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

Zheng Sun Chi-Wang Shu Brown University

Numerical Analysis and Scientific Computing mathscidoc:1804.25002

Annals of Mathematical Sciences and Applications, 2, 255-284, 2017
In this paper, we analyze the stability of the fourth order Runge-Kutta method for integrating semi-discrete approximations of time-dependent 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 four-stage 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.
Stability analysis; Runge-Kutta method; method of lines; spectral Galerkin method; discontinuous Galerkin method
[ Download ] [ 2018-04-16 09:15:12 uploaded by chiwangshu ] [ 602 downloads ] [ 0 comments ]
@inproceedings{zheng2017stability,
  title={Stability of the fourth order Runge-Kutta method for time-dependent partial differential equations},
  author={Zheng Sun, and Chi-Wang Shu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180416091512681574037},
  booktitle={Annals of Mathematical Sciences and Applications},
  volume={2},
  pages={255-284},
  year={2017},
}
Zheng Sun, and Chi-Wang Shu. Stability of the fourth order Runge-Kutta method for time-dependent partial differential equations. 2017. Vol. 2. In Annals of Mathematical Sciences and Applications. pp.255-284. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180416091512681574037.
Please log in for comment!
 
 
Contact us: office-iccm@tsinghua.edu.cn | Copyright Reserved