On an approximate Godunov scheme

We are interested in the numerical resolution of hyperbolic systems of cons ervation laws which do not allow any analytical calculation and for which i t is difficult to use classical schemes such as Roe's scheme. We introduce a new finite volume scheme called VFRoe. As the Roe scheme, it is based on the local resolution of a linearized Riemann problem. The numerical flux is defined following the Godunov scheme, as the physical flux evaluated at th e interface value of the linearized solver. The VFRoe scheme is conservativ e and consistent without fulfilling any Roe's type condition. Some numerica l tests on shock tube problems and two-phase flows problems are presented.