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18.336spring2009 lecture13 03/19/09

InitialValueProblems(IVP)

ut =Lu in]0, T[ PDEu=u0 on {0} initialconditionu=g on]0, T[ boundaryconditionwhereLdifferentialoperator.

2

Ex.:L=PoissonequationLu=b uadvectionequation

heatequation

u)Lu=2(2 notdoneyet.biharmonicequationLu=F|u|Eikonalequation

beamequationnonlinearlevelsetequation

etc.StationarysolutionofIVP: Lu= 0 in (ifitexists) u=g on

Later:secondorderproblems systems

u 0 1 uutt =uxx

t v = 1 0 x v(waveequation)

Semi-DiscretizationInspace(methodoflines):

Approximateu(, t)byu(t)ApproximateLubyA u(forlinearproblems)[FD,FE,spectral]

dsystemofODE:

dtu=Au

Intime:Approximatetimederivativebystep:d

u(x, t)u(x, t+ t)u(x, t)

[explicitEuler]dt tStationaryproblem:

unew(x) =u(x) + tLu(x) = (I+ tL)u(x)NeedtoknowaboutODEsolvers.

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c1...cr

a11 ... a1r...ar1

. . .

......arr

= c AbT

b1 ... br0

EE:IE: 1

Explicitmidpoint: 012

Heuns: 01

RK4:012121

Implicittrapezoidal: 12MultistepMethods:

r

r

111

k1 =f(yn)yn+1

=yn +tk1k1 =f(

=yn+1 yn +tk1)

yn+1 =yn +tk1120 1112 1212

16121

1213

113 16

PDECrank-Nicolson

jyn+j = t jf(yn+j)j=0 j=0ExplicitAdams-Bashforth:

yn+1 = yn + tf(yn) = EE O(t)yn+2 = yn+1 + t [ 3f(yn+1)1f(yn)] O(t2)

2 2...

ImplicitAdams-Moultion:yn+1 = yn + t (

2

1f(yn) +12(yn+1)) =trapezoidal O(t2)

yn+2 = yn+1 + t (5 f(yn+2) + 8 f(yn+1) 1 f(yn)) O(t3)12 12 12

...BDF(backwarddifferentiation):

yn+1 = yn + tf(yn+1) = IE O(t)3yn+2 4yn+1 +yn = 2tf(yn+2) O(t2)

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LinearODESystemsy =A yy(0)= y

solution: y(t)=exp(tA) ysolutionstable,ifRe(i(A))

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MIT OpenCourseWarehttp://ocw.mit.edu

18.336 Numerical Methods for Partial Differential Equations

Spring 2009

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