Superconvergence of the semi-discrete local discontinuous galerkin method for nonlinear KDV-type problems

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Abstract

In this paper, we present and analyze a superconvergent local discontinuous Galerkin (LDG) scheme for the numerical solution of nonlinear KdV-type partial differential equations. Optimal a priori error estimates for the LDG solution and for the two auxiliary variables that approximate the firstand second-order derivative are derived in the L2-norm for the semi-discrete formulation. The order of convergence is proved to be p + 1, when piecewise polynomials of degree at most p are used. We further prove that the derivative of the LDG solution is superconvergent with order p + 1 towards the derivative of a special projection of the exact solution. We use this results to prove that the LDG solution is superconvergent with order p + 3/2 toward a special Gauss-Radau projection of the exact solution. Finally, several numerical examples are given to validate the theoretical results. Our proofs are valid for arbitrary regular meshes using Pp polynomials with p ≥ 1 and under the condition that |f'(u)| possesses a uniform positive lower bound, where f(u) is the nonlinear flux function. Our experiments demonstrate that our results hold true for KdV equations with general flux functions.

Original languageEnglish (US)
Pages (from-to)19-54
Number of pages36
JournalDiscrete and Continuous Dynamical Systems - Series B
Volume24
Issue number1
DOIs
StatePublished - Jan 1 2019

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Local Discontinuous Galerkin Method
Discontinuous Galerkin
Superconvergence
Galerkin methods
Derivatives
Polynomials
Fluxes
Exact Solution
Projection
Derivative
Partial differential equations
A Priori Error Estimates
Second-order Derivatives
Optimal Error Estimates
Auxiliary Variables
Piecewise Polynomials
KdV Equation
Order of Convergence
Korteweg-de Vries Equation
Gauss

Keywords

  • Local Discontinuous Galerkin Method Kdv Equations
  • Stability A Priori Error Estimates Superconvergence Gauss-Radau Projections

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

Cite this

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title = "Superconvergence of the semi-discrete local discontinuous galerkin method for nonlinear KDV-type problems",
abstract = "In this paper, we present and analyze a superconvergent local discontinuous Galerkin (LDG) scheme for the numerical solution of nonlinear KdV-type partial differential equations. Optimal a priori error estimates for the LDG solution and for the two auxiliary variables that approximate the firstand second-order derivative are derived in the L2-norm for the semi-discrete formulation. The order of convergence is proved to be p + 1, when piecewise polynomials of degree at most p are used. We further prove that the derivative of the LDG solution is superconvergent with order p + 1 towards the derivative of a special projection of the exact solution. We use this results to prove that the LDG solution is superconvergent with order p + 3/2 toward a special Gauss-Radau projection of the exact solution. Finally, several numerical examples are given to validate the theoretical results. Our proofs are valid for arbitrary regular meshes using Pp polynomials with p ≥ 1 and under the condition that |f'(u)| possesses a uniform positive lower bound, where f(u) is the nonlinear flux function. Our experiments demonstrate that our results hold true for KdV equations with general flux functions.",
keywords = "Local Discontinuous Galerkin Method Kdv Equations, Stability A Priori Error Estimates Superconvergence Gauss-Radau Projections",
author = "Mahboub Baccouch",
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doi = "10.3934/dcdsb.2018104",
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N2 - In this paper, we present and analyze a superconvergent local discontinuous Galerkin (LDG) scheme for the numerical solution of nonlinear KdV-type partial differential equations. Optimal a priori error estimates for the LDG solution and for the two auxiliary variables that approximate the firstand second-order derivative are derived in the L2-norm for the semi-discrete formulation. The order of convergence is proved to be p + 1, when piecewise polynomials of degree at most p are used. We further prove that the derivative of the LDG solution is superconvergent with order p + 1 towards the derivative of a special projection of the exact solution. We use this results to prove that the LDG solution is superconvergent with order p + 3/2 toward a special Gauss-Radau projection of the exact solution. Finally, several numerical examples are given to validate the theoretical results. Our proofs are valid for arbitrary regular meshes using Pp polynomials with p ≥ 1 and under the condition that |f'(u)| possesses a uniform positive lower bound, where f(u) is the nonlinear flux function. Our experiments demonstrate that our results hold true for KdV equations with general flux functions.

AB - In this paper, we present and analyze a superconvergent local discontinuous Galerkin (LDG) scheme for the numerical solution of nonlinear KdV-type partial differential equations. Optimal a priori error estimates for the LDG solution and for the two auxiliary variables that approximate the firstand second-order derivative are derived in the L2-norm for the semi-discrete formulation. The order of convergence is proved to be p + 1, when piecewise polynomials of degree at most p are used. We further prove that the derivative of the LDG solution is superconvergent with order p + 1 towards the derivative of a special projection of the exact solution. We use this results to prove that the LDG solution is superconvergent with order p + 3/2 toward a special Gauss-Radau projection of the exact solution. Finally, several numerical examples are given to validate the theoretical results. Our proofs are valid for arbitrary regular meshes using Pp polynomials with p ≥ 1 and under the condition that |f'(u)| possesses a uniform positive lower bound, where f(u) is the nonlinear flux function. Our experiments demonstrate that our results hold true for KdV equations with general flux functions.

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