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.
Degond, P., Deluzet, F., & Doyen, D. (2017). Asymptotic-Preserving Particle-In-Cell methods for the Vlasov–Maxwell system in the quasi-neutral limit. Journal of Computational Physics, 330, 467–492. https://doi.org/10.1016/j.jcp.2016.11.018