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 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.

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