## Abstract

We obtain sharp pointwise error estimates for relaxation approximation to scalar conservation laws with piecewise smooth solutions. We first prove that the first-order partial derivatives for the perturbation solutions are uniformly upper bounded (the so-called Lip^{+} stability). A one-sided interpolation inequality between classical L^{1} error estimates and Lip^{+} stability bounds enables us to convert a global L^{1} result into a (nonoptimal) local estimate. Optimal error bounds on the weighted error then follow from the maximum principle for weakly coupled hyperbolic systems. The main difficulties in obtaining the Lip^{+} stability and the optimal pointwise errors are how to construct appropriate "difference functions" so that the maximum principle can be applied.

Original language | English |
---|---|

Pages (from-to) | 870-886 |

Number of pages | 17 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 32 |

Issue number | 4 |

DOIs | |

Publication status | Published - Jul 2000 |

## Scopus Subject Areas

- Analysis
- Computational Mathematics
- Applied Mathematics

## User-Defined Keywords

- Conservation laws
- Error estimates
- Maximum principle
- One-sided interpolation inequality
- Optimal convergence rate
- Relaxation method