Development of a Riemann solver for non conservative product for plasmas in thermal non-equilibrium

This thesis deals with the modelling of plasmas in thermal non-equilibrium and the numerical simulation of shocks. A simplified version of the multicomponent model, developed by [Graille\&al], is presented where a nonconservative product term is found in the electron energy equation and represents an issue for the traditional conservative schemes. Among the introduced available solving strategies, a deeper focus is brought on a method that requires a decoupling of the system and that allows an analytical travelling wave solution for the electron variables. A numerical treatment of the nonconservative product, benefiting from this new approach and developed by [Wargnier\&al], is then introduced. The contribution is divided into two parts: 1) an analysis of the impact on the physical validity of the solution that appears to violate the second principle of thermodynamics. 2) the rehabilitation of the two-temperatures relaxation process in the simplified model and its impact on the expression and the benefits of the aforementioned numerical treatment.