Conservative semi-Lagrangian schemes for Vlasov equations (2010)

by N Crouseilles, M Mehrenberger, E Sonnendrücker
Venue:J. Comput. Phys
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Guiding-center simulations on curvilinear meshes using semi-Lagrangian conservative methods, in "Issue dedicated to the proceedings of the second edition of the Conference Numerical models for fusion

by Nicolas Crouseilles, Michel Mehrenberger, Eric Sonnendrücker , 2012
"... Abstract. The purpose of this work is to design simulation tools for magnetised plasmas in the ITER project framework. The specific issue we consider is the simulation of turbulent transport in the core of a Tokamak plasma, for which a 5D gyrokinetic model is generally used, where the fast gyromotio ..."
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Abstract. The purpose of this work is to design simulation tools for magnetised plasmas in the ITER project framework. The specific issue we consider is the simulation of turbulent transport in the core of a Tokamak plasma, for which a 5D gyrokinetic model is generally used, where the fast gyromotion of the particles in the strong magnetic field is averaged in order to remove the associated fast time-scale and to reduce the dimension of 6D phase space involved in the full Vlasov model. Very accurate schemes and efficient parallel algorithms are required to cope with these still very costly simulations. The presence of a strong magnetic field constrains the time scales of the particle motion along and accross the magnetic field line, the latter being at least an order of magnitude slower. This also has an impact on the spatial variations of the observables. Therefore, the efficiency of the algorithm can be improved considerably by aligning the mesh with the magnetic field lines. For this reason, we study the behavior of semi-Lagrangian solvers in curvilinear coordinates. Before tackling the full gyrokinetic model in a future work, we consider here the reduced 2D Guiding-Center model. We introduce our numerical algorithm and provide some numerical results showing its good properties. 1. Introduction. In
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High order Runge-Kutta-Nyström splitting methods for the Vlasov-Poisson equation

by Nicolas Crouseilles, Erwan Faou, Michel Mehrenberger , 2011
"... In this work, we derive the order conditions for fourth order time splitting schemes in the case of the 1D Vlasov-Poisson system. Computations to obtain such conditions are motivated by the specific Poisson structure of the Vlasov-Poisson system: this structure is similar to Runge-Kutta-Nyström syst ..."
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In this work, we derive the order conditions for fourth order time splitting schemes in the case of the 1D Vlasov-Poisson system. Computations to obtain such conditions are motivated by the specific Poisson structure of the Vlasov-Poisson system: this structure is similar to Runge-Kutta-Nyström systems. The obtained conditions are proved to be the same as RKN conditions derived for ODE up to the fourth order. Numerical results are performed and show the benefit of using high order splitting schemes in that context. 1
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Discontinuous Galerkin Semi-Lagrangian Method for Vlasov-Poisson, CEMRACS’10 research achievements: Numerical modeling of fusion

by N. Crouseilles, M. Mehrenberger, F. Vecil - ESAIM Proceedings, October 2011
"... Abstract. We present a discontinuous Galerkin scheme for the numerical approximation of the onedimensional periodic Vlasov-Poisson equation. The scheme is based on a Galerkin-characteristics method in which the distribution function is projected onto a space of discontinuous functions. We present co ..."
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Abstract. We present a discontinuous Galerkin scheme for the numerical approximation of the onedimensional periodic Vlasov-Poisson equation. The scheme is based on a Galerkin-characteristics method in which the distribution function is projected onto a space of discontinuous functions. We present comparisons with a semi-Lagrangian method to emphasize the good behavior of this scheme when applied to Vlasov-Poisson test cases. Résumé. Une méthode de Galerkin discontinu est proposée pour l’approximation numérique de l’équation de Vlasov-Poisson 1D. L’approche est basée sur une méthode Galerkin-caractéristiques où la fonction de distribution est projetée sur un espace de fonctions discontinues. En particulier, la méthode est comparée à une méthode semi-Lagrangienne pour l’approximation de l’équation de Vlasov-Poisson.
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Conservative and non-conservative methods based on Hermite weighted essentially-non-oscillatory reconstruction for Vlasov equations

by Chang Yang, Francis Filbet - Université de Lyon & Inria, Institut Camille Jordan, EPI Kaliffe, 43 boulevard 11 novembre 1918, F-69622 Villeurbanne cedex, FRANCE E-mail address, F. Filbet: filbet@math.univ-lyon1.fr Department of Mathematics, Harbin Institute of Technology, 92 West Da
"... Abstract. We introduce a WENO reconstruction based on Hermite interpo-lation both for semi-Lagrangian and finite difference methods. This WENO re-construction technique allows to control spurious oscillations. We develop third and fifth order methods and apply them to non-conservative semi-Lagrangia ..."
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Abstract. We introduce a WENO reconstruction based on Hermite interpo-lation both for semi-Lagrangian and finite difference methods. This WENO re-construction technique allows to control spurious oscillations. We develop third and fifth order methods and apply them to non-conservative semi-Lagrangian schemes and conservative finite difference methods. Our numerical results will be compared to the usual semi-Lagrangian method with cubic spline recon-struction and the classical fifth order WENO finite difference scheme. These reconstructions are observed to be less dissipative than the usual weighted essentially non-oscillatory procedure. We apply these methods to transport equations in the context of plasma physics and the numerical simulation of turbulence phenomena.
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KINETIC/FLUID MICRO-MACRO NUMERICAL SCHEMES FOR VLASOV-POISSON-BGK EQUATION USING PARTICLES

by Anaïs Crestetto, Nicolas Crouseilles, Mohammed Lemou
"... This 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 [3] where asymptotic preserving schemes have been derived in the fluid limit. In [3], a uniform grid was used to approximat ..."
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This 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 [3] where asymptotic preserving schemes have been derived in the fluid limit. In [3], 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 itdegenerates into a uniformly(with respect to the Knudsen number) consistent (and deterministic) approximation of the limiting equation in the fluid regime.

