Asymptotic-Preserving Particle-In-Cell method for the Vlasov–Poisson system near quasineutrality

Abstract

This paper deals with the numerical resolution of the Vlasov-Poisson system in a nearly quasineutral regime by Particle-In-Cell (PIC) methods. In this regime, classical PIC methods are subject to stability constraints on the time and space steps related to the small Debye length and large plasma frequency. Here, we propose an "Asymptotic-Preserving" PIC scheme which is not subject to these limitations. Additionally, when the plasma period and Debye length are small compared to the time and space steps, this method provides a consistent PIC discretization of the quasineutral Vlasov equation. We perform several one-dimensional numerical experiments which provide a solid validation of the method and its underlying concepts, and compare the method with classical PIC and Direct-Implicit methods.

Multiscale Schemes for the BGK--Vlasov--Poisson System in the Quasi-Neutral and Fluid Limits. Stability Analysis and First Order Schemes

“…Thus, in order to derive numerical schemes which work independently on the Debye length scale λ, one idea is to discretize the reformulated system instead of the original one. This is what has been done in [11], [1,18,19], in which different numerical schemes efficient in the quasi-neutral limit have been developed.…”

“…This is exactly the same situation as in the incompressible Euler equations in which the pressure is the Lagrange multiplier of the divergence-free constraint. Thus, in this case, in order to recover an explicit equation for the potential ϕ, one idea consists in the reformulation of the system P ε,λ (see [11] for the fluid case and [18], [1], [19] for the kinetic one). Let us begin integrating (4a) with respect to the velocity variable, using the quasi-neutrality constraint, it leads to the divergence-free constraint for the momentum…”

“…In this first work, in which we consider the extension of the AP methodologies to the case of multiple scales, we present a first order in time scheme and a relative linear stability analysis which proves stability for small values of the Debye length. The idea is based on two main ingredients: first, the reformulation of the Poisson equation (introduced in [10] and then used in [11] and [1,19]) and second, the construction of IMEX schemes for collisional kinetic equation (studied for instance in [26,28]). In additiom, we analyze the state of the art of the schemes which are able to handle the quasi-neutrality constraint and we show that the splitting approach, typical of particle methods as PIC methods [3], or semi-Lagrangian approaches [1] may suffer from incompatibility with the quasi-neutrality if no special care is taken.…”

Modelling and simulating a multispecies plasma

“…The scheme presented hereafter was introduced by Belaouar, Crouseilles, Degond and Sonnendrücker [1] for the resolution of the one species Vlasov Poisson system with a semi-Lagrangian method. The two species case was treated in [7] with a PIC method. Let us discretize equation (2.7) semi-implicitly in time,…”

Asymptotic-preserving well-balanced scheme for the electronic M1 model in the diffusive limit: Particular cases

“…The derivation of well-balanced schemes also helps to design AP schemes [38,39] (see also [1,9,10,18,21,22,37,41] for details on well-balanced schemes in different frameworks). The AP frame was also largely extended to the quasi-neutral limit [26][27][28][29][30]43]. In [7], an HLLC scheme is proposed to solve the M 1 model of radiative transfer in two space dimensions.…”