A particle micro-macro decomposition based numerical scheme for collisional kinetic equations in the diffusion scaling

In this work, we derive particle schemes, based on micro-macro decomposition, for linear kinetic equations in the diffusion limit. Due to the particle approximation of the micro part, a splitting between the transport and the collision part has to be performed, and the stiffness of both these two parts prevent from uniform stability. To overcome this difficulty, the micro-macro system is reformulated into a continuous PDE whose coefficients are no longer stiff, and depend on the time step $\Delta t$ in a consistent way. This non-stiff reformulation of the micro-macro system allows the use of standard particle approximations for the transport part, and extends the work in [5] where a particle approximation has been applied using a micro-macro decomposition on kinetic equations in the fluid scaling. Beyond the so-called asymptotic-preserving property which is satisfied by our schemes, they significantly reduce the inherent noise of traditional particle methods, and they have a computational cost which decreases as the system approaches the diffusion limit

Kinetic/fluid micro-macro numerical schemes for Vlasov-Poisson-BGK equation using particles

International audienceThis work is devoted to the numerical simulation of the Vlasov equation in the fluid limit using particles. To that purpose, we first perform a micro-macro decomposition as in \cite{benoune} where asymptotic preserving schemes have been derived in the fluid limit. In \cite{benoune}, a uniform grid was used to approximate both the micro and the macro part of the full distribution function. Here, we modify this approach by using a particle approximation for the kinetic (micro) part, the fluid (macro) part being always discretized by standard finite volume schemes. There are many advantages in doing so: $(i)$ the so-obtained scheme presents a much less level of noise compared to the standard particle method; $(ii)$ the computational cost of the micro-macro model is reduced in the fluid regime since a small number of particles is needed for the micro part; $(iii)$ the scheme is asymptotic preserving in the sense that it is consistent with the kinetic equation in the rarefied regime and it degenerates into a uniformly (with respect to the Knudsen number) consistent (and deterministic) approximation of the limiting equation in the fluid regime

Asymptotic-Preserving scheme based on a Finite Volume/Particle-In-Cell coupling for Boltzmann- BGK-like equations in the diffusion scaling

International audienceThis work is devoted to the numerical simulation of the collisional Vlasov equation in the diffusion limit using particles. To that purpose, we use a micro-macro decomposition technique introduced by Bennoune, Lemou and Mieussens. Whereas a uniform grid was used to approximate both the micro and the macro part of the full distribution function in their article, we use here a particle approximation for the kinetic (micro) part, the fluid (macro) part being always discretized by standard finite volume schemes. There are many advantages in doing so: (i) the so-obtained scheme presents a much less level of noise compared to the standard particle method; (ii) the computational cost of the micro-macro model is reduced in the diffusion limit since a small number of particles is needed for the micro part; (iii) the scheme is asymptotic preserving in the sense that it is consistent with the kinetic equation in the rarefied regime and it degenerates into a uniformly (with respect to the Knudsen number) consistent (and deterministic) approximation of the limiting equation in the diffusion regime

Multi-Water-Bag Model And Method Of Moments For The Vlasov Equation

International audienceThe kinetic Vlasov-Poisson model is very expensive to solve numerically. It can be approximated by a multi-water-bag model in order to reduce the complexity. This model amounts to solve a set of Burgers equations, which can be done easily by finite volume methods. However, the solution is naturally multivalued (filamentation). The multivalued solution can be computed by a moment method. We present here several numerical experiments

Resolution of the Vlasov-Maxwell system by PIC Discontinuous Galerkin method on GPU with OpenCL

International audienceWe present an implementation of a Vlasov-Maxwell solver for multicore processors. The Vlasov equation describes the evolution of charged particles in an electromagnetic field, solution of the Maxwell equations. The Vlasov equation is solved by a Particle-In-Cell method (PIC), while the Maxwell system is computed by a Discontinuous Galerkin method. We use the OpenCL framework, which allows our code to run on multicore processors or recent Graphic Processing Units (GPU). We present several numerical applications to two-dimensional test cases

Space-only hyperbolic approximation of the Vlasov equation

International audienceWe construct an hyperbolic approximation of the Vlasov equation in which the dependency on the velocity variable is removed. The resulting model enjoys interesting conservation and entropy properties. It can be numerically solved by standard schemes for hyperbolic systems. We present numerical results for one-dimensional classical test cases in plasma physics: Landau damping, two-stream instabilit

An axisymmetric PIC code based on isogeometric analysis

International audienceIsogeometric analysis has been developed recently to use basis functions resulting from the CAO description of the computational domain for the finite element spaces. The goal of this study is to develop an axisymmetric Finite Element PIC code in which specific spline Finite Elements are used to solve the Maxwell equations and the same spline functions serve as shape function for the particles. The computational domain itself is defined using splines or NURBS

Reduced Vlasov-Maxwell simulations

International audienceIn this paper we review two different numerical methods for Vlasov-Maxwell simulations. The first method is based on a coupling between a Discontinuous Galerkin (DG) Maxwell solver and a Particle-In-Cell (PIC) Vlasov solver. The second method only uses a DG approach for the Vlasov and Maxwell equations. The Vlasov equation is first reduced to a space-only hyperbolic system thanks to a reduction method proposed recently by the authors. The two numerical methods are implemented using OpenCL in order to achieve high performance on recent Graphic Processing Units (GPU)

Semaine d'Etude Mathématiques et Entreprises 2 : Analyse multivariées pour la production d'aluminium

Ce rapport présente l'étude statistique, menée au cours de la deuxième Semaine d'Étude Maths-Entreprises, d'un problème industriel rencontré par Rio Tinto Alcan. Productrice d'aluminium par électrolyse, cette entreprise cherche à expliquer des fluctuations de procédé. À partir d'un ensemble de mesures sur les anodes et sur les cuves à électrolyse, nous proposons d'utiliser des méthodes d'analyse multivariée pour construire des modèles explicatifs. Le but étant de permettre aux usines d'éviter les périodes avec des fluctuations. Dans une première section, nous présentons le problème et ses enjeux. Nous détaillons dans les sections suivantes les différentes méthodes explorées et les résultats obtenus : l'analyse du coefficient de corré- lation en présence d'un déphasage et l'auto-corrélation, l'analyse en composantes principales, les arbres de décisions, le clustering et la régression linéaire. Des résultats complémentaires sont donnés en annexe

High order numerical methods for Vlasov-Poisson models of plasma sheaths

This article is a report of the CEMRACS 2022 project, called HIVLASHEA, standing for "High order methods for Vlasov-Poisson models for sheaths". A two-species Vlasov-Poisson model is described together with some numerical simulations, permitting to exhibit the formation of a plasma sheath. The numerical simulations are performed with two different methods: a first order classical finite difference (FD) scheme and a high order semi-Lagrangian (SL) scheme with Strang splitting; for the latter one, the implementation of (non-periodic) boundary conditions is discussed. The codes are first evaluated on a one-species case, where an analytical solution is known. For the two-species case, cross comparisons and the influence of the numerical parameters for the SL method are performed in order to have an idea of a reference numerical simulation. Aknowledgements Centre de Calcul Intensif d'Aix-Marseille is acknowledged for granting access to its high performance computing resources