有限差分2 Introduction to Finite Difference Methods

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1、IntroductiontoFiniteDi¤erenceMethodsApril22,20071TheOneDimensionalHeatEquationOneofthesimplestparabolic(orheat)equaionsistheonedimensionalproblemut(x;t)=uxx(x;t);t2(0;T];x2(0;a);(1)u(t;0)=h1(t);u(t;a)=h2(t);u(0;x)=f(x):Becausevaluesofthefunctionuarespeci…edattheendpointsx=0andx=aofthespaceint

2、ervale,wecallthisproblemaDirichletproblem.Ifvaluesofthederivativeuxarespeci…edattheboundariesx=0andx=a,theproblemiscalledaNeumannproblem.Toapproximatethesolutionofthisequation,weconstructaspace-timegrid.Thatis,wedividetherectangleR=f(x;t):0xa;0tTgintoagridofn1bym1rectangleswithsieds4x=a=

3、nand4t=T=m:Startingfromthebottomrowt=t1=0,thesolutionisu(xi;t1)=f(xi);i=1;2;:::;n:Toapproximatethesolutionatthenextgridline,thedi¤erentialequationisapproxi-matedbyadi¤erenceequationasfollows.Thetimederivatvieutisapproximatedbythedi¤erenceformulau(x;t+4t)u(x;t)ut(x;t)4tandthespacederivativeis

4、approximatedbyu(x+4x;t)2u(x;t)+u(x4x;t)uxx(x;t)4x2WecanshowusingTaylorseriesexpansionsthattheseapproximationsareoforder4tintimeand4x2inspace.Denotingbyutheapproximatevalueofu(x;y)thedi¤erentialijijequation(1)isapproximatedbythedi¤erenceequationui;j+1ui;jui+1;j2ui;j+ui1;j=:4t4x2110.90.80

5、.70.60.50.40.30.20.1000.10.20.30.40.50.60.70.80.91Substitutingr=4t=4x2(forconventience)andsolvingforu;weobtaini;j+1ui;j+1=rui+1;j(12r)ui;j+rui1;j:(2)ExampleFortheheatequationwithf(x)=4xx2;h(x)=h(x)=0;a=1;r=:45,12the…gurebelowshows10timestepsoftheevolutionofthesolution.Theinitialsolutionde

6、caystowardszerowithtimeasexpected.Repeatingthecalculationwithr=:6producestheresultsshownbelowInthiscasetheinitialsolutionbuildsuptwardsin…nity.Thisisatypicalcaseoftheinstabilityofthenumericalschemeusedtoapproximatethesolution.Wecanshowthatthesolutionisstableifandonlyifr:5:Theinstabilityinthep

7、reviousexamplecanbepredictedfrommathematicalanalysisbyamethodcalledNeumannstabilitymethod.Theideaistoconsiderthesolutionofthedi¤erenceequationasasuperpositionofFourierbasicfunctions(modes)andrequirethatnoneofthebasicfunctionsshouldblowu