Considerthefollowingdifferentialequation
d2u
−2u3=0;2dx
0≤x≤9;
u(0)=1,
u(9)=0.1
•Applythefinite-differencemethodtothisequationtogetalinearizeddifferenceequa-tionatgridpointiawayfromtheboundary.Notethatasecond-orderdifference
approximationforthesecond-derivativeis
dudx2
2
i
ui−1−2ui+ui+12
=+O∆x
∆x2
•Assemblethediscretesystemofequationsforafour-pointgridintoamatrixsystemoftheform
[A]{u}={b}where
{u}={u1u2u3u4}T
•DevelopaMATLABprogramtosolvethefinite-differenceequationsonagridwithNpoints.Applythiscodetoobtainthesolutionona4-pointgrid(∆x=3).Fortheinitialguess,usealinearvariationbetweenthetwoboundaryvalues.Convergeyoursolutionuntiltheresidualisbelow10−6.Plottheresidualsvs.iterationnumber.Hint:InMATLAB,initializeallelementsof[A]tozero.Forrowiof[A]when2≤i≤N−1,youneedtosetonlytheelementsAi,i−1,Ai,iandAi,i+1.
•Plotthefinite-differencesolutionobtainedonthe4-pointgridandcompareitwiththeexactsolution
1
uexact=
x+1•UseyourMATLABprogramtoobtainthesolutionona7-pointgrid(∆x=1.5).Plotthesolutionandcompareitwiththesolutionforthe4-pointgridandtheexactsolution.
1
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