The periodic centered-difference kinetic matrix $T_h$ is real symmetric, and the sampled real potential $V_h$ is a real diagonal matrix. Therefore $-iT_h$ and $-iV_h$ are skew-Hermitian, so each matrix exponential in the <Strang splitting> is a <unitary matrix>. Their product is unitary as well. Hence
$$
\lVert u^{n+1}\rVert_2^2
=\lVert u^n\rVert_2^2,
\qquad
\sum_m|u_m^n|^2=\text{constant}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 341 / 5 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-341/5/c/solution