Superconvergence of the local discontinuous Galerkin method for the sine-Gordon equation in one space dimension

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Abstract

In this paper, we present superconvergence results for the local discontinuous Galerkin (LDG) method for the sine-Gordon nonlinear hyperbolic equation in one space dimension. We identify a special numerical flux and a suitable projection of the initial conditions for the LDG scheme for which the L2-norm of the LDG solution and its spatial derivative are of order p+1, when piecewise polynomials of degree at most p are used. Our numerical experiments demonstrate optimal order of convergence. We further prove superconvergence toward particular projections of the exact solutions. More precisely, we prove that the LDG solution and its spatial derivative are O(hp+3∕2) super close to particular projections of the exact solutions, while computational results show higher O(hp+2) convergence rate. Our analysis is valid for arbitrary regular meshes and for Pp polynomials with arbitrary p≥1. Numerical experiments validating these theoretical results are presented.

Original languageEnglish (US)
Pages (from-to)292-313
Number of pages22
JournalJournal of Computational and Applied Mathematics
Volume333
DOIs
StatePublished - May 1 2018

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sine-Gordon equation
Local Discontinuous Galerkin Method
Sine-Gordon Equation
Discontinuous Galerkin
Superconvergence
Galerkin methods
Polynomials
Projection
Derivatives
Exact Solution
Numerical Experiment
Nonlinear Hyperbolic Equation
Derivative
Experiments
Piecewise Polynomials
Order of Convergence
Arbitrary
Fluxes
Convergence Rate
Computational Results

Keywords

  • Error estimates
  • Local discontinuous Galerkin method
  • Projections
  • Sine-Gordon equation
  • Superconvergence

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

Cite this

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title = "Superconvergence of the local discontinuous Galerkin method for the sine-Gordon equation in one space dimension",
abstract = "In this paper, we present superconvergence results for the local discontinuous Galerkin (LDG) method for the sine-Gordon nonlinear hyperbolic equation in one space dimension. We identify a special numerical flux and a suitable projection of the initial conditions for the LDG scheme for which the L2-norm of the LDG solution and its spatial derivative are of order p+1, when piecewise polynomials of degree at most p are used. Our numerical experiments demonstrate optimal order of convergence. We further prove superconvergence toward particular projections of the exact solutions. More precisely, we prove that the LDG solution and its spatial derivative are O(hp+3∕2) super close to particular projections of the exact solutions, while computational results show higher O(hp+2) convergence rate. Our analysis is valid for arbitrary regular meshes and for Pp polynomials with arbitrary p≥1. Numerical experiments validating these theoretical results are presented.",
keywords = "Error estimates, Local discontinuous Galerkin method, Projections, Sine-Gordon equation, Superconvergence",
author = "Mahboub Baccouch",
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T1 - Superconvergence of the local discontinuous Galerkin method for the sine-Gordon equation in one space dimension

AU - Baccouch, Mahboub

PY - 2018/5/1

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N2 - In this paper, we present superconvergence results for the local discontinuous Galerkin (LDG) method for the sine-Gordon nonlinear hyperbolic equation in one space dimension. We identify a special numerical flux and a suitable projection of the initial conditions for the LDG scheme for which the L2-norm of the LDG solution and its spatial derivative are of order p+1, when piecewise polynomials of degree at most p are used. Our numerical experiments demonstrate optimal order of convergence. We further prove superconvergence toward particular projections of the exact solutions. More precisely, we prove that the LDG solution and its spatial derivative are O(hp+3∕2) super close to particular projections of the exact solutions, while computational results show higher O(hp+2) convergence rate. Our analysis is valid for arbitrary regular meshes and for Pp polynomials with arbitrary p≥1. Numerical experiments validating these theoretical results are presented.

AB - In this paper, we present superconvergence results for the local discontinuous Galerkin (LDG) method for the sine-Gordon nonlinear hyperbolic equation in one space dimension. We identify a special numerical flux and a suitable projection of the initial conditions for the LDG scheme for which the L2-norm of the LDG solution and its spatial derivative are of order p+1, when piecewise polynomials of degree at most p are used. Our numerical experiments demonstrate optimal order of convergence. We further prove superconvergence toward particular projections of the exact solutions. More precisely, we prove that the LDG solution and its spatial derivative are O(hp+3∕2) super close to particular projections of the exact solutions, while computational results show higher O(hp+2) convergence rate. Our analysis is valid for arbitrary regular meshes and for Pp polynomials with arbitrary p≥1. Numerical experiments validating these theoretical results are presented.

KW - Error estimates

KW - Local discontinuous Galerkin method

KW - Projections

KW - Sine-Gordon equation

KW - Superconvergence

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