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1、AMATH581PracticalScienti?cComputing?J.NathanKutz?January4,2005AbstractThiscourseisasurveyofpracticalnumericalsolutiontechniquesforordinaryandpartialdi?erentialequations.Emphasiswillbeontheimple-mentationofnumericalschemestopracticalproblemsintheengineeringandphysicalsciences.Me
2、thodsforpartialdi?erentialequationswillin-clude?nitedi?erence,?niteelementandspectraltechniques.FullusewillbemadeofMATLABanditsprogrammingfunctionality.?ThesenotesareintendedastheprimarysourceofinformationforAMATH581.Thenotesareimcompleteandmaycontainerrors.Anyotheruseasidefrom
3、classroompurposesandpersonalresearchpleasecontactmeatkutz@amath.washington.edu.cJ.N.Kutz,Autumn2003(Version1.1)?DepartmentofAppliedMathematics,Box352420,UniversityofWashington,Seattle,WA98195-2420(kutz@amath.washington.edu).1AMATH581(cJ.N.Kutz)2Contents1InitialandBoundaryValueP
4、roblemsofDi?erentialEquations51.1Initialvalueproblems:Euler,Runge-KuttaandAdamsmethods51.2Erroranalysisfortime-steppingroutines..............121.3Boundaryvalueproblems:theshootingmethod..........171.4Implementationofshootingandconvergencestudies.......231.5Boundaryvalueproblems
5、:directsolveandrelaxation......282FiniteDi?erenceMethods342.1Finitedi?erencediscretization...................342.2DirectsolutionmethodsforAx=b.................392.3IterativesolutionmethodsforAx=b................442.4Fast-PoissonSolvers:theFourierTransform...........492.5Compari
6、sonofsolutiontechniquesforAx=b:rulesofthumb..522.6Overcomingcomputationaldi?culties...............573TimeandSpaceSteppingSchemes:MethodofLines613.1Basictime-steppingschemes.....................623.2Time-steppingschemes:explicitandimplicitmethods......663.3Stabilityanalysis.....
7、......................713.4Comparisonoftime-steppingschemes...............753.5Optimizingcomputationalperformance:rulesofthumb.....784SpectralMethods844.1Fast-FourierTransformsandCosine/Sinetransform.......844.2ChebychevPolynomialsandTransform..............884.3Spectralmethodim
8、plementation..................934.4Pseudo-spectraltech
