### Abstract

In this paper, we propose and analyze a superconvergent discontinuous Galerkin (DG) method for nonlinear second-order initial-value problems for ordinary differential equations. Optimal a priori error estimates for the solution and for the auxiliary variable that approximates the first-order derivative are derived in the L^{2}-norm. 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 p-degree DG solutions are O(h^{2p+1}) superconvergent at the downwind points. Finally, we prove that the DG solutions are superconvergent with order p+2 to a particular projection of the exact solutions. The proofs are valid for arbitrary nonuniform regular meshes and for piecewise P^{p} polynomials with arbitrary p≥1. Computational results indicate that the theoretical orders of convergence and superconvergence are optimal.

Original language | English (US) |
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Pages (from-to) | 160-179 |

Number of pages | 20 |

Journal | Applied Numerical Mathematics |

Volume | 115 |

DOIs | |

State | Published - May 1 2017 |

### Keywords

- A priori error estimates
- Discontinuous Galerkin method
- Nonlinear second-order ordinary differential equations equation
- Superconvergence

### ASJC Scopus subject areas

- Numerical Analysis
- Computational Mathematics
- Applied Mathematics