Section: New Results

Simulations of highly oscillatory Vlasov-type models

Participants : Emmanuel Frénod [Univ. Bretagne-Sud] , Sever Hirstoaga.

We continued our exploration of a new time-stepping method based on an exponential integrator.

First, we have improved the algorithm introduced in [11] for solving a multi-scale 1d-1d Vlasov-Poisson system within a Particle-In-Cell method, in order to do accurate long time simulations. As an exponential integrator, the new scheme (see [10] ) allows to use large time steps compared to the size of oscillations in the solution. More precisely, the new idea is to push each particle with its computed period. Our simulations show that using precise periods for each particle and at each macroscopic time step results in a more accurate scheme in long times.

Then, similar ideas are used for a 2d-2d multi-scale Vlasov-Poisson system (see [27] ). We propose in a Particle-In-Cell framework a robust time-stepping method that works uniformly when the small parameter (the smallest scale) vanishes. We first verify our scheme in the framework of a proposed analytic solution with fast oscillations in time and we show that the scheme works for any initial condition. Then we test the method in the nonlinear case of a Vlasov-Poisson simulation. The scheme is able to use large time steps with respect to the typical size of the solution's fast oscillations. In addition, we show numerically that the method has accurate long time behaviour and that it is asymptotic preserving with respect to the limiting Guiding Center system.