arXiv subjects
Publications and source records attributed to Bastien Grosse.
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.
In this article, we design a fully spectral method in both space and velocity for a linear inhomogeneous kinetic equation with mass, momentum and energy conservation. We focus on the linear BGK equation with a confinement potential $\Phi$, even if the method could be applied to different collision operators. It is based upon the projection on Hermite polynomials in velocity and orthonormal polynomials with respect to the weight $e^{-$\Phi$}$ in space. The potential $\Phi$ is assumed to be a polynomial. It is, to the author's knowledge, the first scheme which preserves hypocoercive behavior in addition to the conservation laws. These different properties are illustrated numerically on both quadratic and double well potential.
In this paper, we prove a quantitative approximation result by orthonormal polynomials associated to an exponential weight of the form e -$Φ$ , where $Φ$ is an even polynomial with positive leading coefficient. This result is a consequence of a recursion relation for the orthonormal polynomials and of the strong Poincar{é} inequality. Simulations are provided at the end of the article, on smooth, non-smooth functions as well as in the Gaussian and the double well case.