With the aim of solving in a four dimensional phase space a multi-scale VlasovPoisson system, we propose in a Particle-In-Cell framework a robust time-stepping method that works uniformly when the small parameter vanishes. As an exponential integrator, 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.

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Abstract. We study the dynamics of charged particles under the influence of a strong magnetic field by numerically solving the Vlasov-Poisson and guiding center models. By using appropriate data structures, we implement an efficient (from the memory access point of view) particle-in-cell method which enables simulations with a large number of particles. We present numerical results for classical one-dimensional Landau damping and two-dimensional Kelvin-Helmholtz test cases. The implementation also relies on a standard hybrid MPI/OpenMP parallelization. Code performance is assessed by the observed speedup and attained memory bandwidth.

We change a previous time-stepping algorithm for solving a multi-scale Vlasov-Poisson system within a Particle-In-Cell method, in order to do accurate long time simulations. As an exponential integrator, the new scheme allows to use large time steps compared to the size of oscillations in the solution.

Abstract. In this article we study a charge-conserving finite-element particle scheme for the Maxwell-Vlasov system that is based on a div-conforming representation of the electric field and we propose a high-order deposition algorithm for smooth particles with piecewise polynomial shape. The numerical performances of the method are assessed with an academic beam test-case, and it is shown that for an appropriate choice of the particle parameters the efficiency of the resulting method overcomes that of similar finite-element schemes using point particles.Résumé. Cet article présente un schémaéléments-finis conservant la charge pour le système de Vlasov-Maxwell basé sur une représentation div-conforme du champélectrique, ainsi qu'un algorithme d'ordreélevé pour déposer le courant porté par des particules régulières polynômiales par morceaux. Les performances numériques du schéma couplé sontévaluéesà l'aide d'un cas test académique de faisceau, et nous montrons que pour un choix approprié des paramètres des particules, cette méthode se révèle plus efficace que d'autres schémas similaires couplés avec des particules ponctuelles.

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We study a Lie Transform method for a charged beam under the action of a radial external electric field. The aim of the Lie transform method that is used here is to construct a change of variable which transforms the 2D kinetic problem into a 1D problem. This reduces the dimensionality of the problem and make it easier to solve numerically. After applying the Lie transform method, we truncate the expression of the characteristics of the Vlasov equation and the expression of the Poisson equation in the Lie coordinate system and we develop a numerical method for solving the truncated model and we study its efficiency for the simulation of long time beam evolution.

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