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.

LanguageEnglish (US)
Pages19-54
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.",
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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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