Strong stability of explicit Runge-Kutta time discretizations
For researchers in numerical analysis of hyperbolic conservation laws, this work provides a systematic tool to assess stability of high-order Runge-Kutta methods, though it is incremental in extending known energy method ideas.
This paper develops a framework for analyzing the strong stability of explicit Runge-Kutta time discretizations for semi-negative autonomous linear systems, using the energy method and computer assistance. It characterizes features of strongly stable schemes and provides a necessary and sufficient condition for strong stability of RK methods of odd linear order.
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.