Artificial Viscosity to Get Both Robustness and Discrete Entropy Inequalities

In the present work, we consider the numerical approximation of the weak solutions of first-order system of evolution laws supplemented with entropy inequalities. The systems under consideration are hyperbolic as soon as a conservation form is satisfied, but such stability property may be lost for non-conservative systems. Here, we show that the robustness and the entropy stability of any finite volume numerical scheme can be restored by introducing a suitable artificial numerical viscosity.