Discontinuous Galerkin methods for nonlinear scalar hyperbolic conservation laws:divided difference estimates and accuracy enhancement

Meng, Xiong and Ryan, Jennifer (2017) Discontinuous Galerkin methods for nonlinear scalar hyperbolic conservation laws:divided difference estimates and accuracy enhancement. Numerische Mathematik, 136 (1). 27–73. ISSN 0029-599X

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Abstract

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 alpha-th order (1 <= \alpha <= k+1) divided difference of the DG error in the L2-norm is of order k+(3-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 k+(3-alpha)/2 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 (3k/2+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 (2k+1)th order superconvergence in negative-order norm holds. Numerical experiments are given that confirm the theoretical results.

Item Type: Article
Additional Information: © The Author(s) 2016 Open Access: This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Uncontrolled Keywords: discontinuous galerkin method,nonlinear hyperbolic conservation laws,negative-order norm error estimates,post-processing,divided difference,siac filtering
Faculty \ School: Faculty of Science > School of Mathematics
Depositing User: Pure Connector
Date Deposited: 24 Sep 2016 00:06
Last Modified: 11 Jun 2020 00:01
URI: https://ueaeprints.uea.ac.uk/id/eprint/59831
DOI: 10.1007/s00211-016-0833-y

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