Microscopically implicit-macroscopically explicit schemes for the BGK equation

In this work a new class of numerical methods for the BGK model of kinetic equations is introduced. The schemes proposed are implicit with respect to the distribution function, while the macroscopic moments are evolved explicitly. In this fashion, the stability condi- tion on the time step coincides with a macroscopic CFL, evaluated using estimated values for the macroscopic velocity and sound speed. Thus the stability restriction does not depend on the relaxation time and it does not depend on the microscopic velocity of ener- getic particles either. With the technique proposed here, the updating of the distribution function requires the solution of a linear system of equations, even though the BGK model is highly non linear. Thus the proposed schemes are particularly effective for high or moderate Mach numbers, where the macroscopic CFL condition is comparable to accuracy requirements. We show results for schemes of order 1 and 2, and the generalization to higher order is sketched

A convergent method for linear half-space kinetic equations

We give a unified proof for the well-posedness of a class of linear half-space equations with general incoming data and construct a Galerkin method to numerically resolve this type of equations in a systematic way. Our main strategy in both analysis and numerics includes three steps: adding damping terms to the original half-space equation, using an inf-sup argument and even-odd decomposition to establish the well-posedness of the damped equation, and then recovering solutions to the original half-space equation. The proposed numerical methods for the damped equation is shown to be quasi-optimal and the numerical error of approximations to the original equation is controlled by that of the damped equation. This efficient solution to the half-space problem is useful for kinetic-fluid coupling simulations

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 thrusters