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

Xiong Meng, Jennifer K. Ryan

Research output: Contribution to journalArticle

3 Citations (Scopus)
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Abstract

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 (1≤ α≤ k+1) divided difference of the DG error with upwind fluxes is of order k+(3-α)/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-α)/2 for the negative-order norm, indicating that some particular compact kernels can be used to extract at least (3k/2+1)-th order superconvergence for nonlinear systems of conservation laws.
Numerical experiments are shown to demonstrate the theoretical results.
Original languageEnglish
Pages (from-to)125–155
JournalIMA Journal of Numerical Analysis
Volume38
Issue number1
Early online date20 Feb 2017
DOIs
Publication statusPublished - 25 Jan 2018

Keywords

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

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