Apply the <implicit midpoint rule>, equivalently the <Crank--Nicolson method>, to the semidiscrete equation:
$$
i\frac{\mathbf u^{n+1}-\mathbf u^n}{\Delta t}
=H_h\frac{\mathbf u^{n+1}+\mathbf u^n}{2}.
$$
It has order two. Its amplification matrix is the <Cayley transform of a Hermitian matrix>[Cayley transform]
$$
Q=\left(I+\frac{i\Delta t}{2}H_h\right)^{-1}
\left(I-\frac{i\Delta t}{2}H_h\right).
$$
Because $H_h$ is Hermitian, $Q$ is unitary. Hence $\|\mathbf u^{n+1}\|_2=\|\mathbf u^n\|_2$ exactly.

Solution

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

{c}
{wiki=Crank–Nicolson_method}

The Crank--Nicolson method applies the <trapezoidal rule> to a semidiscrete evolution equation. For a linear system $u'=Lu$ its update is
$$
\left(I-\frac h2L\right)u^{n+1}=\left(I+\frac h2L\right)u^n.
$$

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