Optimal error estimates of the local discontinuous galerkin method for the two-dimensional sine-gordon equation on cartesian grids

Research output: Contribution to journalArticle

Abstract

The sine-Gordon equation is one of the basic equations in modern nonlinear wave theory. It has applications in many areas of physics and mathematics. In this paper, we develop and analyze an energy-conserving local discontinuous Galerkin (LDG) method for the two-dimensional sine-Gordon nonlinear hyperbolic equation on Cartesian grids. We prove the energy conserving property, the L 2 stability, and optimal L 2 error estimates for the semi-discrete method. More precisely, we identify special numerical fluxes and a suitable projection of the initial conditions for the LDG scheme to achieve p + 1 order of convergence for both the potential and its gradient in the L 2 -norm, when tensor product polynomials of degree at most p are used. We present several numerical examples to validate the theoretical results. Our numerical examples show the sharpness of the O(h p +1) estimate.

Original languageEnglish (US)
Pages (from-to)436-462
Number of pages27
JournalInternational Journal of Numerical Analysis and Modeling
Volume16
Issue number3
StatePublished - Jan 1 2019

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sine-Gordon equation
Local Discontinuous Galerkin Method
Cartesian Grid
Sine-Gordon Equation
Optimal Error Estimates
Galerkin methods
Tensors
Physics
Polynomials
Fluxes
Nonlinear Hyperbolic Equation
Numerical Examples
Discontinuous Galerkin
Sharpness
Nonlinear Waves
Order of Convergence
Energy
Tensor Product
Error Estimates
Initial conditions

Keywords

  • A priori error estimates
  • Cartesian grids
  • Energy conservation
  • L stability
  • Local discontinuous galerkin method
  • Sine-Gordon equation

ASJC Scopus subject areas

  • Numerical Analysis

Cite this

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title = "Optimal error estimates of the local discontinuous galerkin method for the two-dimensional sine-gordon equation on cartesian grids",
abstract = "The sine-Gordon equation is one of the basic equations in modern nonlinear wave theory. It has applications in many areas of physics and mathematics. In this paper, we develop and analyze an energy-conserving local discontinuous Galerkin (LDG) method for the two-dimensional sine-Gordon nonlinear hyperbolic equation on Cartesian grids. We prove the energy conserving property, the L 2 stability, and optimal L 2 error estimates for the semi-discrete method. More precisely, we identify special numerical fluxes and a suitable projection of the initial conditions for the LDG scheme to achieve p + 1 order of convergence for both the potential and its gradient in the L 2 -norm, when tensor product polynomials of degree at most p are used. We present several numerical examples to validate the theoretical results. Our numerical examples show the sharpness of the O(h p +1) estimate.",
keywords = "A priori error estimates, Cartesian grids, Energy conservation, L stability, Local discontinuous galerkin method, Sine-Gordon equation",
author = "Mahboub Baccouch",
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N2 - The sine-Gordon equation is one of the basic equations in modern nonlinear wave theory. It has applications in many areas of physics and mathematics. In this paper, we develop and analyze an energy-conserving local discontinuous Galerkin (LDG) method for the two-dimensional sine-Gordon nonlinear hyperbolic equation on Cartesian grids. We prove the energy conserving property, the L 2 stability, and optimal L 2 error estimates for the semi-discrete method. More precisely, we identify special numerical fluxes and a suitable projection of the initial conditions for the LDG scheme to achieve p + 1 order of convergence for both the potential and its gradient in the L 2 -norm, when tensor product polynomials of degree at most p are used. We present several numerical examples to validate the theoretical results. Our numerical examples show the sharpness of the O(h p +1) estimate.

AB - The sine-Gordon equation is one of the basic equations in modern nonlinear wave theory. It has applications in many areas of physics and mathematics. In this paper, we develop and analyze an energy-conserving local discontinuous Galerkin (LDG) method for the two-dimensional sine-Gordon nonlinear hyperbolic equation on Cartesian grids. We prove the energy conserving property, the L 2 stability, and optimal L 2 error estimates for the semi-discrete method. More precisely, we identify special numerical fluxes and a suitable projection of the initial conditions for the LDG scheme to achieve p + 1 order of convergence for both the potential and its gradient in the L 2 -norm, when tensor product polynomials of degree at most p are used. We present several numerical examples to validate the theoretical results. Our numerical examples show the sharpness of the O(h p +1) estimate.

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