On the well-balance property of Roe's method for nonconservative hyperbolic systems.: Applications to shallow-water systems

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 [Bermudez and Vazquez-Cendon, Comput. Fluids 23 ( 1994) 1049-1071]; in the case of two layer flows, they are compared with the numerical scheme presented by [Castro, Macias and Pares, ESAIM: M2AN 35 ( 2001) 107-127].