A Well-Balanced Scheme Using Non-Conservative Products Designed for Hyperbolic Systems of Conservation Laws With Source Terms

The aim of this paper is to present a new kind of numerical processing for hyperbolic systems of conservation laws with source terms. This is achieved by means of a non-conservative reformulation of the zero-order terms of the right-hand side of the equations. In this context, we decided to use the results of Dal Maso, Le Floch and Murat about non-conservative products, and the generalized Roe matrices introduced by Toumi to derive a first-order linearized well-balanced scheme in the sense of Greenberg and Le Roux. As a main feature, this approach is able to preserve the right asymptotic behavior of the original inhomogeneous system, which is not an obvious property. Numerical results for the Euler equations are shown to handle correctly these equilibria in various situations.