Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 4 / 39A / a / Solution

Let $L=\mathrm{tridiag}(1,-2,1)$. The zero Dirichlet boundary values introduce no forcing terms, so the scheme isThe common orthogonal sine eigenvectors have $L$-eigenvalues $-4{\sin }^2[\pi j/(2(M+1))]$. Thus $B$ is invertible for every $\mu >0$, and each amplification eigenvalue isThe matrix $B^{-1}C$ is orthogonally diagonalizable, so its powers have Euclidean norm at most one, uniformly in the mesh and time step. Therefore Crank-Nicolson diffusion scheme is stable for every $\mu >0$. Negative amplification factors at large $\mu$ can create temporal sign oscillations, which do not contradict this norm stability.