High-order RKDG Methods for Computational Electromagnetics

In this talk, we devise a new Runge-Kutta Discontinuous Galerkin (RKDG) method that achieves full high-order convergence in time and space while keeping the timestep proportional to the spatial mesh-size. To this end, we derive an extension to non-autonomous linear systems of the mth-order, m-stage strong stability preserving Runge-Kutta (SSP-RK) scheme with low storage described in Gottlieb et al. [2]. With this time-integration scheme, and if polynomials of degree k are used in the space discretization, our RKDG method can be made to converge with overall order m = k + 1, for any k > 0. In particular, the scheme allows for a high-order accurate treatment of the inhomogeneous (time-dependent) terms that enter the semi-discrete problem on account of the physical boundary conditions. Therefore, we can implement the exact non-reflecting boundary conditions derived by Grote, Keller [3] and Sofronov [4]. Numerical results in two space dimensions are presented that confirm the predicted convergence properties.