## Abstract

In this paper we address the questions of the convergence rate for approximate solutions to conservation laws with piecewise smooth solutions in a weighted W^{1,1} space. Convergence rate for the derivative of the approximate solutions is established under the assumption that a weak pointwise-error estimate is given. In other words, we are able to convert weak pointwise-error estimates to optimal error bounds in a weighted W^{1,1} space. For convex conservation laws, the assumption of a weak pointwise-error estimate is verified by Tadmor [SIAM J. Numer. Anal., 28 (1991), pp. 891-906]. Therefore, one immediate application of our W^{1,1}- convergence theory is that for convex conservation laws we indeed have W^{1,1}-error bounds for the approximate solutions to conservation laws. Furthermore, the Script O sign(ε)-pointwise-error estimates of Tadmor and Tang [SIAM J. Numer. Anal., 36 (1999), pp. 1739-1758] are recovered by the use of the W^{1,1}-convergence result.

Original language | English |
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Pages (from-to) | 1483-1495 |

Number of pages | 13 |

Journal | SIAM Journal on Numerical Analysis |

Volume | 38 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2001 |

## Scopus Subject Areas

- Numerical Analysis
- Computational Mathematics
- Applied Mathematics

## User-Defined Keywords

- Conservation laws
- Error estimates
- Optimal convergence rate
- Viscosity approximation