Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 341 / 1 / ii

For the Dahlquist test equation, put $z=h\lambda$. Eliminating the stages of the Runge-Kutta method gives the stability functionIts poles are $2\pm i\sqrt{2}$, strictly in the right half-plane. To prove A-stability directly, write $z=-r+iy$ with $r\ge 0$. With $D(z)=1-2z/3+z^2/6$ and $N(z)=1+z/3$, elementary expansion givesTherefore $|R(z)|\le 1$ throughout the closed left half-plane; the inequality is strict except at $z=0$. The method is A-stable. Moreover $R(z)\to 0$ as $|z|\to \infty$, so the Radau IIA method is also L-stable.