Space-Time Discontinuous Galerkin Method for Maxwell's Equations (Communications in Computational Physics, 2013)
A fully discrete discontinuous Galerkin method is introduced for solvingtime-dependent Maxwell's equations. Distinguished from the Runge-Kutta discontinuous Galerkin method (RKDG) and the finite element time domain method (FETD), inour scheme, discontinuous Galerkin methods are used to discretize not only the spatialdomain but also the temporal domain. The proposed numerical scheme is proved to beunconditionally 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 andspatial approximation, respectively. Numerical results in both 2-D and 3-D are provided to validate the theoretical prediction. An ultra-convergence of order $(∆t)^{2r+1}$ intime step is observed numerically for the numerical fluxes w.r.t. temporal variable atthe grid points.