In this note, we connect two different topics from linear algebra and numerical analysis: hypocoercivity of semi-dissipative matrices and strong stability for explicit Runge--Kutta schemes. Linear autonomous ODE systems with a non-coercive matrix are called hypocoercive if they still exhibit uniform exponential decay towards the steady state. Strong stability is a property of time-integration schemes for ODEs that preserve the temporal monotonicity of the discrete solutions. It is proved that explicit Runge--Kutta schemes are strongly stable with respect to semi-dissipative, asymptotically stable matrices if the hypocoercivity index is sufficiently small compared to the order of the scheme. Otherwise, the Runge--Kutta schemes are in general not strongly stable. As a corollary, explicit Runge--Kutta schemes of order $p\in 4\N$ with $s=p$ stages turn out to be \emph{not} strongly stable. This result was proved in \cite{AAJ23}, filling a gap left open in \cite{SunShu19}. Here, we present an alternative, direct proof.

翻译：本文连接了线性代数与数值分析中的两个不同主题：半耗散矩阵的低共轭性（hypocoercivity）以及显式龙格-库塔格式的强稳定性（strong stability）。若具有非强制性矩阵的线性自治常微分方程系统仍能对稳态呈现一致指数衰减，则称该系统具有低共轭性。强稳定性是常微分方程时间积分格式的一种性质，它确保离散解保持时间单调性。本文证明：当低共轭性指数足够小于格式阶数时，显式龙格-库塔格式对于半耗散且渐近稳定的矩阵是强稳定的；反之，龙格-库塔格式通常不具有强稳定性。作为推论，具有$s=p$个阶段的$p\in 4\N$阶显式龙格-库塔格式被证明是\emph{不}强稳定的。该结果在文献\cite{AAJ23}中得以证明，填补了文献\cite{SunShu19}中遗留的空白。本文给出了一种替代性的直接证明方法。