Convergence of the Upwind Interface Source method for hyperbolic conservation laws

Benoît Perthame (1) , Chiara Simeoni (1)

1 DMA - Département de Mathématiques et Applications - ENS Paris

This paper deals with typical questions arising in the analysis of numerical approximations for scalar conservation laws with a source term. We focus our attention on semi-discrete finite volume schemes, in the general case of a nonuniform spatial mesh. To define appropriate discretizations of the source term, we introduce the formalism peculiar to the Upwind Interface Source method and we establish conditions on the numerical functions so that the discrete solver preserves the steady state solutions. Then we formulate a rigorous definition of consistency, adapted to the class of well-balanced schemes, for which we are able to prove a Lax-Wendroff type convergence theorem. Some examples of numerical methods are discussed, in order to validate the arguments we propose.

Mots clés: Lax-Wendroff theorem; consistency; scalar conservation laws; entropy inequalities; nonuniform grids; well-balanced schemes; source terms

Domaines: Analyse numérique [math.NA]

Benoît Perthame, Chiara Simeoni. Convergence of the Upwind Interface Source method for hyperbolic conservation laws. Proceedings of the Ninth International Conference on Hyperbolic Problems, Mar 2002, CalTech, Pasadena, United States. pp.61-78, ⟨10.1007/978-3-642-55711-8_5⟩. ⟨hal-00922680v2⟩