Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 68 / 4 / b / Solution

For a Runge-Kutta method, the stability function is $R(z)=1+z{b}^T(I-zA{)}^{-1}e$. Solving the stage equations on the Dahlquist test equation givesThe denominator has roots of a polynomial $3\pm i\sqrt{3}$, both in the open right half-plane. If $z=x+iy$, direct expansion givesSince $|z{|}^2+12>0$, the modulus of $R$ is at most one exactly when $x\le 0$. ThusThe method is A-stable. Its stability function has unit modulus on the imaginary axis and tends to one as $z\to -\infty$, so it is not L-stable.