Asymptotic-Preserving Particle-In-Cell methods for the Vlasov–Maxwell system in the quasi-neutral limit

Abstract

In this article, we design Asymptotic-Preserving Particle-In-Cell methods for the Vlasov-Maxwell system in the quasi-neutral limit, this limit being characterized by a Debye length negligible compared to the space scale of the problem. These methods are consistent discretizations of the Vlasov-Maxwell system which, in the quasi-neutral limit, remain stable and are consistent with a quasi-neutral model (in this quasi-neutral model, the electric field is computed by means of a generalized Ohm law). The derivation of Asymptotic-Preserving methods is not straightforward since the quasi-neutral model is a singular limit of the Vlasov-Maxwell model. The key step is a reformulation of the Vlasov-Maxwell system which unifies the two models in a single set of equations with a smooth transition from one to another. As demonstrated in various and demanding numerical simulations, the Asymptotic-Preserving methods are able to treat efficiently both quasi-neutral plasmas and non-neutral plasmas, making them particularly well suited for complex problems involving dense plasmas with localized non-neutral regions. E, because this field varies on much larger scales than the charge density. It is thus preferable to assume the charge neutrality (this is the so-called plasma approximation) and to compute the electric field by other means. Examples of such quasi-neutral descriptions can be found, for instance, in the Langmuir model [26,38], the electric field being given by the Boltzmann approximation, or in the Magneto-Hydro-Dynamic (MHD) models [7] and the kinetic quasi-neutral models described in [31,35,1], for which the electric field is provided by an Ohm law. As outlined in [12, chapter 3], "in a plasma it is possible to assume n i = n e and ∇ · E = 0 at the same time", leading to the following guideline: "Do not use Poisson's equation to obtain E unless it is unavoidable !" which defines the path followed in the present work.In many plasma problems, the non-neutral areas are very localized but crucial for the global evolution of the plasma. Furthermore, the extent of these regions evolves with time. That is the case, for instance, of Plasma Opening Switches (POS) [55,54,47] where an electromagnetic wave interacts with a dense plasma, leading to the formation of non-neutral sheaths. Quasi-neutral models are not appropriate to describe all the complex physics that occurs in these non-neutral regions. Non-neutral models, such as the Vlasov-Maxwell model or the Euler-Maxwell model, are not appropriate either because the quasi-neutral limit is a singular limit (some equations degenerate in the quasi-neutral limit). In particular, explicit discretizations of nonneutral models are subject to stability conditions on the mesh size and the time step that are all the more restrictive than the Debye length and the plasma period are small.A first way to cope with these multi-physics problems is to split the domain into non-neutral and quasineutral areas, and to use suitable models in each area (see [48] for example). Another way is to devel...

Asymptotic-Preserving methods and multiscale models for plasma physics

“…Different space discretization have been considered. For kinetic description, either Particle-In-Cell [51,52,47] or semi-Lagrangian [6] discretizations have been implemented, while, for fluid descriptions finite volume (on Cartesian meshes) are used [39,55]. In this last series of works dedicated to the Euler-Maxwell system, an exact consistency with the Maxwell-Gauss equation can be obtained.…”

“…Pioneering works have been first devoted to the Euler-Poisson system [39], then extended to kinetic electrostatic descriptions by means of the Vlasov-Poisson system [51,52,6]. Electromagnetic fields have been considered in the frame of the bi-fluid isothermal Euler-Maxwell system in [55] (extended to the M1-Maxwell model in [78]) and finally with the Vlasov-Maxwell system [47]. In Sec.…”

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