Finite volume evolution Galerkin methods for the shallow water equations with dry beds

有限体积法

FiniteVolumeEvolutionGalerkinMethodsforthe

ShallowWaterEquationswithDryBeds

ˇov´AndreasBollermann1,?,SebastianNoelle1andMariaLuk´aca-

Medvid’ov´a2

1

arXiv:1501.03628v1 [math.NA] 15 Jan 20152IGPM,RWTHAachen,Templergraben55,52062Aachen,Germany.DepartmentofMathematics,UniversityofTechnologyHamburg,Schwarzenbergstraße95,21073Hamburg,GermanyAbstract.WepresentanewFiniteVolumeEvolutionGalerkin(FVEG)schemeforthesolutionoftheshallowwaterequations(SWE)withthebottomtopographyasasourceˇov´term.OurnewschemewillbebasedontheFVEGmethodspresentedin(Luk´aca,NoelleandKraft,J.Comp.Phys.221,2007),butaddsthepossibilitytohandledryboundaries.Themostimportantaspectistopreservethepositivityofthewaterheight.Wepresentageneralapproachtoensurethisforarbitrary?nitevolumeschemes.Themainideaistolimittheoutgoing?uxesofacellwhenevertheywouldcreatenegativewaterheight.Physically,thiscorrespondstotheabsenceof?uxesinthepresenceofvacuum.Well-balancingisthenre-establishedbysplittinggravitationalandgravitydrivenpartsofthe?ux.Moreover,anewentropy?xisintroducedthatimprovesthereproductionofsonicrarefactionwaves.AMSsubjectclassi?cations:65M08,76B15,76M12,35L50PACS:02.60.Cb,47.11.Df,92.10.SxKeywords:Well-balancedschemes,Dryboundaries,Shallowwaterequations,EvolutionGalerkinschemes,Sourceterms1IntroductionTheshallowwaterequations(SWE)areamathematicalmodelforthemovementofwaterundertheactionofgravity.Mathematicallyspoken,theyformasetofhyperboliccon-servationlaws,whichcanbeextendedbysourcetermslikethein?uenceofthebottomtopography,frictionorwindforces.Inthiscase,wewillspeakofabalancelaw.Forsimplicity,thisworkwillconsiderthevariationofthebottomastheonlysourceterm.

Manyimportantpropertiesofthemodelrelyonthefactthatthewaterheightisstrictlypositive.Despitethis,typicalrelevantproblemsincludetheoccurrenceofdryareas,likedambreakproblemsortherun-upofwavesatacoast,withtsunamisasthe?Correspondingauthor.Emailaddress:bollermann@igpm.rwth-aachen.de(A.Bollermann)

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mostimpressiveexample.Soforsimulationsoftheseproblems,wehavetodevelopnu-mericalschemesthatcanhandlethe(possiblymoving)shorelineinastableandef?cientway.Anothercrucialpointinsolvingbalancelawsisthetreatmentofthesourceterms.Forprecisesolutions,itisnecessarytoevaluatethesourceterminsuchawaythatcertainsteadystatesarekeptnumerically,i.e.thenumerical?uxandthenumericalsourcetermcanceleachotherexactlyforequilibriumsolutions.

