NMM: Week 5 - Finite Differences: PDE's

This section explores discretizing partial differential equations, using the wave equation as a test case. Stability criteria (Courant-Friedrichs-Lewy condition) and numerical methods are demonstrated.

Hand computations:

It appears you don't have a PDF plugin for this browser. You can click here to download the PDF

For problem 8.1 (f) we wish to solve the 1D wave equation using explicit finite difference method along a bounded X axis with one nonzero node at t = 0. We first create a 2d mesh to store the wave u = (x[j],t[n]) and load initial conditions. We use Dirichlet conditions to define wave behavior at the boundaries, so u[0,1] = u[n*x,1] = 0. From the notes above setting up the finite differencing scheme, we needed to compute the first time step differently because it depends upon a point outside the time mesh. Rearranged we get: u[j,n+1] = u[j,0] - 0.5*ir^2(u[j+1,n]-2u[j,n]+u[j-1,n])

8.1(f) - 1D Wave equation video:

NMM 2014