Annals of Mathematical Sciences and Applications

Volume 2 (2017)

Number 2

Guest Editors: Tai-Chia Lin (National Taiwan University), Wen-Wei Lin (National Chiao Tung University), Tony Wen-Hann Sheu (National Taiwan University), Weichung Wang (National Taiwan University), Chih-wen Weng (National Chiao Tung University), and Salil Vadhan (Harvard University).

Stability of the fourth order Runge–Kutta method for time-dependent partial differential equations

Pages: 255 – 284

DOI: https://dx.doi.org/10.4310/AMSA.2017.v2.n2.a3

Authors

Zheng Sun (Division of Applied Mathematics, Brown University, Providence, Rhode Island, U.S.A.)

Chi-Wang Shu (Division of Applied Mathematics, Brown University, Providence, Rhode Island, U.S.A.)

Abstract

In this paper, we analyze the stability of the fourth order Runge–Kutta method for integrating semi-discrete approximations of timedependent partial differential equations. Our study focuses on linear problems and covers general semi-bounded spatial discretizations. A counter example is given to show that the classical fourstage fourth order Runge–Kutta method can not preserve the one-step strong stability, even though the ordinary differential equation system is energy-decaying. But with an energy argument, we show that the strong stability property holds in two steps under an appropriate time step constraint. Based on this fact, the stability extends to general well-posed linear systems. As an application, we utilize the results to examine the stability of the fourth order Runge–Kutta approximations of several specific method of lines schemes for hyperbolic problems, including the spectral Galerkin method and the discontinuous Galerkin method.

Keywords

stability analysis, Runge–Kutta method, method of lines, spectral Galerkin method, discontinuous Galerkin method

2010 Mathematics Subject Classification

65L06, 65M12, 65M20

Received 12 August 2016

Published 10 August 2017