Finite difference and pseudospectral methods appli

iFiniteDifferenceandPseudospectralMethodsappliedtotheShallowWaterEquationsinSphericalCoordinatesbyDAVIDWARRENMERRILLB.A.,BatesCollege,1992AthesissubmittedtotheFacultyoftheGraduateSchooloftheUniversityofColoradoinpartialfulfillmentoftherequirementforthedegreeofMastersofArtsDepartmentofMathematics1997iiThisthesisfortheMastersofArtsdegreebyDavidWarrenMerrillhasbeenapprovedfortheDepartmentofMathematicsby__________________________________BengtFornberg__________________________________RichardHolley__________________________________ArlanRamsay__________________________________RuedigerJakob-ChienDate_______________iiiMerrill,DavidWarren(M.A.,Mathematics)FiniteDifferenceandPseudospectralMethodsAppliedtotheShallowWaterEquationsinSphericalCoordinatesThesisdirectedbyProfessorBengtFornbergTheshallowwaterequationsareasetofequationsusedtomodelmanyfluidflows.Theyareparticularlywellsuited-andoftenused-totestnumericaltechniquesforweatherprediction.Inthisstudywecarrythroughtwonumericaltestcasesinvolvingtheshallowwaterequations.Wedothisusingthreedifferentnumericalmethodsforcalculatingderivatives;secondandfourthorderfinitedifferences,andapseudospectralmethod.Wewillshowthat,byusingourpseudospectralmethod,i.wegetsimilaraccuracycomparedtoaparticularimplementationofaspectraltransformmethodbasedonsphericalharmonictechniquesandfarhigheraccuracythanwithfinitedifferences(forthesametwotestcases),ii.theoperationcountforeachtimestepismuchlowercomparedtotheabovementionedspectraltransformmethod,iii.ourlongitude-latitudegridallowsforaparticularlyeasyformulationofthecodeand,iv.polesingularitieswillnotcauseanydifficulties.ivAcknowledgmentsIwouldliketoexpressmysincereappreciationtoBengtFornberg,mythesisadvisor,forgivinghistimetomegenerously,alwayspatientlyexplainingnewanddifficultconcepts,andalwaysbeingencouragingduringthepastyear.IwouldalsoliketothankRuedigerJakob-Chienformeetingwithuseveryfewweeksduringthelastsixmonths.Hegaveusinsightsonhowthetestcasesshouldwork,helpedfindabugintheRunge-Kuttacode,andmademanycommentsonvariousdraftsofthethesis.IwouldliketothankmyfriendsintheAppliedMathDepartment,RudyHorne,ChristinaPerez,BernardDeconinck,andChrisHigginson,forlettingmesharemyexcitementandfrustrationswiththemduringthepastyear.Finally,Iwouldliketothankmywife,EmmaHolder,forputtingupwiththestressofpreparingforthedefense,thanksbabe.vTableofContentsI.Introduction11.1TheShallowWaterEquations31.2TheNumericalMethod81.2.1TheDomain81.2.2GridPointEquations91.2.3SpatialSecondOrderFiniteDifferences101.2.4SpatialFourthOrderFiniteDifferences141.2.5ThePseudospectralMethod181.2.6PseudospectralDerivatives221.2.7Smoothing251.2.8TimeStepping301.2.9OperationCount311.2.10ErrorAnalysis33II.TwoCases362.1Williamson'sFirstCase362.1.1Acomparisonofthedifferentnumericalmethods382.1.2ErrorAnalysis432.2Williamson'sSecondCase462.1.1ErrorAnalysis47III.Conclusion48BIBLIOGRAPHY49viAPPENDIX51A.SoftwareandGraphicsdescription51B.CodeListing51B1.Case151B2.Case255B3.Calculatingtheanalyticsolution60B4.Transformingcoordinates62B5.Calculatingderivatives63B6.FastFourierTransform67B7.Smoothing68B8.Calculatingnorms69B9.Thesurfaceintegral71B10.Formattingoutput72viiTABLES1.2.2-1LatitudeateachgridrowwithM=5.91.2.7-1Numberofmodesalteredpergridrow.281.2.9-1GridresolutionandOperationcountforvariousM.321.2.9-2GridresolutionandOperationcountforvariousn.32FIGURES1.2.1-1Unrollingthesphereontoatwodimensionalgrid.81.2.3-1FD2schemeforφderivatives.101.2.3-2FD2edgeφderivatives.111.2.3-3FD2schemeforθderivatives.111.2.3-4FD2edgeθderivatives.131.2.4-1FD4schemeforφderivatives.141.2.4-2FD4rightedgeφderivatives.151.2.4-3FD4φderivativesonegridpointinfromrightedge.151.2.4-4FD4φderivativesonegridpointinfromleftedge.151.2.4-5FD4schemeforθderivatives.161.2.4-6FD4derivativesforθalongtopandbottomedge.171.2.5-1PSderivativesforφ.191.2.5-2PSderivativesforθ.211.2.5-3WhatS1containsatthebeginningoftheIloop.211.2.6-1ModesinarraysAandB.231.2.6-2NewarrangementofmodesinarraysAandB.231.2.7-1ModesinarraysWKSP(*,*).261.2.7-2NewarrangementofmodesinWKSP(*,*).27viii1.2.10-1Rotationofcosinebellwithα=0.341.2.10-2Transformationtonew′φ,′θ()coordinatesystem.342.1-1Flowstructureonthesphereforα=π2.372.1-2InitialcosinebellforCase1(M=32).382.1.1-1CosinebellafteronerevolutionwithFD2scheme(M=32).392.1.1-2CosinebellafteronerevolutionwithFD4scheme(M=32).392.1.1-3CosinebellafteronerevolutionwithPSscheme(M=32)402.1.1-4InitialcosinebellforCase1(M=16).412.1.1-5CosinebellafteronerevolutionwithFD2scheme(M=16).412.1.1-6CosinebellafteronerevolutionwithFD4scheme(M=16).422.1.1-7CosinebellafteronerevolutionwithPSscheme(M=16)422.1.2-1l1,(dotted)l2,(dashed)andl∞(solid)normsforPSscheme(M=16).432.1.2-2l1,(dotted)l2,(dashed)andl∞(solid)normsforWilliamson(T31).442.1.2-3l∞normforα=0M=32,PSscheme.452.2.1-1l1,(dotted)l2,(dashed)andl∞(solid)normsforCase2PSrun(M=32).471FiniteDifferenceandPseudospectralMethodsappliedtotheShallowWaterEquationsinSphericalCoordinates.Chapter1IntroductionTheshallowwaterequationsareasetofequationsusedtomodelmanyfluidflows.Theyareparticularlyw