Numerical Solution of Ordinary Differential Equati

NumericalSolutionofOrdinaryDifferentialEquationsE.S¨uliApril3,20131Contents1Picard’stheorem12One-stepmethods42.1Euler’smethodanditsrelatives:theθ-method....................42.2Erroranalysisoftheθ-method.............................72.3Generalexplicitone-stepmethod............................92.4Runge–Kuttamethods..................................132.5AbsolutestabilityofRunge–Kuttamethods......................193Linearmulti-stepmethods213.1Constructionoflinearmulti-stepmethods.......................223.2Zero-stability.......................................243.3Consistency........................................263.4Convergence........................................293.4.1Necessaryconditionsforconvergence......................303.4.2Sufficientconditionsforconvergence......................333.5Maximumorderofazero-stablelinearmulti-stepmethod..............373.6Absolutestabilityoflinearmultistepmethods.....................433.7Generalmethodsforlocatingtheintervalofabsolutestability............453.7.1TheSchurcriterion................................453.7.2TheRouth–Hurwitzcriterion..........................463.8Predictor-correctormethods...............................483.8.1Absolutestabilityofpredictor-correctormethods...............503.8.2Theaccuracyofpredictor-correctormethods.................524Stiffproblems534.1Stabilityofnumericalmethodsforstiffsystems....................544.2Backwarddifferentiationmethodsforstiffsystems..................574.3Gear’smethod......................................585NonlinearStability596Boundaryvalueproblems626.1Shootingmethods....................................626.1.1Themethodofbisection.............................636.1.2TheNewton–Raphsonmethod.........................636.2Matrixmethods......................................666.2.1Linearboundaryvalueproblem.........................666.2.2Nonlinearboundaryvalueproblem.......................696.3Collocationmethod....................................70Preface.Thepurposeoftheselecturenotesistoprovideanintroductiontocompu-tationalmethodsfortheapproximatesolutionofordinarydifferentialequations(ODEs).Onlyminimalprerequisitesindifferentialandintegralcalculus,differentialequationthe-ory,complexanalysisandlinearalgebraareassumed.ThenotesfocusontheconstructionofnumericalalgorithmsforODEsandthemathematicalanalysisoftheirbehaviour,cov-eringthematerialtaughtintheM.Sc.inMathematicalModellingandScientificCompu-tationintheeight-lecturecourseNumericalSolutionofOrdinaryDifferentialEquations.Thenotesbeginwithastudyofwell-posednessofinitialvalueproblemsforafirst-orderdifferentialequationsandsystemsofsuchequations.Thebasicideasofdiscretisationanderroranalysisarethenintroducedinthecaseofone-stepmethods.Thisisfollowedbyanextensionoftheconceptsofstabilityandaccuracytolinearmulti-stepmethods,includingpredictorcorrectormethods,andabriefexcursionintonumericalmethodsforstiffsystemsofODEs.Thefinalsectionsaredevotedtoanoverviewofclassicalalgorithmsforthenumericalsolutionoftwo-pointboundaryvalueproblems.Syllabus.Approximationofinitialvalueproblemsforordinarydifferentialequations:one-stepmethodsincludingtheexplicitandimplicitEulermethods,thetrapeziumrulemethod,andRunge–Kuttamethods.Linearmulti-stepmethods:consistency,zero-stabilityandconvergence;absolutestability.Predictor-correctormethods.Stiffness,stabilityregions,Gear’smethodsandtheirimplementation.Nonlinearstability.Boundaryvalueproblems:shootingmethods,matrixmethodsandcollocation.ReadingList:[1]H.B.Keller,NumericalMethodsforTwo-pointBoundaryValueProblems.SIAM,Philadelphia,1976.[2]J.D.Lambert,ComputationalMethodsinOrdinaryDifferentialEquations.Wiley,Chichester,1991.FurtherReading:[1]E.Hairer,S.P.Norsett,andG.Wanner,SolvingOrdinaryDifferentialEqua-tionsI:NonstiffProblems.Springer-Verlag,Berlin,1987.[2]P.Henrici,DiscreteVariableMethodsinOrdinaryDifferentialEquations.Wiley,NewYork,1962.[3]K.W.Morton,NumericalSolutionofOrdinaryDifferentialEquations.OxfordUniversityComputingLaboratory,1987.[4]A.M.StuartandA.R.Humphries,DynamicalSystemsandNumericalAnalysis.CambridgeUniversityPress,Cambridge,1996.1Picard’stheoremOrdinarydifferentialequationsfrequentlyoccurasmathematicalmodelsinmanybranchesofscience,engineeringandeconomy.Unfortunatelyitisseldomthattheseequationshavesolutionsthatcanbeexpressedinclosedform,soitiscommontoseekapproximatesolutionsbymeansofnumericalmethods;nowadaysthiscanusuallybeachievedveryin-expensivelytohighaccuracyandwithareliableboundontheerrorbetweentheanalyticalsolutionanditsnumericalapproximation.Inthissectionweshallbeconcernedwiththeconstructionandtheanalysisofnumericalmethodsforfirst-orderdifferentialequationsoftheformy′=f(x,y)(1)forthereal-valuedfunctionyoftherealvariablex,wherey′≡dy/dx.Inordertoselectaparticularintegralfromtheinfinitefamilyofsolutioncurvesthatconstitutethegeneralsolutionto(1),thedifferentialequationwillbeconsideredintandemwithaninitialcondition:giventworealnumbersx0andy0,weseekasolutionto(1)forxx0suchthaty(x0)=y0.(2)Thedifferentialequation(1)togetherwiththeinitialcondition(2)iscalledaninitialvalueprobl