A posteriori error estimates for a discontinuous Galerkin method applied to one-dimensional nonlinear scalar conservation laws

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Abstract

In this paper, new a posteriori error estimates for a discontinuous Galerkin (DG) formulation applied to nonlinear scalar conservation laws in one space dimension are presented and analyzed. These error estimates are computationally simple and are obtained by solving a local problem with no boundary condition on each element of the mesh. We first show that the leading error term on each element for the solution is proportional to a (p+1)-degree Radau polynomial, when p-degree piecewise polynomials with p≤1 are used. This result allows us to prove that, for smooth solutions, these 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+5/4. Finally, we prove that the global effectivity indices in the L2-norm converge to unity at O(h1/2) rate. Our computational results indicate that the observed numerical convergence rates are higher than the theoretical rates.

Original languageEnglish (US)
Pages (from-to)1-21
Number of pages21
JournalApplied Numerical Mathematics
Volume84
DOIs
Publication statusPublished - Oct 2014

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Keywords

  • A posteriori error estimation
  • Discontinuous Galerkin method
  • Nonlinear conservation laws
  • Superconvergence

ASJC Scopus subject areas

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

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