Volume 14, Issue 4
Space-Time Discontinuous Galerkin Method for Maxwell's Equations

Ziqing Xie, Bo Wang & Zhimin Zhang

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

Published online: 2013-10

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  • 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 $\mathcal{O}((∆t)^{r+1}+h^{k+1/2})$ 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 $(∆t)^{2r+1}$ in time step is observed numerically for the numerical fluxes w.r.t. temporal variable at the grid points.

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@Article{CiCP-14-916, author = {}, title = {Space-Time Discontinuous Galerkin Method for Maxwell's Equations}, journal = {Communications in Computational Physics}, year = {2013}, volume = {14}, number = {4}, pages = {916--939}, 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 $\mathcal{O}((∆t)^{r+1}+h^{k+1/2})$ 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 $(∆t)^{2r+1}$ in time step is observed numerically for the numerical fluxes w.r.t. temporal variable at the grid points.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.230412.271212a}, url = {http://global-sci.org/intro/article_detail/cicp/7186.html} }
TY - JOUR T1 - Space-Time Discontinuous Galerkin Method for Maxwell's Equations JO - Communications in Computational Physics VL - 4 SP - 916 EP - 939 PY - 2013 DA - 2013/10 SN - 14 DO - http://doi.org/10.4208/cicp.230412.271212a UR - https://global-sci.org/intro/article_detail/cicp/7186.html KW - AB -

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 $\mathcal{O}((∆t)^{r+1}+h^{k+1/2})$ 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 $(∆t)^{2r+1}$ in time step is observed numerically for the numerical fluxes w.r.t. temporal variable at the grid points.

Ziqing Xie, Bo Wang & Zhimin Zhang. (2020). Space-Time Discontinuous Galerkin Method for Maxwell's Equations. Communications in Computational Physics. 14 (4). 916-939. doi:10.4208/cicp.230412.271212a
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