### Abstract

In this paper, new a posteriori error estimates for a discontinuous Galerkin (DG) formulation applied to nonlinear scalar conservation laws in one space dimension are presented and analyzed. These error estimates are computationally simple and are obtained by solving a local problem with no boundary condition on each element of the mesh. We first show that the leading error term on each element for the solution is proportional to a (p+1)-degree Radau polynomial, when p-degree piecewise polynomials with p≤1 are used. This result allows us to prove that, for smooth solutions, these error estimates at a fixed time converge to the true spatial errors in the ^{L2}-norm under mesh refinement. The order of convergence is proved to be p+5/4. Finally, we prove that the global effectivity indices in the ^{L2}-norm converge to unity at O(h1^{/2}) rate. Our computational results indicate that the observed numerical convergence rates are higher than the theoretical rates.

Original language | English (US) |
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Pages (from-to) | 1-21 |

Number of pages | 21 |

Journal | Applied Numerical Mathematics |

Volume | 84 |

DOIs | |

Publication status | Published - Oct 2014 |

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### Keywords

- A posteriori error estimation
- Discontinuous Galerkin method
- Nonlinear conservation laws
- Superconvergence

### ASJC Scopus subject areas

- Numerical Analysis
- Computational Mathematics
- Applied Mathematics