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∈4ℕ with s = p stages turn out to be not strongly stable. This result was proved in [4], filling a gap left open in [8]. Here, we present an alternative, direct proof.