Polynomial Inequalities and Optimal Stability of Numerical Integrators

A numerical integrator for $\dot{x}=f(x)$ is called \emph{stable} if, when applied to the 1D Dahlquist test equation $\dot{x}=\lambda x,\lambda\in\mathbb{C}$ with fixed timestep $h>0$, the numerical solution remains bounded as the number of steps tends to infinity. It is well known that no explicit integrator may remain stable beyond certain limits in $\lambda$. Furthermore, these stability limits are only tight for certain specific integrators (different in each case), which may then be called `optimally stable'. Such optimal stability results are typically proven using sophisticated techniques from complex analysis, leading to rather abstruse proofs. In this article, we pursue an alternative approach, exploiting connections with the Bernstein and Markov brothers inequalities for polynomials. This simplifies the proofs greatly and offers a framework which unifies the diverse results that have been obtained.