### 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.

Language | English (US) |
---|---|

Pages | 19-54 |

Number of pages | 36 |

Journal | Discrete and Continuous Dynamical Systems - Series B |

Volume | 24 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2019 |

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### 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

**Superconvergence of the semi-discrete local discontinuous galerkin method for nonlinear KDV-type problems.** / Baccouch, Mahboub.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Superconvergence of the semi-discrete local discontinuous galerkin method for nonlinear KDV-type problems

AU - Baccouch, Mahboub

PY - 2019/1/1

Y1 - 2019/1/1

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.

KW - Local Discontinuous Galerkin Method Kdv Equations

KW - Stability A Priori Error Estimates Superconvergence Gauss-Radau Projections

UR - http://www.scopus.com/inward/record.url?scp=85055203165&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85055203165&partnerID=8YFLogxK

U2 - 10.3934/dcdsb.2018104

DO - 10.3934/dcdsb.2018104

M3 - Article

VL - 24

SP - 19

EP - 54

JO - Discrete and Continuous Dynamical Systems - Series B

T2 - Discrete and Continuous Dynamical Systems - Series B

JF - Discrete and Continuous Dynamical Systems - Series B

SN - 1531-3492

IS - 1

ER -