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Guidingcenter simulations on curvilinear meshes using semiLagrangian conservative methods, in "Issue dedicated to the proceedings of the second edition of the Conference Numerical models for fusion
, 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 timescale 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 semiLagrangian solvers in curvilinear coordinates. Before tackling the full gyrokinetic model in a future work, we consider here the reduced 2D GuidingCenter model. We introduce our numerical algorithm and provide some numerical results showing its good properties. 1. Introduction. In
High order RungeKuttaNyström splitting methods for the VlasovPoisson equation
, 2011
"... In this work, we derive the order conditions for fourth order time splitting schemes in the case of the 1D VlasovPoisson system. Computations to obtain such conditions are motivated by the specific Poisson structure of the VlasovPoisson system: this structure is similar to RungeKuttaNyström 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 VlasovPoisson system. Computations to obtain such conditions are motivated by the specific Poisson structure of the VlasovPoisson system: this structure is similar to RungeKuttaNyström 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
Discontinuous Galerkin SemiLagrangian Method for VlasovPoisson, CEMRACS’10 research achievements: Numerical modeling of fusion
 ESAIM Proceedings, October 2011
"... Abstract. We present a discontinuous Galerkin scheme for the numerical approximation of the onedimensional periodic VlasovPoisson equation. The scheme is based on a Galerkincharacteristics 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 VlasovPoisson equation. The scheme is based on a Galerkincharacteristics method in which the distribution function is projected onto a space of discontinuous functions. We present comparisons with a semiLagrangian method to emphasize the good behavior of this scheme when applied to VlasovPoisson test cases. Résumé. Une méthode de Galerkin discontinu est proposée pour l’approximation numérique de l’équation de VlasovPoisson 1D. L’approche est basée sur une méthode Galerkincaractéristiques où la fonction de distribution est projetée sur un espace de fonctions discontinues. En particulier, la méthode est comparée à une méthode semiLagrangienne pour l’approximation de l’équation de VlasovPoisson.
Conservative and nonconservative methods based on Hermite weighted essentiallynonoscillatory reconstruction for Vlasov equations
 Université de Lyon & Inria, Institut Camille Jordan, EPI Kaliffe, 43 boulevard 11 novembre 1918, F69622 Villeurbanne cedex, FRANCE Email address, F. Filbet: filbet@math.univlyon1.fr Department of Mathematics, Harbin Institute of Technology, 92 West Da
"... Abstract. We introduce a WENO reconstruction based on Hermite interpolation both for semiLagrangian and finite difference methods. This WENO reconstruction technique allows to control spurious oscillations. We develop third and fifth order methods and apply them to nonconservative semiLagrangia ..."
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Abstract. We introduce a WENO reconstruction based on Hermite interpolation both for semiLagrangian and finite difference methods. This WENO reconstruction technique allows to control spurious oscillations. We develop third and fifth order methods and apply them to nonconservative semiLagrangian schemes and conservative finite difference methods. Our numerical results will be compared to the usual semiLagrangian method with cubic spline reconstruction and the classical fifth order WENO finite difference scheme. These reconstructions are observed to be less dissipative than the usual weighted essentially nonoscillatory procedure. We apply these methods to transport equations in the context of plasma physics and the numerical simulation of turbulence phenomena.
KINETIC/FLUID MICROMACRO NUMERICAL SCHEMES FOR VLASOVPOISSONBGK EQUATION USING PARTICLES
"... 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 micromacro 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 micromacro 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 soobtained scheme presents a much less level of noise compared to the standard particle method; (ii) the computational cost of the micromacro 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 VlasovAmpère equations
 Commun. Comput. Phys
"... In this report, a charge preserving numerical resolution of the 1D VlasovAmpère equation is achieved, with a forward SemiLagrangian 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 VlasovAmpère equation is achieved, with a forward SemiLagrangian 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 selfconsistently rules the electric field evolution. In order to ensure having proper physical solutions, it is necessary that the scheme preserves charge numerically. BSpline deposition will be used for the interpolation step. The solving of the characteristics will be made with a RungeKutta 2 method and with a CauchyKovalevsky procedure.
Blockstructured adaptive mesh refinement algorithms for vlasov simulation
 CoRR
"... Direct discretization of continuum kinetic equations, like the Vlasov equation, are underutilized because the distribution function generally exists in a highdimensional (>3D) space and computational cost increases geometrically with dimension. We propose to use highorder finitevolume techni ..."
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Direct discretization of continuum kinetic equations, like the Vlasov equation, are underutilized because the distribution function generally exists in a highdimensional (>3D) space and computational cost increases geometrically with dimension. We propose to use highorder finitevolume techniques with blockstructured 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 singledimension block structured AMR to the multidimension 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 highorder spatial and temporal discretizations are discussed. Preliminary results for a standard 1D+1V VlasovPoisson 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 DriftKinetic simulations.
"... apport de recherche ISSN 02496399 ISRN INRIA/RR7467FR+ENGinria00540948, version 1 30 Nov 2010Test of some numerical limiters for the conservative PSM scheme for 4D DriftKinetic simulations. ..."
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apport de recherche ISSN 02496399 ISRN INRIA/RR7467FR+ENGinria00540948, version 1 30 Nov 2010Test of some numerical limiters for the conservative PSM scheme for 4D DriftKinetic simulations.
DISCONTINOUS GALERKIN METHODS FOR VLASOV MODELS OF PLASMA By
, 2012
"... iAbstract The VlasovPoisson equations describe the evolution of a collisionless plasma, represented through a probability density function (PDF) that selfinteracts via an electrostatic force. One of the main difficulties in numerically solving this system is the severe timestep restriction that ..."
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iAbstract The VlasovPoisson equations describe the evolution of a collisionless plasma, represented through a probability density function (PDF) that selfinteracts via an electrostatic force. One of the main difficulties in numerically solving this system is the severe timestep restriction that arises from parts of the PDF associated with moderatetolarge velocities. The dominant approach in the plasma physics community for removing these timestep restrictions is the socalled particleincell (PIC) method, which discretizes the distribution function into a set of macroparticles, while the electric field is represented on a mesh. Several alternatives to this approach exist, including fully Lagrangian, fully Eulerian, and socalled semiLagrangian methods. The focus of this work is the semiLagrangian approach, which begins with a gridbased 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 semiLagrangian and a hybrid semiLagrangian method for solving the Vlasov Poisson equations, based on highorder discontinuous Galerkin (DG) spatial