Section: New Results

Asymptotic preserving schemes

Participant : Nicolas Crouseilles.

In [18] , we extend the micro-macro decomposition based numerical schemes developed previously to the collisional Vlasov-Poisson model in the diffusion and high-field asymptotics. In doing so, we first write the Vlasov-Poisson model as a system that couples the macroscopic (equilibrium) part with the remainder part. A suitable discretization of this micro-macro model enables to derive an asymptotic preserving scheme in the diffusion and high-field asymptotics. In addition, two main improvements are presented: On the one hand a self-consistent electric field is introduced, which induces a specific discretization in the velocity direction, and represents a wide range of applications in plasma physics. On the other hand, as suggested in a previous reference, we introduce a suitable reformulation of the micro-macro scheme which leads to an asymptotic preserving property with the following property: It degenerates into an implicit scheme for the diffusion limit model when $\epsilon \to 0$, which makes it free from the usual diffusion constraint $\Delta t=𝒪\left(\Delta x^2\right)$ in all regimes. Numerical examples are used to demonstrate the efficiency and the applicability of the schemes for both regimes.

In [45] , a Two-Scale Macro-Micro decomposition of the Vlasov equation with a strong magnetic field is derived. This consists in writing the solution of this equation as a sum of two oscillating functions with circonscribed oscillations. The first of these functions has a shape which is close to the shape of the Two-Scale limit of the solution and the second one is a correction built to offset this imposed shape.