Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 341 / Section A / 3 / c

Apply the implicit midpoint rule, equivalently the Crank--Nicolson method, to the semidiscrete equation:It has order two. Its amplification matrix is the Cayley transformBecause $H_h$ is Hermitian, $Q$ is unitary. Hence $\Vert {\mathbf{u}}^{n+1}{\Vert }_2=\Vert {\mathbf{u}}^n{\Vert }_2$ exactly.