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14
Convergence of an adaptive semiLagrangian scheme for the VlasovPoisson system
, 2007
"... An adaptive semiLagrangian scheme for solving the Cauchy problem associated to the periodic 1+1dimensional VlasovPoisson system in the twodimensional phase space is proposed and analyzed. A key feature of our method is the accurate evolution of the adaptive mesh from one time step to the next on ..."
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Cited by 8 (3 self)
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An adaptive semiLagrangian scheme for solving the Cauchy problem associated to the periodic 1+1dimensional VlasovPoisson system in the twodimensional phase space is proposed and analyzed. A key feature of our method is the accurate evolution of the adaptive mesh from one time step to the next one, based on a rigorous analysis of the local regularity and how it gets transported by the numerical flow. The accuracy of the scheme is monitored by a prescribed tolerance parameter ε which represents the local interpolation error at each time step, in the L ∞ metric. The numerical solutions are proved to converge in L ∞ towards the exact ones as ε and ∆t tend to zero provided the initial data is Lipschitz and has a finite total curvature, or in other words, that it belongs to W 1, ∞ ∩W 2,1. The rate of convergence is O(∆t 2 + ε/∆t), which should be compared to the results of Besse, who recently established [6] similar rates for a uniform semiLagrangian scheme, but requiring that the initial data are in
Convergence analysis of a discontinuous Galerkin/Strang splitting approximation for the Vlasov–Poisson equations
, 2012
"... Abstract. A rigorous convergence analysis of the Strang splitting algorithm with a discontinuous Galerkin approximation in space for the Vlasov–Poisson equations is provided. It is shown that under suitable assumptions the error is of order O (τ2 + hq + hq/τ), where τ is the size of a time step, h i ..."
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Abstract. A rigorous convergence analysis of the Strang splitting algorithm with a discontinuous Galerkin approximation in space for the Vlasov–Poisson equations is provided. It is shown that under suitable assumptions the error is of order O (τ2 + hq + hq/τ), where τ is the size of a time step, h is the cell size, and q the order of the discontinuous Galerkin approximation. In order to investigate the recurrence phenomena for approximations of higher order as well as to compare the algorithm with numerical results already available in the literature a number of numerical simulations are performed. Key words. Strang splitting, discontinuous Galerkin approximation, convergence analysis, Vlasov–Poisson equations, recurrence AMS subject classifications. 65M12, 82D10, 65L05, 65M60 1. Introduction. In
Filbet, Analysis of a High Order Finite Volume Scheme for the VlasovPoisson System, preprint
"... Abstract. We propose a second order finite volume scheme to discretize the onedimensional VlasovPoisson system with boundary conditions. For this problem, a rather general initial and boundary data lead to a unique solution with bounded variations but such a solution becomes discontinuous when th ..."
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Cited by 2 (1 self)
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Abstract. We propose a second order finite volume scheme to discretize the onedimensional VlasovPoisson system with boundary conditions. For this problem, a rather general initial and boundary data lead to a unique solution with bounded variations but such a solution becomes discontinuous when the external voltage is large enough. We prove that the numerical approximation converges to the weak solution and show the efficiency of the scheme to simulate beam propagation with several particle species.
Numerical approximation of self consistent Vlasov models for lowfrequency electromagnetic phenomena
 Int. J. Appl. Math. Comput. Sci
"... We present a new numerical method to solve the VlasovDarwin and VlasovPoisswell systems which are approximations of the VlasovMaxwell equation in the asymptotic limit of the infinite speed of light. These systems model lowfrequency electromagnetic phenomena in plasmas, and thus “light waves ” ar ..."
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Cited by 1 (0 self)
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We present a new numerical method to solve the VlasovDarwin and VlasovPoisswell systems which are approximations of the VlasovMaxwell equation in the asymptotic limit of the infinite speed of light. These systems model lowfrequency electromagnetic phenomena in plasmas, and thus “light waves ” are somewhat supressed, which in turn allows the numerical discretization to dispense with the CourantFriedrichsLewy condition on the time step. We construct a numerical scheme based on semiLagrangian methods and time splitting techniques. We develop a fourdimensional phase space algorithm for the distribution function while the electromagnetic field is solved on a twodimensional Cartesian grid. Finally, we present two nontrivial test cases: (a) the wave Landau damping and (b) the electromagnetic beamplasma instability. For these cases our numerical scheme works very well and is in agreement with analytic kinetic theory.
Solving the vlasov equation in complex geometries
 ESAIM: Proc
"... Abstract. This paper introduces an isoparametric analysis to solve the Vlasov equation with a semiLagrangian scheme. A VlasovPoisson problem modeling a heavy ion beam in an axisymmetric configuration is considered. Numerical experiments are conducted on computational meshes targeting different ge ..."
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Abstract. This paper introduces an isoparametric analysis to solve the Vlasov equation with a semiLagrangian scheme. A VlasovPoisson problem modeling a heavy ion beam in an axisymmetric configuration is considered. Numerical experiments are conducted on computational meshes targeting different geometries. The impact of the computational grid on the accuracy and the computational cost are shown. The use of analytical mapping or Bézier patches does not induce a too large computational overhead and is quite accurate. This approach successfully couples an isoparametric analysis with a semiLagrangian scheme, and we expect to apply it to a gyrokinetic Vlasov solver. Résumé. Nous présentons ici une analyse isoparamétrique pour résoudre l’équation de Vlasov a ̀ l’aide d’un schéma SemiLagrangien. Le cas test d’un faisceau axisymétrique d’ions lourds est étudie ́ dans le cadre du système VlasovPoisson. Des tests numériques sont effectués sur différents maillages afin d’étudier diverses géométries. L’impact du choix de maillage sur la précision numérique et le coût de calcul est quantifié. L’utilisation de mapping analytique ou de patches de Bézier ne semble pas trop coûteux et permet une précision numérique suffisante. Le couplage de l’analyse isoparamétrique au schéma SemiLagrangien est donc réussi, nous espérons pouvoir appliquer cette méthode a ̀ des solveurs de l’équation de Vlasov gyrocinétique.
THEME
"... 3.1. Kinetic models for plasma and beam physics 2 3.1.1. Models for plasma and beam physics 3 ..."
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3.1. Kinetic models for plasma and beam physics 2 3.1.1. Models for plasma and beam physics 3
ProjectTeam Calvi Scientific Computing and Visualization
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The semiLagrangian method on curvilinear grids
, 2015
"... We study the semiLagrangian method on curvilinear grids. The classical backward semiLagrangian method [1] preserves constant states but is not mass conservative. Natural reconstruction of the field permits nevertheless to have at least first order in time conservation of mass, even if the spatia ..."
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We study the semiLagrangian method on curvilinear grids. The classical backward semiLagrangian method [1] preserves constant states but is not mass conservative. Natural reconstruction of the field permits nevertheless to have at least first order in time conservation of mass, even if the spatial error is large. Interpolation is performed with classical cubic splines and also cubic Hermite interpolation with arbitrary reconstruction order of the derivatives. High odd order reconstruction of the derivatives is shown to be a good ersatz of cubic splines which do not behave very well as time step tends to zero. A conservative semiLagrangian scheme along the lines of [2] is then described; here conservation of mass is automatically satisfied and constant states are shown to be preserved up to first order in time.1 1
HIGHORDER HAMILTONIAN SPLITTING FOR VLASOV–POISSON EQUATIONS
, 2015
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.