A New Lax–Wendroff Discontinuous Galerkin Method with Superconvergence

Abstract

Superconvergence of discontinuous Galerkin (DG) methods for hyperbolic conservation laws has been intensively studied in different settings in the past. For example, the numerical solution by a semi-discrete DG scheme is superconvergent with order $$2k+1$$2k+1 in the negative-order norms, when the solution is globally smooth. Hence the accuracy of the numerical solution can be enhanced to $$(2k+1)$$(2k+1)th order accuracy by simply applying a carefully designed post-processor (Cockburn et al. in Math Comput 72:577---606, 2003). In this paper, we investigate superconvergence for the DG schemes coupled with Lax---Wendroff (LW) time discretization (LWDG). Through numerical experiments, we find that the original LWDG scheme developed in Qiu et al. (Comput Methods Appl Mech Eng 194:4528---4543, 2005) does not exhibit superconvergence properties mentioned above. In order to restore superconvergence, we propose to use the techniques from the local DG scheme to reconstruct high order spatial derivatives, while, in the original LWDG formulation, the high order derivatives are obtained by direct differentiation of the numerical solution. A collection of 1-D and 2-D numerical experiments are presented to verify superconvergence properties of the newly proposed LWDG scheme. We also perform Fourier analysis via symbolic computations to investigate the superconvergence of the proposed scheme.