Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 341 / 2 / b

Use the stability function $R(z)=(z^2+6z+12)/(z^2-6z+12)$. Its denominator vanishes only at $3\pm i\sqrt{3}$, both in the open right half-plane. For $z=x+iy$,This is nonnegative whenever $x\le 0$, so $|R(z)|\le 1$ throughout the closed left half-plane. The internal stage equations are also solvable there, since $\det (I-zA)=1-z/2+z^2/12$. Therefore the method is A-stable.

The inequality is strict in the open left half-plane, while $|R(iy)|=1$. Moreover $R(z)\to 1$ as $|z|\to \infty$, so the method is not L-stable. These conclusions follow directly from the definitions of A-stability and L-stability, rather than from the order of the method.