Polynomial Inequalities and Optimal Stability of Numerical Integrators
Luke Shaw
TL;DR
The paper addresses the problem of determining sharp stability limits for explicit numerical integrators applied to the Dahlquist test equation $\dot{x}=\lambda x$, by recasting stability as a polynomial extremal problem. It replaces complex-analytic proofs with a polynomial framework grounded in the Bernstein and Markov inequalities, yielding simple, expository derivations. The main contributions include unified proofs of the three canonical optimal-stability results for general, parabolic, and hyperbolic regimes, with explicit optimal polynomials: $P(z)=(1+z/m)^m$ (general), $P(z)=T_m(1+z/m^2)$ (parabolic), and a Chebyshev-based construction for the hyperbolic case, all tying stability to polynomial extremal problems. The approach clarifies the role of classical polynomial inequalities in numerical stability, offering pedagogical value and applicability to explicit Runge-Kutta and related methods.
Abstract
A numerical integrator for $\dot{x}=f(x)$ is called \emph{stable} if, when applied to the 1D Dahlquist test equation $\dot{x}=λx,λ\in\mathbb{C}$ 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 $λ$. 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.