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 $\lambda$ 的某些界限之外无法保持稳定性。此外，这些稳定性界限仅对某些特定积分器（不同情况下各异）是紧致的，这些积分器可称为“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术进行证明，导致论证过程较为晦涩。本文采用一种替代方法，通过利用多项式Bernstein不等式与Markov兄弟不等式之间的联系，极大地简化了证明过程，并提供了一个统一现有各类结果的框架。