Math 6610 • Analysis of Numerical Methods I

Heat Equation Derive the Crank-Nicolson scheme for the following problem \[ u_t=u_{xx},\qquad t>0,\qquad 0<x<1, \] with the boundary conditions \[ u(0,t)=u(1,t)=0,\qquad t\ge 0, \] and the initial data \[ u(x,0)=u_0(x),\qquad 0\le x\le 1, \] for some given function \(u_0\). Use the usual discretization with uniform mesh points \(x_j=jh\) where \(h=1/N\) and \(j=0,1,2,\ldots ,N\). Show that the method is second order by deriving the truncation error. What other conditions do we need to guarantee the convergence of the numerical solution? Derive the stability condition in \(\ell _2\) norm by using Fourier analysis. For any value of \(\nu =\Delta t/(\Delta x)^2\), do we always have the maximum principle satisfied? If we have some initial data such that \(0\le u_0(x)\le 1\), derive the condition that should be imposed on \(\nu \) so that \(0\le u_j^n\le 1\) also holds for all \(j\) and \(n>0\).