Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 341 / 2 / b / Solution

The stability function of a Runge-Kutta method is $R(z)=1+z{b}^T(I-zA{)}^{-1}1$. Substitution givesWriting the numerator and denominator as $N(z)$ and $D(z)$,The denominator has zeros $3\pm i\sqrt{3}$, both in the right half-plane. Therefore $|R(z)|\le 1$ throughout the closed left half-plane, with strict inequality in its interior. The method is A-stable. Since $R(z)\to 1$ at infinity, it is not L-stable.