber,whichwillbecomeimportantinthe nextchapter.
InFig. 2.6 wehavepresentedthestencil,whichisthesetofpointsthatareusedinthis approximation,fortheforwardEuler,usinganupwindapproach,scheme.Itshouldbenoted thatforwardEulerreferstotheforwardapproximationofthetemporalcomponent.Forall ofthestencilplotsthatweshowinthischapter,thegreencirclesarethosethatareusedto createthatfinitedifferencescheme.
FIGURE2.6 IllustrationofthestencilfortheupwindforwardEulerfinitedifferencescheme.
Iftheadvectivevelocityis u< 0,thenwecandefineadownwindapproximationtothe spatialderivative,andifweusetheforwardEulerapproximationforthetimecomponent, thenweobtain
Toimplementthisnumericalschemepresentedin(2.30),werearrangeitas
Notes:(1)Thereasonwhytheschemein(2.30)isnotreferredtoasthebackwardEuleris becausethatnamebelongstotheimplicitversionsofthetimederivativethatweshallpresent later.(2)Eqs.(2.28)and(2.30)arereferredtoasthe differenceequations forthecontinuous partialdifferentialequation.
Throughoutthischapterweshallassesstheperformanceofsomeoftheadvectionschemes withtheadvectionofastepfunction,aswellasaGaussianbellcurve.Beforethatweinvestigatesomepropertiesofeachschemeinthenextchapter.Aswesawearlier,thecharacteristics
2.2.FinitedifferenceapproximationstotheEulerianformulationoftheadvectionequation 17
tellusthattheshapesshouldsimplybemovingalong,withoutanydistortions,withadvectionspeed u,withoutslowingdownorspeedingup.
UpwindforwardEulerwiththebellcurve Forthefirstsetofplotsforthistestcase,weshallconsideranadvectionspeedof u = 1, andspatialandtemporalgridspacingof x = t = 0 1 units.Thebellcurveisdefinedby
wheretheparameter μ determineswherethebellisinitiallycentered,and σ determinesthe spreadofthecurve.WeshouldnotethatthisissimilartothedefinitionoftheGaussian distribution,exceptthatwedonothavethenormalizingfactorof 1
2πσ
Forthedemonstrationsthatwepresentinthischapter,weuse μ = 5,implyingthatthebell willinitiallybecenteredat x = 5,and σ = 0.5.TheinitialprofileispresentedinFig. 2.7.
FIGURE2.7 Plotofthetruebellcurveforthesituationwhen μ = 5 and σ = 0 5 at t = 0
GiventheinformationpresentedhereweknowthattheCourantnumberforthisconfigurationis C ≡ u t x = 1.Weareusingaspatialdomainof x ∈ [0, 10],andatemporaldomainof 0 ≤ t ≤ 10.Thismeansthatthereare101 x gridpointsand101 t pointsforthisconfiguration. Weareusingaperiodicityconditionastheboundaryconditionsforeitherendofthedomain,
2.Eulerianmodelingofadvectionproblems andwiththeadvectionspeedof 1,thisthenmeansthatthebellcurvecantakeonecomplete circuitofthedomaininthistime.
Weareabletoplotthetruesolutionateverytimestepasweknowthatitisgivenby
where n istheindexforthespecifictimestepthatthenumericalmodeisat.Thiswillenable ustoquantifythe error associatedwitheachnumericalscheme.
InFig. 2.8 wehaveplottedboththetrueandnumericalsolutionsfortheconfiguration shownatevery10timesteps.Weshouldpointoutthatthereisnoerrorinthisplot,simply since,when C = 1,theupwindforwardEulerapproximatesthecontinuousbellfunctionvery well.
FIGURE2.8 Plotofthetruebellcurve,inred,andthenumericalsolutionfromtheupwindforwardEulerscheme for tn = 10 t, 20 t,..., 90 t steps.
Wewillnowlookattheeffectsofchangingthespatialstepsize.InFig. 2.9 wehavepresentedasimilarplottothatinFig. 2.8,butnowwehavethesituationwhere C = 1 25,which ariseskeeping t = 0 1 buthavingincreasedtheresolutionofthe x directionto x = 0 08 Whilethischangemaynotseemlikeabigdeal,wecanseefromFig. 2.9 thatweareno longermaintainingthestructureofthebellcurves.Wecanclearlyseesomedampingofthe curvewithin20timesteps,andthat,bythetimewearriveat70timessteps(panel7),wecan clearlyseethatthesolutionisstartingtodisplaysomefeaturebehindthebellcurve,andby 90timesstepswecanseedistortionsbothbehindthewaveandalsotravelingfurtherbehind thecurve.Thiscouldindicatearelationshipbetween C , t ,and x .
FIGURE2.9 PlotoftheupwindforwardEulerschemeforthebellcurvewhentheCourantnumberis C = 1 25.
Thenextsetofplotsthatwepresentforthissituationwiththisnumericalschemearefor thecasewhenwehave C = 0 5.Herewearetryingtosavesomestoragespaceandalsospeed upthenumericalintegration.Wehavekept t = 0.1,butnow x = 0.2.Thefirstfeature tonoticeinFig. 2.10 isthattheupwindforwardEulerschemeshowwhatisreferredtoas damping ofthebellcurveastimegoeson;onapositivenote,wecanseethattheschemeisnot outofphase withthetruesolution.Therefore,byreducingthenumberofspatialgridpoints, wehaveactivatedanotherfeatureofthenumericalschemethatwasnotpresentwhenwe hada higherspatialresolution.
Asmentionedearlier,thisupwindforwardEulerissupposedtobeusedwhen u> 0,and wenowpresentaplotoftheaffectsofapplyingthisschemewhen u< 0.InFig. 2.11 wehave plottedtheresultsat10and20timesstepsofapplyingtheupwindforwardEulerscheme when u< 0.Weseethatafter10timessteps,thenumericalsolutionsisstartingtogooutof phasewiththetruesolution,andisalsointroducingan amplitudeerror,aswellasovershoots. However,only10timestepslater,weseethatthenumericalsolutionisnothinglikethetrue solutionandisstartingtoblowup.Therewasnopointinproducingoneofthesetsof9plots forthiscaseastheschemebecamecompletelyunstableafterafewtimesteps.
Wenowmoveontothediscontinuouscaseofthestepfunctionwiththesamesetupused heretoseetheimpactthatdiscontinuitieshaveontheperformanceoftheupwindforward Eulerscheme.
