arXiv subjects
Publications and source records attributed to Mehdi Badsi.
We prove existence and uniqueness in BV for a Vlasov-Poisson boundary value problem which takes into account a nonlinear boundary condition at the boundary on the electric potential. It models the dynamical interaction between a plasma and an electrically isolated metallic wall. The nonlinear boundary condition is the so-called floating potential equation of plasma physics.
We propose and study a fully implicit finite volume scheme for the pressureless Euler-Poisson-Boltzmann equations on the one dimensional torus. Especially, we design a consistent and dissipative discretization of the force term which yields an unconditional energy decay. In addition, we establish a discrete analogue of the modulated energy estimate around constant states with a small velocity. Numerical experiments are carried to illustrate our theoretical results and to assess the accuracy of our scheme. A test case of the literature is also illustrated.
Plasma sheaths are inhomogeneous stationary states that form when a plasma is in contact with an absorbing wall. We prove linear and non linear stability of a kinetic sheath stationary state for a Vlasov-Poisson type system in a bounded interval. Notably, in the linear setting, we obtain exponential decay of the fluctuation provided the rate of injection of particles at equilibrium is smaller than the rate of absorption at the wall. In the non linear setting, we prove a similar result for small enough equilibrium and small localized perturbation of the equilibrium.
In this article, we study a collisionless kinetic model for plasmas in the neighborhood of a cylindrical metallic Langmuir probe. This model consists in a bi-species Vlasov-Poisson equation in a domain contained between two cylinders with prescribed boundary conditions. The interior cylinder models the probe while the exterior cylinder models the interaction with the plasma core. We prove the existence of a weak-strong solution for this model in the sense that we get a weak solution for the 2 Vlasov equations and a strong solution for the Poisson equation. The first parts of the article are devoted to explain the model and proceed to a detailed study of the Vlasov equations. This study leads to a reformulation of the Poisson equation as a 1D non-linear and non-local equation and we prove it admits a strong solution using an iterative fixed-point procedure.
We are interested in developing a numerical method for capturing stationary sheaths, that a plasma forms in contact with a metallic wall. This work is based on a bi-species (ion/electron) Vlasov-Amp{\`e}re model proposed in [3]. The main question addressed in this work is to know if classical numerical schemes can preserve stationary solutions with boundary conditions, since these solutions are not a priori conserved at the discrete level. In the context of high-order semi-Lagrangian method, due to their large stencil, interpolation near the boundary of the domain requires also a specific treatment.