Discontinuous Galerkin methods for nonlinear scalar hyperbolic conservation laws: divided difference estimates and accuracy enhancement

In this paper, an analysis of the accuracy-enhancement for the discontinuous Galerkin (DG) method applied to one-dimensional scalar nonlinear hyperbolic conservation laws is carried out. This requires analyzing the divided difference of the errors for the DG solution. We therefore first prove that the -th order divided difference of the DG error in the norm is of order when upwind fluxes are used, under the condition that possesses a uniform positive lower bound. By the duality argument, we then derive superconvergence results of order in the negative-order norm, demonstrating that it is possible to extend the Smoothness-Increasing Accuracy-Conserving filter to nonlinear conservation laws to obtain at least th order superconvergence for post-processed solutions. As a by-product, for variable coefficient hyperbolic equations, we provide an explicit proof for optimal convergence results of order in the norm for the divided differences of DG errors and thus th order superconvergence in negative-order norm holds. Numerical experiments are given that confirm the theoretical results.