## Abstract

Hyperbolic conservation laws with source terms arise in many applications, especially as a model for geophysical flows because of the gravity, and their numerical approximation leads to specific difficulties. In the context of finite-volume schemes, many authors have proposed to upwind sources at interfaces, the U.S.I. method, while a cell-centered treatment seems more natural. This note gives a general mathematical formalism for such schemes. We define consistency and give a stability condition for the U.S.I. method. We relate the notion of consistency to the "well-balanced" property, but its stability remains open, and we also study second-order approximations, as well as error estimates. The general case of a nonuniform spatial mesh is particularly interesting, motivated by two-dimensional problems set on unstructured grids.

Original language | English (US) |
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Pages (from-to) | 309-316 |

Number of pages | 8 |

Journal | Applied Mathematics Letters |

Volume | 17 |

Issue number | 3 |

DOIs | |

State | Published - Mar 2004 |

Externally published | Yes |

### Bibliographical note

Funding Information:This work was partially supported by the ACI (Minist~re de la Recherche, France): Mod4lisation de processus hydrauliques & surface libre en pr4sence de singularit4s (http://www-rocq. in_via, fr/m3n/CatNat/), and by HYKE European programme HPRN-CT-2002-00282 (http://~zw.hyke. org). The authors would like to thank F. Bouchut and Th. Gallou~t for helpful discussions.

## Keywords

- Conservation laws
- Error estimates
- Finite-volume schemes
- Second-order approximations
- Upwinding source terms

## ASJC Scopus subject areas

- Applied Mathematics