Discontinuous Galerkin Method For Solving Time-dependent Maxwell Equations

In this paper, discontinuous Galerkin(DG) methods for solving time de-pendent Maxwell's equations are investigated. Both semi-discrete numerical scheme and space-time fully discrete scheme based on DG methods are pro-posed for solving Maxwell's equations. Simple media and some important kinds of dispersive media (cold plasma, Lorentz medium and Debye medium, etc) are discussed respectively.In this paper a semi-discrete scheme based on DG method for solving Maxwell's equations in simple media is established first. Corresponding theo-retical analysis and numerical results are given. We prove that this scheme is stable and has a convergence rate of order O(hk+1/2) in L2-norm. In order to obtain numerical results, we use continuous finite element method to solve the reduced system of ordinary differential equations obtained after the spatial discretization. Numerical results show a convergence rate of order O(hk+1), which is 1/2-order higher than our theoretical prediction.Next, we turn to discuss some important kinds of dispersive media (cold plasma, Lorentz medium and Debye medium, etc). A unified mathematical model is proposed to generalize some physical models who govern the electro-magnetic wave propagation in these dispersive media. Semi-discrete numerical scheme is constructed. Theoretical analysis and numerical results show the same results as we have obtained for simple media.Besides the investigation on semi-discrete numerical schemes, we mainly investigate a space-time fully discrete scheme based on DG method for solving time-dependent Maxwell's equations. Maxwell's equations in simple media and dispersive media are discussed, respectively. Distinguished from the Runge-Kutta discontinuous Galerkin method (RKDG) and the finite element time domain method (FETD), in our space-time fully discrete scheme, DG meth-ods are used to discretize not only the spatial domain but also the temporal domain. The proposed numerical scheme is proved to be unconditionally sta-ble, and a convergence rate of order O(τr+1+hk+1/2) is established under the L2-norm. Numerical results in 2-D and 3-D are provided to validate the the- oretical prediction. Furthermore, an ultra-convergence of orderτ2r+1 in time step is observed for the numerical fluxes w.r.t. temporal variable at the grid points.