American Institute of Mathematical Sciences

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October  2007, 8(3): 677-693. doi: 10.3934/dcdsb.2007.8.677

Efficient time discretization for local discontinuous Galerkin methods

Yinhua Xia 1, , Yan Xu 1, and  Chi-Wang Shu 2,

1. 

Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China, China
2. 

Division of Applied Mathematics, Brown University, Providence, RI 02912, United States

Received  July 2006 Revised  January 2007 Published  July 2007

In this paper, we explore three efficient time discretization techniques for the local discontinuous Galerkin (LDG) methods to solve partial differential equations (PDEs) with higher order spatial derivatives. The main difficulty is the stiffness of the LDG spatial discretization operator, which would require a unreasonably small time step for an explicit local time stepping method. We focus our discussion on the semi-implicit spectral deferred correction (SDC) method, and study its stability and accuracy when coupled with the LDG spatial discretization. We also discuss two other time discretization techniques, namely the additive Runge-Kutta (ARK) method and the exponential time differencing (ETD) method, coupled with the LDG spatial discretization. A comparison is made among these three time discretization techniques, to conclude that all three methods are efficient when coupled with the LDG spatial discretization for solving PDEs containing higher order spatial derivatives. In particular, the SDC method has the advantage of easy implementation for arbitrary order of accuracy, and the ARK method has the smallest CPU cost in our implementation.
Keywords: local discontinuous Galerkin method, additive Runge-Kutta method, higher order spatial derivatives., Spectral deferred correction method, exponential time differencing method.
Mathematics Subject Classification: Primary: 65L06, 65M6.
Citation: Yinhua Xia, Yan Xu, Chi-Wang Shu. Efficient time discretization for local discontinuous Galerkin methods. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 677-693. doi: 10.3934/dcdsb.2007.8.677
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