Strong stability of explicit Runge-Kutta time discretizations

TL;DR

This paper develops an energy-based framework to analyze the strong stability of explicit Runge-Kutta methods for linear systems, including high-order schemes, with conditions for odd-order methods.

Contribution

It introduces a novel, computer-aided energy method framework for analyzing strong stability of explicit RK methods, including high-stage and high-order schemes.

Findings

Strong stability can be characterized using the proposed energy framework.

Numerical experiments support the stability conditions for various RK methods.

A necessary and sufficient condition is provided for odd-order RK methods' strong stability.

Abstract

Motivated by studies on fully discrete numerical schemes for linear hyperbolic conservation laws, we present a framework on analyzing the strong stability of explicit Runge-Kutta (RK) time discretizations for semi-negative autonomous linear systems. The analysis is based on the energy method and can be performed with the aid of a computer. Strong stability of various RK methods, including a sixteen-stage embedded pair of order nine and eight, has been examined under this framework. Based on numerous numerical observations, we further characterize the features of strongly stable schemes. A both necessary and sufficient condition is given for the strong stability of RK methods of odd linear order.