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 α-th order (1 ≤ α≤ k+ 1) divided difference of the DG error in the L² norm is of order k+32-α2 when upwind fluxes are used, under the condition that | f^′(u) | possesses a uniform positive lower bound. By the duality argument, we then derive superconvergence results of order 2k+32-α2 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 (32k+1)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 k+ 1 in the L² norm for the divided differences of DG errors and thus (2 k+ 1) th order superconvergence in negative-order norm holds. Numerical experiments are given that confirm the theoretical results.