Efficient asymptotic preserving schemes for BGK and ES-BGK models on cartesian grids

This work is devoted to the study of complex flows where hydrodynamic and rarefied regimes coexist. This kind of flows are found in vacuum pumps or hypersonic re-entries of space vehicles where the distance between gas molecules is so large that their microscopic behaviour differ from the average behaviour of the flow and has be taken into account. We then consider two models of the Boltzmann equation viable for such flows: the BGK model dans the ES-BGK model. We first devise a new wall boundary condition ensuring a smooth transition of the solution from the rarefied regime to the hydrodynamic regime. We then describe how this boundary condition (and boundary conditions in general) can be enforced with second order accuracy on an immersed body on Cartesian grids preserving the asymptotic limit towards compressible Euler equations. We exploit the ability of Cartesian grids to massive parallel computations (HPC) to drastically reduce the computational time which is an issue for kinetic models. A new approach considering local velocity grids is then presented showing important gain on the computational time (up to 80$\%$). 3D simulations are also presented showing the efficiency of the methods. Finally, solid particle transport in a rarefied flow is studied. The kinetic model is coupled with a Vlasov-type equation modeling the passive particle transport solved with a method based on remeshing processes. As application, we investigate the realistic test case of the pollution of optical devices carried by satellites due to incompletely burned particles coming from the altitude control thruster