### 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) |
---|---|

Pages (from-to) | 1-21 |

Number of pages | 21 |

Journal | Applied Numerical Mathematics |

Volume | 84 |

DOIs | |

State | 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

### Cite this

**A posteriori error estimates for a discontinuous Galerkin method applied to one-dimensional nonlinear scalar conservation laws.** / Baccouch, Mahboub.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - A posteriori error estimates for a discontinuous Galerkin method applied to one-dimensional nonlinear scalar conservation laws

AU - Baccouch, Mahboub

PY - 2014/10

Y1 - 2014/10

N2 - 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.

AB - 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.

KW - A posteriori error estimation

KW - Discontinuous Galerkin method

KW - Nonlinear conservation laws

KW - Superconvergence

UR - http://www.scopus.com/inward/record.url?scp=84901940202&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84901940202&partnerID=8YFLogxK

U2 - 10.1016/j.apnum.2014.04.001

DO - 10.1016/j.apnum.2014.04.001

M3 - Article

AN - SCOPUS:84901940202

VL - 84

SP - 1

EP - 21

JO - Applied Numerical Mathematics

JF - Applied Numerical Mathematics

SN - 0168-9274

ER -