TY - JOUR

T1 - Upwinding sources at interfaces in conservation laws

AU - Katsaounis, Th

AU - Perthame, B.

AU - Simeoni, C.

N1 - 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.

PY - 2004/3

Y1 - 2004/3

N2 - 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.

AB - 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.

KW - Conservation laws

KW - Error estimates

KW - Finite-volume schemes

KW - Second-order approximations

KW - Upwinding source terms

UR - http://www.scopus.com/inward/record.url?scp=1642394095&partnerID=8YFLogxK

U2 - 10.1016/S0893-9659(04)90068-7

DO - 10.1016/S0893-9659(04)90068-7

M3 - Article

AN - SCOPUS:1642394095

VL - 17

SP - 309

EP - 316

JO - Applied Mathematics Letters

JF - Applied Mathematics Letters

SN - 0893-9659

IS - 3

ER -