Past exam of the numerical linear algebra and numerical solution of differential equations course of the University of Oxford / 2020 / 2 / c

For a constant-coefficient linear multistep method, zero-stability means uniform boundedness of the propagation of starting perturbations as $h\to 0$. Equivalently, every solution of the homogeneous linear recurrencesatisfies $|v_n|\le K{\max }_{0\le j<k}|v_j|$ for all $n\ge 0$, with $K$ independent of $n$ and the starting values. The equivalent root condition for a multistep method is that every root of a polynomial $\zeta$ of the degree-$k$ polynomial $\rho$ satisfies $|\zeta |\le 1$, and every root on $|\zeta |=1$ is simple. Indeed, a root of multiplicity $m$ produces recurrence modes $n^q{\zeta }^n$, $0\le q<m$: repeated roots strictly inside the disk are harmless, whereas roots outside it or repeated roots on its boundary produce unbounded modes.