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

Xiong Meng, Jennifer Ryan

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9 Citations (Scopus)
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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.
Original languageEnglish
Pages (from-to)27–73
Number of pages47
JournalNumerische Mathematik
Volume136
Issue number1
Early online date8 Aug 2016
DOIs
Publication statusPublished - May 2017

Keywords

  • Discontinuous Galerkin method
  • Nonlinear hyperbolic conservation laws
  • Negative-order norm error estimates
  • Post-processing
  • Divided difference
  • SIAC filtering

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