Forward and backward relations are established 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 it is possible to associate an entropy flux vector splitting. It is proved that the converse is true: any entropy flux vector splitting can be interpreted by a kinetic model, and an explicit characterization of entropy satisfying flux vector splitting schemes can be obtained.
A new proof of discrete entropy inequalities under a sharp CFL condition that generalizes the monotonicity criterion in the scalar case is deduced. 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. The construction of finite velocity relaxation systems for gas dynamics is deduced, and a HLLC scheme is obtained for which it is possible to prove positiveness of density and internal energy, and discrete entropy inequalities.