Hypocoercivity for Linear ODEs and Strong Stability for Runge--Kutta Methods

Abstract

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.