### 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 language | English (US) |
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Pages (from-to) | 19-54 |

Number of pages | 36 |

Journal | Discrete and Continuous Dynamical Systems - Series B |

Volume | 24 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 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