Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 341 / 5 / a

Let ${\Phi }_h$ be the one-step map of a numerical method. It is a time-symmetric numerical method when reversing the step exactly reverses the update:

Let ${\phi }_h$ be the exact flow map and suppose the method has order $p$ with a nonzero leading local truncation error:Inverting this expansion changes the sign of its leading perturbation, sowhere transport by the exact flow only changes $d$ by $O(h)$ and hence does not affect the leading parity. On the other hand, replacing $h$ by $-h$ in the first expansion givesTime symmetry equates these expressions, so $(-1{)}^{p+1}=-1$. Hence $p+1$ is odd and