A Godunov-type finite volume method for the system

AGodunov-TypeFiniteVolumeMethodforSystemsofShallowWaterEquationsS.ChippadaC.DawsonM.MartinezM.WheelerCRPC-TR97687January1997CenterforResearchonParallelComputationRiceUniversity6100SouthMainStreetCRPC-MS41Houston,TX77005submitted2/97AGodunov-typeFiniteVolumeMethodfortheSystemofShallowWaterEquationsS.Chippada,C.N.Dawson,M.L.Martinez,andM.F.WheelerJanuary23,1997AbstractAnitevolumebasednumericalalgorithmhasbeendevelopedforthenumericalsolutionofthesystemofshallowwaterequations.ThealgorithmisaGodunovtypemethodandsolvestheRiemannproblemapproximatelyusingRoe’stechnique.Thealgorithmisdevelopedin2-Dwitharbitrarytriangulationsandconservesallprimaryvariablessuchasmassandmomentum.Theprocedureisimplementedonsomesimpletestcasesandsomecomplexcoastalowproblems.Thealgorithmisshowntoproduceexcellentresultswithoutspuriousoscillationsandagreesverywellwithknownanalyticalresultsandpredictionsmadebywaveequationformulationsoftheshallowwaterequations.ThebasicGodunovmethodisalsoextendedtosecond-orderaccuracythroughaslope-limitertypealgorithm.1INTRODUCTIONTheShallowWaterEquations(SWE)areusedtodescribefreesurfacehydrodynamicsinverticallywell-mixedwaterbodieswherethehorizontallengthscalesaremuchgreaterthantheuiddepth(i.e.,longwavelengthphenomena).TheSWEareobtainedbyassuminghydrostaticpressuredistribution,andbyintegratingthethree-dimensionalincompressibleNavier-Stokesequationsalongthedepthoftheuidbody.Mostofthenaturallyoccurringuidowsareturbulent,andauniformvelocityproleisassumedintheverticaldirection.Withtheseassumptions,thethree-dimensionalfreeboundaryproblemreducestoatwo-dimensionalxedboundaryproblemwiththeprimaryvariablesbeingtheverticalaveragesofthehorizontaluidvelocitiesandtheuiddepth(seeWeiyan(1992)foraderivationoftheshallowwaterequations).TheSWEcanbeusedtostudymanyphysicalphenomenaofinterest,suchasstormsurges,tidaluctuations,tsunamiwaves,forcesactingono-shorestructures,andcontaminantandsalinitytransport(Kinnmark,1985).NumericalsolutionoftheSWEismadechallengingduetomanyfactors.TheSWEareasystemofcouplednonlinearconservationlawswhichneedtobesolvedoncomplicatedphysicaldomainsarisingfromirregularcoast-linesandislands.Thebottomseabed(bathymetry)isoftenveryirregular.ShallowwatersystemsaresubjectedtoawidevarietyofphenomenasuchastheCoriolisforce,thesurfacewindstress,atmosphericpressuregradient,andtidalpotentialforces.Inadditiontothesephysicalfactorsthereare1additionaldicultiesarisingfromthemathematicalnatureoftheSWE.Mostimportantisthecouplingbetweentheuiddepthandthehorizontalvelocityeldwhichcouldleadtospuriousspatialoscillationsifthenumericalalgorithmsarenotchosenwithcare.SeveralnumericalalgorithmshavebeendevelopedovertheyearsfortheSWE.Thesenumericalalgorithmscanbeclassiedintotwobroadcategories.Intherstcategory,theprimitiveformoftheSWEthatareobtainedfromthedirectverticalintegrationofthe3DincompressibleNavier-Stokes,arenumericallysolved.However,astraightforwarduseofequal-orderinterpolationspacesintheniteelementcontextortheuseofnon-staggeredgridsinthenitedierencecontextcanleadtospuriousspatialoscillationsduetothenonlinearcouplingbetweentheuiddepthandthehorizontalvelocityeld.Thesespatialoscillationscanbeminimizedand/oreliminatedthroughtheuseofstaggeredgridsormixedinterpolationspaces.Forexample,KingandNorton(1978)approximatevelocitiesthroughpiecewisequadraticandelevationsusingpiecewiselinearbasisfunctions.SeveralnumericalmethodsbasedontheprimitiveSWEandequalorderapproximationshavealsobeendeveloped(e.g.,Kawaharaetal.(1982),Szymkiewicz(1993),ZienkiewiczandOrtiz(1995)).Ifthespuriousspatialoscillationsaresuppressedthroughcarefulsplittingbetweentheelevationandvelocityeld,thenumericalproceduresbasedonstaggeredorequalorderapproximationsaregenerallyconsideredtobemoreecientfromanimplementationpointofview.Inthesecondcategory,theprimitiveSWEarereformulatedandtherst-orderhyperbolicformoftheprimitivecontinuityequationisreplacedwithasecond-orderwaveequation(LynchandGray(1979),Luettichetal.(1991)).Clearly,theelevation-velocitycouplinghasplayedanimportantroleinthedevelopmentofnumericalalgorithmsforshallowwatersystems.InthispaperwetakeaslightlydierentviewoftheSWE.TheSWEareasystemofconservationlaws.MathematicallytheSWEareverysimilartothecompressibleEulerandNavier-Stokesequationswiththecompressibilitycomingfromthenitespeedofthesurfacegravitywave.ArichvarietyofnumericalmethodshavebeendevelopedforcompressibleEulerequations,andtheseareextendedtotheSWEinthispaper.Inparticular,weconsidertheGodunov-typenitevolumemethodwhichhasbeenshowntobeastable,monotonicprocedure(Godunov1959,Hirsch1990,LeVeque1992).Thisnumericalprocedureconservesmassandmomentumlocally,andcanmodeldiscontinuitiessuchasshocks.AlcrudoandGarcia-Navarro(1993)usedaGodunov-typenitevolumemethodfortheSWE.Thepresentpaperappliestheproceduretounstructuredmeshesbasedonlineartriangles.TheeciencyandaccuracyforrealisticcoastalowproblemsisestablishedandcomparisonsaremadewiththegeneralizedwavecontinuityequationformulationofLuettichetal.(1991).Inx2themathematicalmodelincludingtheboundary