Entropy satisfying flux vector splittings and kinetic BGK models

We establish forward and backward relations between entropy satisfying BGK relaxation models such as those introduced previously by the author and the first order flux vector splitting numerical methods for general systems of conservation laws. Classically, to a kinetic BGK model that is compatible with some family of entropies we can associate an entropy flux vector splitting. We prove that the converse is true: any entropy flux vector splitting can be interpreted by a kinetic model, and we obtain an explicit characterization of entropy satisfying flux vector splitting schemes. We deduce a new proof of discrete entropy inequalities under a sharp CFL condition that generalizes the monotonicity criterion in the scalar case. In particular, this gives a stability condition for numerical kinetic methods with noncompact velocity support. A new interpretation of general kinetic schemes is also provided via approximate Riemann solvers. We deduce the construction of finite velocity relaxation systems for gas dynamics, and obtain a HLLC scheme for which we are able to prove positiveness of density and internal energy, and discrete entropy inequalities.