Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 66 / 2 / b

The Dahlquist equivalence theorem states that a consistent linear multistep method is convergent for suitably consistent starting values exactly when it is zero-stable. The root condition for a multistep method requires every root of $\rho$ to lie in the closed unit disk, with every unit-modulus root simple.

The roots are $1$ and $a$. Thus $|a|\le 1$ is necessary; $a=1$ is excluded because it gives a double root at one. At $a=-1$, the two unit roots are distinct, so the endpoint is allowed. Combined with part (a), this provesAssume a locally Lipschitz vector field, a smooth solution on the fixed time interval, a nearby solvable implicit branch and starting errors of the required order. The global order is three at $a=-1/5$ and two at the other convergent parameter values. Outside this interval, zero-step perturbations already grow through either an exterior root or a unit-root polynomial factor.