Study and discretization of kinetic models and of fluid models at low Mach number
This thesis summarizes our work between 1995 and 2010. It concerns the analysis and the discretization of Fokker-Planck or semi-classical Boltzmann kinetic models and of Euler or Navier-Stokes fluid models at low Mach number. The studied Fokker-Planck equation models the collisions between ions and electrons in a hot plasma, and is here applied to the inertial confinement fusion. The studied semi-classical Boltzmann equations are of two types. The first one models the thermonuclear reaction between a deuterium ion and a tritium ion producing an alpha particle and a neutron particle, and is also in our case used to describe inertial confinement fusion. The second one - known as the Wang-Chang & Uhlenbeck equations - models the transitions between electronic quantified energy levels of uranium and iron atoms in the AVLIS isotopic separation process. We studied the basic properties of these two Boltzmann equations, and, for the Wang-Chang & Uhlenbeck equations, we proposed a kinetic fluid coupling algorithm. This kinetic-fluid coupling algorithm incited us to study the relaxation concept for gas and immiscible fluids mixtures, and to underline connections with classical kinetic theory. Then, we proposed a diphasic low Mach number model without acoustic waves to model the deformation of the interface between two immiscible fluids induced by high heat transfers at low Mach number. And, to increase the accuracy of the results without increasing computational cost, we studied an AMR algorithm on a simplified interface deformation model. These low Mach number studies also incited us to analyse on cartesian meshes the inaccuracy at low Mach number of Godunov schemes. Finally, we justified the LBM algorithm applied to the heat equation.