Inthelastyears,manygroupscontributedtothesolutionofthedif?cultiesdescribedabove.In[?],Audusseet.al.proposedareconstructionprocedurewherethefreesurfaceandwaterheightarereconstructedandthebottomslopesarecomputedfromthese.Thisguaranteesthepositivityofthewaterheightandgivesawell-balancedschemeatthesametime.BegnudelliandSandersdevelopedaschemefortriangularmeshesincludingscalartransportsin[?].Theyproposedastrategyhowtoexactlyrepresentthefreesurfaceinpartiallywettedcells,leadingtoimprovedresultsatthewetting/dryingfront.In[?],Brufauet.al.analysehowtodealwith?owonanadverseslope.Theylocallymodifythebottomtopographyincertainsituationstoavoidunphysicalrun-upsorwavecreationatthedryboundary.Gallardoet.al.discussedvarioussolutionsoftheRiemannproblematthefrontandusedtheminamodi?edRoescheme.TheythenusedthelocalhyperbolicharmonicmethodfromMarquina(cf.[?])inthereconstructionsteptoachievehigherorder,see[?].KurganovandPetrovaproposedacentral-upwindschemethatiswell-balancedandpositivitypreservingin[?].Itisbasedonacontinuous,piecewiselinearapproximationofthebottomtopographyandperformsthecomputationintermsofthefreesurfaceinsteadoftherelativewaterheighttosimplifythewell-balancing.ThelastfeatureisalsoabuildingblockintheworkofLiangandMarche[?].Theyalsoprovideamethodtoextendthiswell-balancingfeaturetosituationsincludingwetting/dryingfronts.LiangandBorthwick[?]usedadaptivequad-treegridstoimprovetheef?ciencyoftheirschemes.Wettinganddryingeffectsarehandledaswellasfrictionterms.Inthecontextofresidualdistributionmethods,RicchiutoandBollermanndevelopedapositiv-itypreservingandwell-balancedschemeforunstructuredtriangulations[?].

ˇov´The?nitevolumeevolutionGalerkin(FVEG)methodsdevelopedbyLuk´aca,Mor-

tonandWarnecke,cf.[?,?,?],havebeensuccessfullyappliedtotheSWEin[?].Theyarebasedontheevaluationofsocalledevolutionoperatorswhichpredictvaluesforthe?nitevolumeupdate.Thankstotheseoperators,theschemestakeintoaccountalldirectionsofwavepropagation,enablingthemtopreciselycatchmultidimensionaleffectsevenonCartesiangrids.Theseschemesshowaverygoodaccuracyevenonrelativelycoarsemeshescomparedtootherstateoftheartschemesandtheyarealsocompetitiveintermsofef?ciency(cf.[?]).

However,theexistingFVEGschemesarenotabletodealwithdryboundaries.Thusinthisworkwewillpresentamethodtopreservethepositivityofthewaterheightwithanarbitrary?nitevolumemethod.Toachievethis,wereducetheout?owondrainingcellssuchthatthewaterheightdoesnotbecomenegative.Wewillthenprovidethemeanstopreservethewell-balancingpropertyunderthepresenceofdryareas,andapplybothtechniquestoanewFVEGmethod.Inaddition,wepresentanewentropy?xfor

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theFVEGschemesthatimprovesthereproductionofsonicrarefactionwaves.

WestartourpaperwithashortpresentationoftheSWEinSection2.Section3de-scribestheFVEGmethodwewillstartfrom.Thearisingdif?cultiesbyintroducingdryareasandmeanstoovercomethemaredescribedinSection4,whichisthemainpartofthepaper.Finally,inSection5,wewillshowselectednumericaltestcasesthatdemon-stratetheperformanceofourschemes.

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2.1

TheShallowWaterEquations

BalanceLawForm

Weconsidertheshallowwatersysteminbalanceform

?u

+?·F(u)=?S(u, x).?t

Theconservedvariablesandthe?uxaregivenby

h

u=?hv1?,

hv2

?

?

hv1

2?2

F(u)=(F1(u)F2(u))=?hv1+ghhv1v2

?

hv2

?

hv1v2?,

h2

hv2+g2?

(2.2)(2.1)

wherehdenotestherelativewaterheight, v=(v1,v2)Tthe?owspeedandgthe(constant)

gravityacceleration.ThesourcetermS(u, x)isgivenby

??0x)???b(

(2.3)S(u, x)=gh?1?

?b( x)

2

withb( x)thelocalbottomheight.Wealsointroducethefreesurfacelevel,ortotalwaterheight,

H( x)=h( x)+b( x)(2.4)andtheso-calledspeedofsound

c=??.

(2.5)

Thisisthevelocityofthegravitywavesandshouldnotbeconfusedwiththephysical

soundspeedinair.

2.2Quasi-linearForm

ForthederivationoftheevolutionoperatorsinSection3.2,itishelpfultorewrite(2.1)inprimitivevariables.Thesystemthentakestheform

wt+A1(w)wx1+A2(w)wx2=t

3

(2.6)

有限体积法

January16,2015with

??h

w=?v1?,

v2?

v1h0

A1=?gv10?,

00v1

?0

t=??gbx1?.

