High-order RKDG methods for computational electromagnetics

In this paper we introduce a new RKDG method for problems of wave propagation that achieves full high-order convergence in time and space. The novelty of the method resides in the way in which it marches in time. It uses an mth-order m-stage, low storage SSP-RK scheme which is an extension to a class of non-autonomous linear systems of a recently designed method for autonomous linear systems. This extension 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. Thus, if polynomials of degree k are used in the space discretization, the RKDG method is of overall order m = k + 1, for any k > 0. Moreover, we also show that the attainment of high-order space-time accuracy allows for an efficient implementation of post-processing techniques that can double the convergence order. We explore this issue in a one-dimensional setting and show that the superconvergence of fluxes previously observed in full space-time DG formulations is also attained in our new RKDG scheme. This allows for the construction of higher-order solutions via local interpolating polynomials. Indeed, if polynomials of degree k are used in the space discretization together with a time-marching method of order 2k + 1, a post-processed approximation of order 2k + 1 is obtained. Numerical results in one and two space dimensions are presented that confirm the predicted convergence properties.