Crank Nicolson Solution to the Heat Equation

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1、CrankNicolsonSolutiontotheHeatEquationME448/548NotesGeraldRecktenwaldPortlandStateUniversityDepartmentofMechanicalEngineeringgerry@me.pdx.eduME448/548:Crank-NicolsonSolutiontotheHeatEquationOverview221.Useniteapproximationsto@u=@tand@u=@x:samecomponentsusedinFTCSandBTCS.222.Averagecontributi

2、onfromtkandtk+1intheapproximationof@u=@x.k+13.Computationalformulais(still)implicit:allumustbesolvedsimultaneously.ikkkInformationfromu,u,anduareusedateachtimestepinthecomputationi1ii+1k+1ofu.i4.SolutionisnotsignicantlymorecomplexthanBTCS.5.LikeBTCS,theCrank-Nicolsonmethodsisunconditionally

3、stablefortheheatequation.226.TruncationerrorisO(
t)+O(Dx).TimeaccuracyisbetterthanBTCSorFTCS.ME448/548:Crank-NicolsonSolutiontotheHeatEquationpage1TheMethod22Evaluatethediusionoperator@u=@xatbothtimestepstk+1andtimesteptk,anduseaweightedaverage"#"#k+1kuk+12uk+1+uk+1uk2uk+ukuiuii1ii+1i1

4、ii+1=+(1)(1)
t
x2
x2where01=0()FTCS=1()BTCSME448/548:Crank-NicolsonSolutiontotheHeatEquationpage2Crank-NicolsonComputationalMoleculet.....................Crank-Nicolsonschemerequiressimultaneousk+1calculationofuatallknodesonthek+1meshlinek1Solutionisknownforthesenodest=0,k=1i=1i1ii+1

5、nxx=0x=LME448/548:Crank-NicolsonSolutiontotheHeatEquationpage3CompareComputationalMoleculesFTCSBTCSCrank-Nicolsonk+1k+1k+1kkki–1ii+1i–1ii+1i–1ii+1ME448/548:Crank-NicolsonSolutiontotheHeatEquationpage4CrankNicolsonApproximationtotheHeatEquationk+11k+1k+1u++uu=2
x2i1
t
x2i2
x2i+1k1kku+u

6、+u(2)2
x2i1
t
x2i2
x2i+1ME448/548:Crank-NicolsonSolutiontotheHeatEquationpage5Crank-NicolsonSystemofEquationsThesystemofequationshasthesamestructureasBTCS2323uk+123a1b10000617d1676uk+17676c2a2b200076276d2767676760c3a3b30076uk+176d376......763..7=6..7(3)600...076.76.76767674000cnx1anx1bnx1

7、56uk+174dnx154nx150000cnxanxk+1dnxunxwherethecoecientsoftheinteriornodes(i=2;3;:::;N1)are2ai=1=
t+=
x=1=
t(bi+ci);2bi=ci==(2
x);kkkdi=ciui1+(1=
t+bi+ci)uibiui+1:ME448/548:Crank-NicolsonSolutiontotheHeatEquationpage6demoCNCode%---Co