Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 1 / 17B (Numerical Analysis)

The local truncation error is the error made by one numerical step started from the exact solution:A one-step method has local order $p+1$ when ${\tau }_{n+1}=O(h^{p+1})$ uniformly for $t_n$ in each fixed bounded time interval.

Let $e_n=y^n-y(t_n)$. Subtracting the exact one-step relation from the numerical method and applying the stated Lipschitz continuity givesIf $\Vert {\tau }_{n+1}\Vert \le C{h}^{p+1}$, iteration yields the discrete Gronwall inequalityFor $nh\le t^{\ast }$,andConsequently

HereFor $0<h\le 1$, the Lipschitz bound on $f$ givesso part (ii) applies with a constant independent of sufficiently small $h$.

The exact solution has the Taylor expansion>Meanwhile,so one numerical step from $y$ isThe local error is therefore $O(h^3)$. Taking $p=2$ in part (ii) proves the required second-order global-error bound. This method is a two-stage Runge-Kutta method.