Section: Research Program

Asymptotic Preserving schemes

We develop robust numerical schemes for kinetic equations that also work in the fluid regime. The goal of this part of the project is to propose a new general and systematic strategy that avoids the inversion of the involved time implicit schemes and that allows to apply the microscopic description without any stability constraint on the numerical parameter $h$.

Development of numerical schemes for stiff problems.

The idea is to combine micro/macro decomposition with penalization techniques for collision operators, leading to completely explicit schemes which are stable in the desired macroscopic limits. The expected schemes should be consistent with the model at both macroscopic and microscopic levels.

However for plasma applications, the Landau-Fokker-Planck operator has a diffusion structure in the velocity variable which induces special behaviors at both mathematical and numerical levels. We will show that the previous methodology can be adapted to overcome the velocity diffusion stiffness in this case. In other words, the obtained numerical schemes are expected to be free of usual diffusion CFL conditions, and will be stable and consistent within the macroscopic limit under consideration. Finally, to decrease the computational cost of the so constructed schemes, which is due to the non local character of the involved collision operator (Boltzmann, Landau, etc), fast computational method for integral operator are needed. On the basis of multi-grid and/or Fast Multi-pole Methods, we shall develop appropriate acceleration methods to our context.

Another important task in this project is to extend the above strategy to the context of a self-consistent Poisson or Maxwell equations. Accurate methods based on finite volume schemes will be developed for high field limit. A stiffness raised by the presence of high electromagnetic fields will also be treated in the same spirit. Such problems are also investigated in the IPSO project (M. Lemou, F. Méhats and N. Crouseilles). Here our strategy is based on a suitable operator decomposition coupled with appropriate IMEX schemes.

Stability and accuracy issue. In the framework of Asymptotic-Preserving (AP) schemes, there are few mathematical justifications of stability and uniform accuracy of such approach. A stability analysis has to be performed to rigorously prove that the numerical scheme is stable for small values of small physical parameters even if the time step does not resolve it. This analysis seems to be tricky for fully nonlinear kinetic equations like the Boltzmann equation. Therefore, we focus on simpler models as discrete velocity models (DVM) which have the same properties as the full Boltzmann operator but deal with a finite set of velocity. In this project we are particularly interested by the long time behavior of the numerical solution when it approaches its stationary state. We plan to apply the entropy-entropy dissipation technique to design new numerical approximations. It gives a specific discretization based on finite volume approximation, which allows to control the numerical entropy production and in some situations, it is often enough to give stability of the numerical solution in the long time asymptotic limit. For general cases, these estimations have to be completed by some discrete functional inequalities.