Discontinuous Galerkin methods for nonlinear scalar hyperbolic conservation laws: divided difference estimates and accuracy enhancement
报告摘要：In this talk, an analysis of the accuracy-enhancement for the discontinuous Galerkin (DG) method applied to one-dimensional scalarnonlinear 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 $alpha$-th order $(1 le alpha le k+1)$ divided difference of the DG error in the$L^2$ norm is of order $k+3/2-alpha/2$ 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 $2k+3/2-alpha/2$ in the negative-order norm, demonstrating that it is possibleto extend the Smoothness-Increasing Accuracy-Conserving filter to nonlinear conservation laws to obtain at least$3k/2+1$th order superconvergence for post-processed solutions. As a by-product,for variable coefficient hyperbolic equations, we provide an explicit proof foroptimal convergence results of order $k+1$ in the $L^2$ norm for the divided differences of DG errors and thus $(2k+1)$th order superconvergence in negative-order norm holds. Numerical experiments are given that confirm the theoretical results.