Finite-Difference Methods Consider the initial boundary value problem \[ v_t=\beta v_{xx},\qquad 0<x<1,\qquad 0<t,\qquad v(x,0)=f(x) \] with Dirichlet boundary conditions \[ v(0,t)=0,\qquad v(1,t)=0. \]
Suppose that the interval \([0,1]\) is partitioned by meshpoints \(x_j=jh\) for \(j=0,1,\ldots ,N\) and \(h=1/N\). A possible discretization of the above problem uses the Crank-Nicolson scheme for the differential equation: \[ \left (I-\frac {\beta k}{2}D_+D_-\right )u_j^{n+1} = \left (I+\frac {\beta k}{2}D_+D_-\right )u_j^n \] for \(j=1,\ldots ,N-1\), combined with the boundary conditions: \[ u_0^n=0\qquad \text {and}\qquad u_N^n=0 \] for each time level \(n\). Initial data is obtained from by setting \(u_j^0=f_j\) for \(j=0,1,\ldots ,N\). (Note that \(D_+\) and \(D_-\) are the usual forward and backward difference operators.)
Set up the matrix system that must be solved in order to advance the numerical solution from time level \(n\) to time level \(n+1\). Show that the suggested discretization is stable in the \(l_2\) norm \[ \lVert u\rVert _2= \left (\sum _{j=0}^N h\lvert u_j\rvert ^2\right )^{1/2} \] for all choices of timestep \(k\) and spacestep \(h\). What other issues should be considered in deciding whether the suggested scheme is a good choice for the problem?