Objectifs :
Collisionless kinetic equations appear, among others, in
plasma physics and astrophysics. In plasma physics, these models
describe accurately the wave-particle interaction which plays a crucial
role in turbulent plasma such as magnetic fusion plasma (ITER project).
In astrophysics these models allow to describe the large-scale structure
of the universe such as clusters of galaxies and the dark matter.
The aim of this lecture is twofold. The first objective is to characterize
the behavior of small perturbated solutions of collisionless equations
around an equilibrium such as damped or instable waves, with some
quantitative estimates (e.g., growth or damping rate of waves amplitude,
waves frequency). Of course, such analytical results are obtained in the
framework of linearized equations. For the study of the nonlinear setting,
in special cases, a rigorous mathematical analysis can sometimes be
performed to obtain some qualitative and quantitative results. For
example, this is the case for the Landau damping in the nonlinear system
of Vlasov-Poisson, for which a rigorous mathematical proof is available.
But in general, for studies in a nonlinear framework, such as the kinetic
turbulence, rigorous mathematical results are so rare or unreachable that
numerical simulations are unavoidable. That is the reason why the design
of stable, convergent and high-order accurate numerical schemes is
needed. It is the second objective of this lecture. Here, we present the
numerical analysis of a family of schemes, known as semi-Lagrangian
schemes, which are very popular in the plasma physics and astrophysics
communities. After defining the principles of such schemes, we study its
stability and show its convergence towards the classical solutions of the
Vlasov-Poisson system. We also obtain high-order a priori error
estimates which yield accuracy of the schemes

Programme :
1) Solutions of the linearized problem: Landau damping versus
instabilities. Proof of Landau damping for the nonlinear system of
Vlasov-Poisson.
2) Analysis of semi-Lagrangian numerical schemes for the Vlasov-
Poisson system.

Pré-requis :
numerical analysis of partial differential equations.

Equipe pédagogique :
Nicolas Besse (OCA, LAGRANGE)

Les Compétences et capacités visées à l'issue de la formation (fiches RNCP)

Arrêté du 22 février 2019 définissant les compétences des diplômés du doctorat et inscrivant le doctorat au répertoire national de la certification professionnelle. https://www.legifrance.gouv.fr/loda/id/JORFTEXT000038200990/

Bloc 1 : Conception et élaboration d’une démarche de recherche et développement, d’études et prospective

- Disposer d'une expertise scientifique tant générale que spécifique d'un domaine de recherche et de travail déterminé

La formation participe à l'objectif suivant :être directement utile pour la réalisation des travaux personnels de recherche