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1、CHAPTER3TREFETHEN1994108Chapter3.FiniteDi erenceApproximations3.1.Scalarmodelequations3.2.Finitedi erenceformulas3.3.Spatialdi erenceoperatorsandthemethodoflines3.4.Implicitformulasandlinearalgebra3.5.Fourieranalysisof nitedi erenceformulas3.6.Fourieranalysi
2、sofvectorandmultistepformulas3.7.NotesandreferencesByasmallsamplewemayjudgeofthewholepiece.
3、MIGUELDECERVANTES,DonQuixotedelaMancha,Chap.1(1615)CHAPTER3TREFETHEN1994109Thischapterbeginsourstudyoftime-dependentpartialdi erentialequa-tions,whosesolutionsvarybot
4、hintime,asinChapter1,andinspace,asinChapter2.Thesimplestapproachtosolvingpartialdi erentialequationsnumericallyistosetuparegulargridinspaceandtimeandcomputeapprox-imatesolutionsonthisgridbymarchingforwardsintime.Theessentialpointisdiscretization.Finitedi eren
5、cemodelingofpartialdi erentialequationsisoneofseveral eldsofsciencethatareconcernedwiththeanalysisofregulardiscretestruc-tures.Anotherisdigitalsignalprocessing,alreadymentionedinChapter1,wherecontinuousfunctionsarediscretizedinasimilarfashionbutforquitedi ere
6、ntpurposes.Athirdiscrystallography,whichinvestigatesthebehav-iorofphysicalstructuresthatarethemselvesdiscrete.Theanalogiesbetweenthesethree eldsareclose,andweshalloccasionallypointthemout.ThereaderwhowishestopursuethemfurtherisreferredtoDiscrete-TimeSignalPro
7、cessing,byA.V.OppenheimandR.V.Schafer,andtoAnIntroductiontoSolidStatePhysics,byC.Kittel.Thischapterwilldescribe vedi erentwaystolookat nitedi erenceformulas
8、asdiscreteapproximationstoderivatives,asconvolution lters,asToeplitzmatrices,asFouriermultipliers,anda
9、sderivativesofpolynomialin-terpolants.Eachofthesepointsofviewhasitsadvantages,andthereadershouldbecomecomfortablewithallofthem.The eldofpartialdi erentialequationsisbroadandvaried,asisin-evitablebecauseofthegreatdiversityofphysicalphenomenathattheseequa-tions
10、model.Muchofthevarietyisintroducedbythefactthatpracticalproblemsusuallyinvolveoneormoreofthefollowingcomplications:multiplespacedimensions,systemsofequations,boundaries,variablecoeci
