Long time simulation of a beam in a periodic focusing channel via a two-scale PIC-method

We study the two-scale asymptotics for a charged beam under the action of a rapidly oscillating external electric field. After proving the convergence to the correct asymptotic state, we develop a numerical method for solving the limit model involving two time scales and validate its efficiency for the simulation of long time beam evolution

Two-dimensional Finite Larmor Radius approximation in canonical gyrokinetic coordinates

In this paper, we present some new results about the approximation of the Vlasov-Poisson system with a strong external magnetic field by the 2D finite Larmor radius model. The proofs within the present work are built by using two-scale convergence tools, and can be viewed as a new slant on previous works of Fr\'enod and Sonnendr\"ucker and Bostan on the 2D finite Larmor Radius model. In a first part, we recall the physical and mathematical contexts. We also recall two main results from previous papers of Fr\'enod and Sonnendr\"ucker and Bostan. Then, we introduce a set of variables which are so-called canonical gyrokinetic coordinates, and we write the Vlasov equation in these new variables. Then, we establish some two-scale convergence and weak-* convergence results

Derivation of a gyrokinetic model. Existence and uniqueness of specific stationary solutions

A finite Larmor radius approximation is derived from the classical Vlasov equation, in the limit of large (and uniform) external magnetic field. We also provide an heuristic derivation of the electroneutrality equation in the finite Larmor radius setting. Existence and uniqueness of a solution is proven in the stationary frame for solutions depending only on the direction parallel to the magnetic field and factorizing in the velocity variables

Expansion of a singularly perturbed equation with a two-scale converging convection term

In many physical contexts, evolution convection equations may present some very large amplitude convective terms. As an example, in the context of magnetic confinement fusion, the distribution function that describes the plasma satisfies the Vlasov equation in which some terms are of the same order as $\epsilon^{-1}$, $\epsilon \ll 1$ being the characteristic gyrokinetic period of the particles around the magnetic lines. In this paper, we aim to present a model hierarchy for modeling the distribution function for any value of $\epsilon$ by using some two-scale convergence tools. Following Fr\'enod \\& Sonnendr\"ucker's recent work, we choose the framework of a singularly perturbed convection equation where the convective terms admit either a high amplitude part or a an oscillating part with high frequency $\epsilon^{-1} \gg 1$. In this abstract framework, we derive an expansion with respect to the small parameter $\epsilon$ and we recursively identify each term of this expansion. Finally, we apply this new model hierarchy to the context of a linear Vlasov equation in three physical contexts linked to the magnetic confinement fusion and the evolution of charged particle beams

On massless electron limit for a multispecies kinetic system with external magnetic field

We consider a three-dimensional kinetic model for a two species plasma consisting of electrons and ions confined by an external nonconstant magnetic field. Then we derive a kinetic-fluid model when the mass ratio $m_e/m_i$ tends to zero. Each species initially obeys a Vlasov-type equation and the electrostatic coupling follows from a Poisson equation. In our modeling, ions are assumed non-collisional while a Fokker-Planck collision operator is taken into account in the electron equation. As the mass ratio tends to zero we show convergence to a new system where the macroscopic electron density satisfies an anisotropic drift-diffusion equation. To achieve this task, we overcome some specific technical issues of our model such as the strong effect of the magnetic field on electrons and the lack of regularity at the limit. With methods usually adapted to diffusion limit of collisional kinetic equations and including renormalized solutions, relative entropy dissipation and velocity averages, we establish the rigorous derivation of the limit model