Divided difference estimates and accuracy enhancement of discontinuous Galerkin methods for nonlinear symmetric systems of hyperbolic conservation laws

In this paper, we investigate the accuracy enhancement for the discontinuous Galerkin (DG) method for solving one-dimensional nonlinear symmetric systems of hyperbolic conservation laws. For nonlinear equations, the divided difference estimate is an important tool that allows for superconvergence of the post-processed solutions in the local L-2 norm. Therefore, we first prove that the L-2 norm of the alpha th-order (1 <= alpha <= k + 1) divided difference of the DG error with upwind fluxes is of order k + 3/2 - alpha/2, provided that the flux Jacobian matrix, f'(u), is symmetric positive definite. Furthermore, using the duality argument, we are able to derive superconvergence estimates of order 2k + 3/2 - alpha/2 for the negative-order norm, indicating that some particular compact kernels can be used to extract at least (3/2k + 1)th-order superconvergence for nonlinear systems of conservation laws. Numerical experiments are shown to demonstrate the theoretical results.