NUMERICAL METHODS FOR NONCONSERVATIVE HYPERBOLIC SYSTEMS: A THEORETICAL FRAMEWORK. ∗ CARLOS PAR ´ ES † Abstract. The goal of this paper is to provide a theoretical framework allowing to extend some general concepts related to the numerical approximation of 1d conservation laws to the more general case of first order quasi-linear hyperbolic systems. In particular this framework is intended to be useful for the design and the analysis of well-balanced numerical schemes for solving balance laws or coupled systems of conservation laws. First, the concept of path-conservative numerical schemes is introduced, which is a generalization of the concept of conservative schemes for systems of conservation laws. Then, we introduce the general definition of Approximate Riemann Solvers and we give the general expression of some well-known families of schemes based on these solvers: Godunov, Roe and Relaxation methods. Finally, the general form of a high order scheme based on a first order path-conservative scheme and a reconstruction operator is presented. Key words. nonconservative products, finite volume method, well-balanced schemes, approxi- mate Riemann solvers, Godunov methods, Roe methods, relaxation methods, high order methods AMS subject classifications. 74S10, 65M06, 35L60, 35L65, 35L67 1. Introduction. The motivating question of this paper was the design of nu- merical schemes for P.D.E. Systems that can be written under the form: ∂ t w + ∂ x F (w)= B(w) · ∂ x w + S(w)∂ x σ, (1.1) where the unkonwn w(x, t) takes values on an open convex set D of R N ; F is a regular function from D to R N ; B is a regular matrix function from D to M N×N (R); S,a function from D to R N ; and σ(x), a known function from R to R. System (1.1) includes as particular cases: systems of conservation laws (B = 0, S = 0); systems of conservation laws with source term or balance laws (B = 0); and coupled system of balance laws as defined in [7]. More precisely, the discretization of the Shallow Water Systems that govern the flow of one shallow layer or two superposed shallow layers of immiscible homogeneous fluids was focused (see http://www.damflow.org) . The corresponding systems can be written respectively as a balance law or a coupled system of two balance laws. Systems with similar characteristics also appear in other flow models such as two- phase flows. It is well known that standard methods that solve correctly systems of conser- vation laws can fail in solving systems of balance laws, specially when approaching equilibria or near to equilibria solutions. Moreover, they can produce unstable meth- ods when they are applied to coupled systems of conservation or balance laws. Many authors have studied the definition of stable numerical schemes for systems or coupled systems of balance laws, which are well-balanced, that is, that preserve some equilib- ria: see [36], [3] , [38], [23], [24], [27], [17], [18], [31], [7], [39], [32], [10], [11], [2], [5], [29], [37], [12] . . . Among the main techniques used in the derivation of well-balanced numerical schemes, one of them consists in choosing first a standard conservative scheme for * This research has been partially supported by the Spanish Government Research project BFM2003-07530-C02-02. † Departamento de An´ alisis Matem´ atico, Facultad de Ciencias, Universidad de M´ alaga, 29071- M´ alaga, Spain (pares@anamat.cie.uma.es). 1