Numerical study of an anisotropic Vlasov equation arising in plasma physics

Abstract

Goal of this paper is to investigate several numerical schemes for the resolution of two anisotropic Vlasov equations. These two toy-models are obtained from a kinetic description of a tokamak plasma confined by strong magnetic fields. The simplicity of our toy-models permits to better understand the features of each scheme, in particular to investigate their asymptotic-preserving properties, in the aim to choose then the most adequate numerical scheme for upcoming, more realistic simulations.

Asymptotic-Preserving Scheme for the Resolution of Evolution Equations with Stiff Transport Terms

“…We were interested in designing a scheme working on a Cartesian grid. One can imagine that for stiff problems of the type (1.8) (or more generally (1.1)), it could be better to adapt the coordinate system, choosing field-aligned variables, and transforming thus the problem into an evolution problem with a strong anisotropy aligned with one coordinate axis, problem which is much simpler to solve (via IMEX schemes for ex., see [18]). Our aim however was rightly to avoid a coordinate transformation and to design a simple scheme based on a Cartesian grid.…”

Asymptotic analysis of the RS-IMEX scheme for the shallow water equations in one space dimension

“…Such an issue can be handled using the classical pre-conditioning techniques as in [6]. Moreover and very recently, the authors in [21] have addressed this point more fundamentally for some toy models related to Vlasov-Maxwell equations. (iv) Mostly, we consider the well-prepared initial data; however, we will also show in Appendix C, that the scheme projects the ill-prepared initial data on the limit manifold for ε 1.…”