Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 63 / 4 / 2

Apply the $a=0$ method to the Dahlquist test equation and write $z=h\lambda$. The recurrence becomesso every amplification root must satisfyChoose $z=-1$, which lies strictly in the left half-plane. The polynomial then gives ${\xi }^2+2\xi -2=0$, henceOne root has modulus $1+\sqrt{3}>1$. Generic initial perturbations excite that growing recurrence mode even though the exact scalar solution decays. The $a=0$ method is not A-stable. There is also a singular implicit coefficient at $z=-\sqrt{2/3}$, another obstruction inside the left half-plane.