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

{title2=$(I+iA)^{-1}(I-iA)$}

If $A$ is a <Hermitian matrix>, then $(I+iA)^{-1}(I-iA)$ is <unitary matrix>[unitary]. This Cayley transform maps each real eigenvalue $\lambda$ to the unit-modulus number $(1-i\lambda)/(1+i\lambda)$.

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 Incoming links: Cayley transform of a Hermitian matrix