This paper is concerned with the numerical approximations of Cauchy problems for
one-dimensional nonconservative hyperbolic systems.
The first goal is to introduce a general concept of well-balancing
for numerical schemes solving this kind of systems. Once this concept stated, we
investigate the well-balance properties of numerical schemes based on the
generalized Roe linearizations introduced by [Toumi, J. Comp. Phys.
102 (1992) 360–373]. Next, this general theory
is applied to obtain well-balanced schemes for solving coupled systems of conservation laws with
source terms. Finally, we focus on applications to shallow water systems: the numerical
schemes obtained and their properties are compared, in the case of one layer flows, with those introduced by
[Bermúdez and Vázquez-Cendón, Comput. Fluids
23 (1994) 1049–1071]; in the case of two layer flows, they are
compared with the numerical scheme presented by [Castro, Macías and Parés, ESAIM: M2AN
35 (2001) 107–127].