On an approximate Godunov scheme

We are interested in the numerical resolution of hyperbolic systems of conservation laws which do not allow any analytical calculation and for which it 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 the interface value of the linearized solver. The VFRoe scheme is conservative and consistent without fulfilling any Roe's type condition. Some numerical tests on shock tube problems and two-phase flows problems are presented.