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

作者：Meng, Xiong; Ryan, Jennifer K.^*

来源：IMA Journal of Numerical Analysis, 2018, 38(1): 125-155.

DOI：10.1093/imanum/drw072

• discontinuous Galerkin method
• nonlinear symmetric systems of hyperbolic conservation laws
• negative-order norm estimates
• post-processing
• divided difference

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.

• 出版日期2018-1
• 单位哈尔滨工业大学