High-order finite volume schemes and entropy inequalities

This work concerns the numerical approximations of the weak solutions of scalar hyperbolic conservation laws. After showing how to bypass the discrete entropy inequality barrier theorems for the linear advection, the derivation of a second-order entropy-satisfying scheme is presented for non-linear equations. The fully discrete stability result is established for regular strictly convex entropy and under a parabolic CFL-like condition. Some numerical experiments are done to assess the accuracy and the stability of the proposed scheme.