Sharp CFL, Discrete Kinetic Formulation, and Entropic Schemes for Scalar Conservation Laws

We consider semidiscrete and fully discrete conservative finite volume schemes approximating the solution to one dimensional scalar conservation law. We show that all E-schemes are associated with a discrete kinetic formulation with a nonnegative kinetic defect measure. This construction provides an alternative proof of the discrete local entropy inequalities with simple expressions of the discrete entropy fluxes. In contrast to the known results which are restricted to CFL of the form λQ ≤ 1/2, our proof holds under “sharp” CFL conditions.