A charge preserving scheme for the numerical resolution of the Vlasov-Ampère equations

by Nicolas Crouseilles, Thomas Respaud - Commun. Comput. Phys
"... In this report, a charge preserving numerical resolution of the 1D Vlasov-Ampère equation is achieved, with a forward Semi-Lagrangian method introduced in [10]. The Vlasov equation belongs to the kinetic way of simulating plasmas evolution, and is coupled with the Poisson’s equation, or equivalently ..."
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In this report, a charge preserving numerical resolution of the 1D Vlasov-Ampère equation is achieved, with a forward Semi-Lagrangian method introduced in [10]. The Vlasov equation belongs to the kinetic way of simulating plasmas evolution, and is coupled with the Poisson’s equation, or equivalently under charge conservation, the Ampère’s one, which self-consistently rules the electric field evolution. In order to ensure having proper physical solutions, it is necessary that the scheme preserves charge numerically. B-Spline deposition will be used for the interpolation step. The solving of the characteristics will be made with a Runge-Kutta 2 method and with a Cauchy-Kovalevsky procedure.
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Space-only hyperbolic approximation of the Vlasov equation

by Nhung Pham, A. Crestetto
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Block-structured adaptive mesh refinement algorithms for vlasov simulation

by J. A. F. Hittinger, J. W. Banks - CoRR
"... Direct discretization of continuum kinetic equations, like the Vlasov equation, are under-utilized because the distribution function generally exists in a high-dimensional (>3D) space and computational cost increases geometrically with dimension. We pro-pose to use high-order finite-volume techni ..."
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Direct discretization of continuum kinetic equations, like the Vlasov equation, are under-utilized because the distribution function generally exists in a high-dimensional (>3D) space and computational cost increases geometrically with dimension. We pro-pose to use high-order finite-volume techniques with block-structured adaptive mesh refinement (AMR) to reduce the computational cost. The primary complication comes from a solution state comprised of variables of different dimensions. We develop the algorithms required to extend standard single-dimension block structured AMR to the multi-dimension case. Specifically, algorithms for reduction and injection operations that transfer data between mesh hierarchies of different dimensions are explained in detail. In addition, modifications to the basic AMR algorithm that enable the use of high-order spatial and temporal discretizations are discussed. Preliminary results for a standard 1D+1V Vlasov-Poisson test problem are presented. Results indicate that there is potential for significant savings for some classes of Vlasov problems. 1

Test of some numerical limiters for the conservative PSM scheme for 4D Drift-Kinetic simulations.

by Jérôme Guterl, Nicolas Crouseilles Virginie Gr, Mehrenberger Eric Sonnendrücker, Jérôme Guterl, Nicolas Crouseilles, Virginie Gr, Guilaume Latu, Michel Mehrenberger, Eric Sonnendrücker, Thème Modélisation, Analyse Numérique
"... apport de recherche ISSN 0249-6399 ISRN INRIA/RR--7467--FR+ENGinria-00540948, version 1- 30 Nov 2010Test of some numerical limiters for the conservative PSM scheme for 4D Drift-Kinetic simulations. ..."
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apport de recherche ISSN 0249-6399 ISRN INRIA/RR--7467--FR+ENGinria-00540948, version 1- 30 Nov 2010Test of some numerical limiters for the conservative PSM scheme for 4D Drift-Kinetic simulations.

DISCONTINOUS GALERKIN METHODS FOR VLASOV MODELS OF PLASMA By

by David C. Seal , 2012
"... iAbstract The Vlasov-Poisson equations describe the evolution of a collisionless plasma, repre-sented through a probability density function (PDF) that self-interacts via an electro-static force. One of the main difficulties in numerically solving this system is the severe time-step restriction that ..."
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iAbstract The Vlasov-Poisson equations describe the evolution of a collisionless plasma, repre-sented through a probability density function (PDF) that self-interacts via an electro-static force. One of the main difficulties in numerically solving this system is the severe time-step restriction that arises from parts of the PDF associated with moderate-to-large velocities. The dominant approach in the plasma physics community for removing these time-step restrictions is the so-called particle-in-cell (PIC) method, which discretizes the distribution function into a set of macro-particles, while the electric field is represented on a mesh. Several alternatives to this approach exist, including fully Lagrangian, fully Eulerian, and so-called semi-Lagrangian methods. The focus of this work is the semi-Lagrangian approach, which begins with a grid-based Eulerian representation of both the PDF and the electric field, then evolves the PDF via Lagrangian dynamics, and finally projects this evolved field back onto the original Eulerian mesh. We present a semi-Lagrangian and a hybrid semi-Lagrangian method for solving the Vlasov Poisson equations, based on high-order discontinuous Galerkin (DG) spatial