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 L2 norm. Therefore, we first prove that the L2 norm of the α-th order (16α 6 k+1) divided difference of the DG error with upwind fluxes is of order k+ 3 2 − 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 − 2 for the negative-order norm, indicating that some particular compact kernels can be used to extract at least ( 3 2 k+1)th order superconvergence for nonlinear systems of conservation laws. Numerical experiments are shown to demonstrate the theoretical results.