?gbx2

?

?

?

v20h

A2=?0v20?

g0v2

?

(2.7)

andthesourceterm

(2.8)

Foreachangleθ∈[0,2π)wede?nethedirection ξ(θ):=(cosθ,sinθ).Assystem(2.1)ishyperbolic,foreachofthesedirectionsanda?xedwthematrix

A(w)= ξ1A1(w)+ ξ2A2(w)

hasrealeigenvalues

λ1= v· ξ?c,

λ2= v· ξ,

λ3= v· ξ+c

(2.10)(2.9)

andafullsetoflinearlyindependenteigenvectors

??1θ?r1=?gcos,θgsin?

?

r2=?sinθ?,

?cosθ

?

?

θ?r3=?gcos.θgsin1

?

(2.11)

2.3LakeatRest

Atrivial,butneverthelessimportantsolutionto(2.1)isthelakeatrestsituation,where

thewaterissteadyandthefreesurfacelevelisconstant,i.e.wehave

v=(0,0)TandH( x)=H0.

From(2.4)weimmediatelyget

(2.12)

?h=??b

andtherefore(with(2.1)–(2.3)and v=(0,0)T)

?

00

?b( x)?2

?gh?+?0?=?gh??1?.2

?b( x)0xgx1

2

2

(2.13)

???

?

?

(2.14)

Aschemeful?llingadiscreteanalogonof(2.14)exactlyiscalledwell-balanced.

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3FVEGSchemes

Finitevolumeschemesareverypopularforsolvinghyperbolicconservationlawsforseveralreasons.Theyrepresenttheunderlyingphysicsinanaturalwayandcanbeim-plementedveryef?ciently.Nevertheless,nearlyallofthemarebasedonthesolutionofone-dimensionalRiemannproblemsandtherewithadimensionalsplitting.Thisintro-ducessomesortofabias:Wavepropagationalignedwiththegridisverywellrepre-sented,whereaswavesobliquetothegridcannotbecaughtasaccurate.

ˇov´InthelastdecadeLuk´acaet.al.developedaclassof?nitevolumeevolutionGalerkin

schemes,seee.g.[?,?,?].TheFVEGschemeisapredictor-correctormethod:Inthepre-dictorstepamultidimensionalevolutionisdone,thecorrectorstepisa?nitevolumeupdate.

Inthissectionwewillrecallthesecondorderschemepresentedin[?].ThismethodwillbethestartingpointforourextensionsforcomputationsincludingdrybedsinSec-tion4.Thereforeweconcentrateonthepropertiesplayingaroleinthiscontextandlimitourselvestothemainideasotherwise.

3.1FiniteVolumeUpdate

Forourcomputations,weuseCartesiangrids,i.e.wedivideourcomputationaldomain?inrectangularcellsCi,separatedbyedgesE.Ontheedges,wehavequadraturepoints xk.Thesubscriptiwillalwaysrefertoacell,whereaskasasubscriptisusedasaglobalindexforquadraturepoints.Ifwetalkaboutthelocalquadraturepointsonasingleedge,weusetheindexjinstead.

Oneachcellwede?netheinitialvalueatas

1u0:=u(0)≈ii|Ci|??Ciu( x,0)d x(3.1)

whereweuseaGaussianquadraturetoapproximatetheintegral.Integrating(2.1)oneachcell,wecanthende?netheupdateas

1uin+1=uin?|Ci|??tn+1????tn?CiF(u( x,t))· nd x+??CiS(u( x,t), x)d xdt??(3.2)

usingtheGausstheorem.HereuindenotescellaverageinCiattimetnand nistheouternormal.Thesolutiononthewholedomainattimetnisthende?nedas

Un( x):=U( x,tn)=uin, x∈Ci.(3.3)

Foranapproximationof(3.2),oneachedgewede?nethreequadraturepoints xj,j=1,2,3,seeFig.1.Thesequadraturepointsarelocatedonthevertices(j=1,3)andthecentre(j=2)ofanedge.The?uxovertheedgeisapproximatedbyusingmidpointruleintimeandSimpson’sruleinspace,hencewewillusetheevolutionoperatorsfromSection3.2

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