Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 341 / 1 / c / Solution

Write $A_n=A({\mathbf{y}}_n)$. The trapezoidal rule isTaking the inner product with ${\mathbf{y}}_{n+1}+{\mathbf{y}}_n$ givesThe two quadratic terms vanish because each $A_j$ is skew-symmetric. Moreover ${\mathbf{y}}_{n+1}^T{A}_n{\mathbf{y}}_n=-{\mathbf{y}}_n^T{A}_n{\mathbf{y}}_{n+1}$. HenceUnlike the implicit midpoint rule, the trapezoidal rule evaluates $A$ at two different states, so the mixed terms need not cancel.