Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 67 / 1 / a / Solution

Introduce one stage $k=f(t_n+h/2,Y)$ with $Y=y_n+hk/2$, and update $y_{n+1}=y_n+hk$. This is the implicit midpoint rule as a one-stage Runge-Kutta method, with Butcher tableauFor a smooth vector field, write $\Delta =y_{n+1}-y_n$. Expanding its stage equation giveswhich matches the exact Taylor expansion through degree two. The method therefore has order at least two. On the Dahlquist test equation $y^{\prime }=\lambda y$, its stability function isThe third coefficient differs from the exponential's $1/6$, so the order is exactly two.

There is no pole in $\mathrm{Re}z\le 0$, andThus $|R(z)|\le 1$ throughout the closed left half-plane:This conclusion is A-stability, not strong damping of arbitrarily stiff modes: $R(z)\to -1$ along the negative real axis.