Necessary and sufficient conditions for strong stability of explicit Runge-Kutta methods

TL;DR

This paper investigates the strong stability of explicit Runge-Kutta methods for linear ODEs, establishing new necessary and sufficient conditions, and resolving an open question about stability for certain orders and stages.

Contribution

It proves that explicit Runge-Kutta schemes of order multiples of 4 with equal stages are not strongly stable and provides conditions involving the stability function and hypocoercivity index for other cases.

Findings

Explicit RK schemes of order p in 4N with s=p stages are not strongly stable.

Necessary and sufficient conditions for strong stability are derived involving the stability function.

The hypocoercivity index of the system matrix plays a role in stability analysis.

Abstract

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.