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.
Bibliographical noteFunding Information:
Bernardo Cockburn gratefully acknowledges support from NSF through grant No. DMS 0107609. Fernando Reitich gratefully acknowledges support from NSF through grant No. DMS-0311763, from AFOSR through contract No. F49620-02-1-0052 and from the Army High Performance Computing Research Center (AHPCRC) under Army Research Laboratory cooperative agreement number DAAD19-01-2-0014.
Disclaimer. Effort sponsored by the Air Force Office of Scientific Research, Air Force Materials Command, USAF, under Grant Number F49620-02-1-0052, and by AHPCRC under the auspices of the Department of the Army, Army Research Laboratory cooperative agreement number DAAD19-01-2-0014. The US Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the author and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Air Force Office of Scientific Research, the Army Research Laboratory or the US Government.
- Discontinuous Galerkin methods
- Maxwell equations
- Wave propagation