Transcription of Finite Difference Methods for Boundary Value Problems
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Finite Difference Methods for Boundary Value Problems
Finite Difference Methods for Boundary Value ProblemsOctober 2, 2013() Finite DifferencesOctober 2, 20131 / 52 GoalsLearn steps to approximate BVPs using the Finite Difference MethodStart with two-point BVP (1D)Investigate common FD approximations foru (x) andu (x) in 1 DUse FD quotients to write a system of Difference equations to solvetwo-point BVPH igher order accurate schemesSystems of first order BVPsUse what we learned from 1D and extend to Poisson s equation in 2D& 3 DLearn how to handle different Boundary conditions() Finite DifferencesOctober 2, 20132 / 52 Steps in the Finite Difference Approach to linear DirichletBVPsOverlay domain with gridChoose Difference quotients to approximate derivatives in DEWrite a Difference equation at each node where there is an unknownSet up resulting system of equations as a matrix problemSolving resulting linear system efficientlyCompute error when solution is known() Finite DifferencesOctober 2, 20133 / 52 Prototype Dirichlet BVP in 1D ddx(p(x)dudx)+q(x)u=f(x)a<x<b,(1)u(a) = u(b) = Whenp(x) =p, a constant, we have pu (x) +q(x)u=f(x)a<x<b,Whenp= 1 andq= 0 we have the Poisson equation u (x) =f(x)a<x<bin one dimension.
Finite di erence quotient for u0(x) The forward or backward di erence quotients for u0(x) are rst order The second centered di erence for u00(x) is second order So we need a second order approximation to u0(x) If we subtract the expansions u(x + …
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