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
StatePublished - Oct 2014

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Scalar Conservation Laws
A Posteriori Error Estimates
Discontinuous Galerkin Method
Galerkin methods
Error Estimates
Conservation
Converge
Norm
Discontinuous Galerkin
Mesh Refinement
Piecewise Polynomials
Order of Convergence
Smooth Solution
Error term
Convergence Rate
Computational Results
Directly proportional
Mesh
Boundary conditions
Polynomial

Keywords

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

ASJC Scopus subject areas

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

Cite this

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title = "A posteriori error estimates for a discontinuous Galerkin method applied to one-dimensional nonlinear scalar conservation laws",
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.",
keywords = "A posteriori error estimation, Discontinuous Galerkin method, Nonlinear conservation laws, Superconvergence",
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N2 - 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.

AB - 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.

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KW - Superconvergence

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