Recovery-Based Error Estimator for the Discontinuous Galerkin Method for Nonlinear Scalar Conservation Laws in One Space Dimension

Research output: Contribution to journalArticle

2 Scopus citations

Abstract

In this paper, we propose and analyze a robust recovery-based error estimator for the original discontinuous Galerkin method for nonlinear scalar conservation laws in one space dimension. The proposed a posteriori error estimator of the recovery-type is easy to implement, computationally simple, asymptotically exact, and is useful in adaptive computations. We use recent results (Meng et al. in SIAM J Numer Anal 50:2336–2356, 2012) to prove that, for smooth solutions, our a posteriori error estimates at a fixed time converge to the true spatial errors in the L2-norm under mesh refinement. The order of convergence is proved to be p + 1, when p-degree piecewise polynomials with p ≥ 1 are used. We further prove that the global effectivity index converges to unity at O(h) rate. Our proofs are valid for arbitrary regular meshes using Pp polynomials with p ≥ 1, under the condition that |f′(u)| possesses a uniform positive lower bound, where f(u) is the nonlinear flux function. We provide several numerical examples to support our theoretical results, to show the effectiveness of our recovery-based a posteriori error estimates, and to demonstrate that our results hold true for nonlinear conservation laws with general flux functions. These experiments indicate that the restriction on f(u) is artificial.

Original languageEnglish (US)
Pages (from-to)459-476
Number of pages18
JournalJournal of Scientific Computing
Volume66
Issue number2
DOIs
StatePublished - Feb 1 2016

Keywords

  • DG method
  • Derivative recovery technique
  • Nonlinear scalar conservation laws
  • Postprocessing
  • Recovery-based a posteriori error estimates

ASJC Scopus subject areas

  • Software
  • Theoretical Computer Science
  • Numerical Analysis
  • Engineering(all)
  • Computational Theory and Mathematics
  • Computational Mathematics
  • Applied Mathematics

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