Invited Lecture
Strong stability of explicit Runge-Kutta time discretizations for semi-negative linear systems
Division of Applied Mathematics, Brown University, Providence, RI 02912, USA
Abstract
We study the strong stability property of explicit Runge-Kutta time discretizations of linear semi-discrete schemes which have semi-negative stability. It is well known that the three stage, third order Runge-Kutta method is strongly stable for semi-negative linear systems, however we show by a simple counter example that the classical four stage, fourth order Runge-Kutta method is not strongly stable, and we also show the (somewhat surprising) result that after two time steps the fourth order Runge-Kutta method is strongly stable. Furthermore, we present a general framework on analyzing the strong stability of explicit Runge-Kutta 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 Runge-Kutta 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 Runge-Kutta methods of odd linear order.
This is joint work with Zheng Sun.
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