54 research outputs found
Global Solutions for the One-Dimensional Vlasov-Maxwell System for Laser-Plasma Interaction
We analyse a reduced 1D Vlasov--Maxwell system introduced recently in the
physical literature for studying laser-plasma interaction. This system can be
seen as a standard Vlasov equation in which the field is split in two terms: an
electrostatic field obtained from Poisson's equation and a vector potential
term satisfying a nonlinear wave equation. Both nonlinearities in the Poisson
and wave equations are due to the coupling with the Vlasov equation through the
charge density. We show global existence of weak solutions in the
non-relativistic case, and global existence of characteristic solutions in the
quasi-relativistic case. Moreover, these solutions are uniquely characterised
as fixed points of a certain operator. We also find a global energy functional
for the system allowing us to obtain -nonlinear stability of some
particular equilibria in the periodic setting
WENO schemes applied to the quasi-relativistic Vlasov--Maxwell model for laser-plasma interaction
In this paper we focus on WENO-based methods for the simulation of the 1D
Quasi-Relativistic Vlasov--Maxwell (QRVM) model used to describe how a laser
wave interacts with and heats a plasma by penetrating into it. We propose
several non-oscillatory methods based on either Runge--Kutta (explicit) or
Time-Splitting (implicit) time discretizations. We then show preliminary
numerical experiments
The Fourier Singular Complement Method for the Poisson problem. Part I: prismatic domains
This is the first part of a threefold article, aimed at solving numerically
the Poisson problem in three-dimensional prismatic or axisymmetric domains. In
this first part, the Fourier Singular Complement Method is introduced and
analysed, in prismatic domains. In the second part, the FSCM is studied in
axisymmetric domains with conical vertices, whereas, in the third part,
implementation issues, numerical tests and comparisons with other methods are
carried out. The method is based on a Fourier expansion in the direction
parallel to the reentrant edges of the domain, and on an improved variant of
the Singular Complement Method in the 2D section perpendicular to those edges.
Neither refinements near the reentrant edges of the domain nor cut-off
functions are required in the computations to achieve an optimal convergence
order in terms of the mesh size and the number of Fourier modes used
The Fourier Singular Complement Method for the Poisson problem. Part II: axisymmetric domains
This paper is the second part of a threefold article, aimed at solving
numerically the Poisson problem in three-dimensional prismatic or axisymmetric
domains. In the first part of this series, the Fourier Singular Complement
Method was introduced and analysed, in prismatic domains. In this second part,
the FSCM is studied in axisymmetric domains with conical vertices, whereas, in
the third part, implementation issues, numerical tests and comparisons with
other methods are carried out. The method is based on a Fourier expansion in
the direction parallel to the reentrant edges of the domain, and on an improved
variant of the Singular Complement Method in the 2D section perpendicular to
those edges. Neither refinements near the reentrant edges or vertices of the
domain, nor cut-off functions are required in the computations to achieve an
optimal convergence order in terms of the mesh size and the number of Fourier
modes used
Polynomial bounds for the solutions of parametric transmission problems on smooth, bounded domains
We consider a \emph{family} of elliptic
second order differential operators on a domain
whose coefficients depend on the space variable and on a probability space. We allow the coefficients of
to have jumps over a fixed interface (independent of
). We obtain polynomial in the norms of the coefficients
estimates on the norm of the solution to the equation with transmission and mixed boundary conditions (we consider
``sign-changing'' problems as well). In particular, we show that, if and
the coefficients are smooth enough and follow a log-normal-type
distribution, then the map
is in , for all . The same is true for the norms
of the inverses of the resulting operators. We expect our estimates to be
useful in Uncertainty Quantification.Comment: We fixed a small .tex problem in the abstract on the site (the
manuscript has not changed
Electromagnetic wave propagation and absorption in magnetised plasmas: variational formulations and domain decomposition
We consider a model for the propagation and absorption of electromagnetic
waves (in the time-harmonic regime) in a magnetised plasma. We present a
rigorous derivation of the model and several boundary conditions modelling wave
injection into the plasma. Then we propose several variational formulations,
mixed and non-mixed, and prove their well-posedness thanks to a theorem by
S\'ebelin et~al. Finally, we propose a non-overlapping domain decomposition
framework, show its well-posedness and equivalence with the one-domain
formulation. These results appear strongly linked to the spectral properties of
the plasma dielectric tensor
Tout ce que vous avez toujours voulu savoir sur Maxwell sans jamais oser le demander
Nous présentons quelques aspects de la théorie mathématique et de la résolution numérique des équations de Maxwell instationnaires, plus particulièrement en vue de la simulation de particules chargées. Deux thématiques servent de fil conducteur à cet article: le traitement la contrainte de divergence et celui des singularités. Les équations habituelles, du premier ordre, peuvent être reformulées en équations du second ordre, plus adaptées à la simulation par éléments finis. On peut d'autre part intégrer différents traitements de la contrainte de divergence, même en présence de données bruitées. Nous esquissons la preuve d'existence et d'unicité de la solution de ces équations. La régularité de cette solution dépend fortement des singularités du domaine de calcul, et influe à son tour sur le choix de la méthode d'éléments finis. Deux méthodes sont examinées en détail: les éléments d'arête et les éléments nodaux; pour ces derniers, deux variantes permettent une prise en compte efficace des singularités. Nous donnons des estimations d'erreurs optimales pour toutes les variantes
Développement asymptotique et approximation de la solution des équations de Maxwell dans un polygone
We present an improved version of the Singular Complement Method (SCM) for Maxwell's equations, which relies on an asymptotic expansion of the solution near non-regular points. This method allows to recover an optimal error estimate when used with Lagrange finite elements; extension to higher-degree elements is possible. It can be applied to static, harmonic, or time-dependent problems
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