Strong stability is a property of time integration schemes for ODEs that preserve temporal monotonicity of solutions in arbitrary (inner product) norms. It is proved that explicit Runge--Kutta schemes of order $p\in 4\mathbb{N}$ with $s=p$ stages for linear autonomous ODE systems are not strongly stable, closing an open stability question from [Z.~Sun and C.-W.~Shu, SIAM J. Numer. Anal. 57 (2019), 1158--1182]. Furthermore, for explicit Runge--Kutta methods of order $p\in\mathbb{N}$ and $s>p$ stages, we prove several sufficient as well as necessary conditions for strong stability. These conditions involve both the stability function and the hypocoercivity index of the ODE system matrix. This index is a structural property combining the Hermitian and skew-Hermitian part of the system matrix.

翻译：强稳定性是常微分方程时间积分方案的一种性质，能保证解在任意（内积）范数下的时间单调性。本文证明：对于线性自治常微分方程组，阶数$p\in 4\mathbb{N}$、阶段数$s=p$的显式Runge-Kutta格式不具有强稳定性，从而解决了[Z.~Sun and C.-W.~Shu, SIAM J. Numer. Anal. 57 (2019), 1158--1182]中提出的一个开放稳定性问题。此外，对于阶数$p\in\mathbb{N}$、阶段数$s>p$的显式Runge-Kutta方法，我们证明了强稳定性的若干充分条件和必要条件。这些条件同时涉及稳定性函数和常微分方程组矩阵的亚等收敛性指标。该指标是结合系统矩阵的埃尔米特部分与反埃尔米特部分的结构性质。