Commun. Comput. Phys., 14 (2013), pp. 916-939.


Space-Time Discontinuous Galerkin Method for Maxwell's Equations

Ziqing Xie 1, Bo Wang 2*, Zhimin Zhang 3

1 School of Mathematics and Computer Science, Guizhou Normal University, Guiyang, Guizhou 550001, China; Key Laboratory of High Performance Computing and Stochastic Information Processing, Ministry of Education of China, Hunan Normal University, Changsha, Hunan 410081, China.
2 College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan 410081, China; Singapore-MIT Alliance, 4 Engineering Drive 3, National University of Singapore, Singapore 117576, Singapore.
3 Department of Mathematics, Wayne State University, Detroit, MI 48202, USA; Guangdong Province Key Laboratory of Computational Science, School of Mathematics and Computational Science, Sun Yat-sen University, Guangzhou, 510275, China.

Received 23 April 2012; Accepted (in revised version) 27 December 2012
Available online 19 March 2013
doi:10.4208/cicp.230412.271212a

Abstract

A fully discrete discontinuous Galerkin method is introduced for solving time-dependent Maxwell's equations. Distinguished from the Runge-Kutta discontinuous Galerkin method (RKDG) and the finite element time domain method (FETD), in our scheme, discontinuous Galerkin methods are used to discretize not only the spatial domain but also the temporal domain. The proposed numerical scheme is proved to be unconditionally stable, and a convergent rate $O((\triangle t)^{r+1} +h^{k+1/2})$ is established under the L^2-norm when polynomials of degree at most r and k are used for temporal and spatial approximation, respectively. Numerical results in both 2-D and 3-D are provided to validate the theoretical prediction. An ultra-convergence of order $(\triangle t)^{2r+1}$ in time step is observed numerically for the numerical fluxes w.r.t. temporal variable at the grid points.

AMS subject classifications: 35Q61, 65M12, 65M60, 65N12

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Key words: Discontinuous Galerkin method, Maxwell's equations, full-discretization, L^2-error estimate, L^2-stability, ultra-convergence.

*Corresponding author.
Email: ziqingxie@yahoo.com.cn (Z. Q. Xie), bowanghn@gmail.com (B. Wang), ag7761@wayne.edu (Z. Zhang)
 

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