Abstract : In this thesis (numerical analysis and scientific computation areas), we are interested in
the knowledge of the numerical entropy dissipation associated with a given numerical scheme. It turns out that generally speaking, this problem is still open while being of crucial interest in many applications.
Our study is composed of two distinct parts.
The first one is concerned with the numerical approximation of the (unsteady in 1D and steady in 2D) solutions of the Navier-Stokes equations involving several independent pressure laws. As in the usual setting of a single pressure law, this system is hyperbolic with genuinely non linear associated fields under classical assumptions.
However, it naturally writes in non conservation form.
The second one deals with the numerical approximation of the unsteady solutions (in 1D) of several systems of conservation laws which are either hyperbolic but with non genuinely non linear (and non linearly degenerate) fields, or mixed hyperbolic-elliptic.
Several models from the physics enter the present framework for which the entropy dissipation plays an important role in the selection of the physical solution. Here we propose several relevant numerical schemes designed on the basis of a precise study of the associated entropy dissipation.