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.

翻译：对于微分方程 $\dot{x}=f(x)$ 的数值积分器，若将其应用于一维 Dahlquist 测试方程 $\dot{x}=\lambda x,\lambda\in\mathbb{C}$（采用固定步长 $h>0$），当步数趋于无穷时数值解保持有界，则称该积分器是\emph{稳定的}。众所周知，显式积分器在 $\lambda$ 超出特定界限后无法保持稳定。此外，这些稳定性界限仅对某些特定的积分器（在不同情形下各不相同）是紧的，这类积分器可称为“最优稳定的”。此类最优稳定性结果通常需借助复分析中的复杂技巧来证明，导致证明过程颇为晦涩。本文采用一种替代方法，利用其与多项式 Bernstein 不等式和 Markov 兄弟不等式的联系。这极大地简化了证明，并提供了一个统一现有各类结果的框架。