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Convergence of an adaptive semi-Lagrangian scheme for the Vlasov-Poisson system
, 2007
"... An adaptive semi-Lagrangian scheme for solving the Cauchy problem associated to the periodic 1+1-dimensional Vlasov-Poisson system in the two-dimensional 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 semi-Lagrangian scheme for solving the Cauchy problem associated to the periodic 1+1-dimensional Vlasov-Poisson system in the two-dimensional 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 semi-Lagrangian 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 Vlasov-Poisson System, preprint
"... Abstract. We propose a second order finite volume scheme to discretize the one-dimensional Vlasov-Poisson 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 dis-continuous 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 one-dimensional Vlasov-Poisson 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 dis-continuous 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 low-frequency electromagnetic phenomena
- Int. J. Appl. Math. Comput. Sci
"... We present a new numerical method to solve the Vlasov-Darwin and Vlasov-Poisswell systems which are approximations of the Vlasov-Maxwell equation in the asymptotic limit of the infinite speed of light. These systems model low-frequency 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 Vlasov-Darwin and Vlasov-Poisswell systems which are approximations of the Vlasov-Maxwell equation in the asymptotic limit of the infinite speed of light. These systems model low-frequency electromagnetic phenomena in plasmas, and thus “light waves ” are somewhat supressed, which in turn allows the numerical discretization to dispense with the Courant-Friedrichs-Lewy condition on the time step. We construct a numerical scheme based on semi-Lagrangian methods and time splitting techniques. We develop a four-dimensional phase space algorithm for the distribution function while the electromagnetic field is solved on a two-dimensional Cartesian grid. Finally, we present two nontrivial test cases: (a) the wave Landau damping and (b) the electromagnetic beam-plasma 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 semi-Lagrangian scheme. A Vlasov-Poisson problem modeling a heavy ion beam in an axisymmetric configu-ration 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 semi-Lagrangian scheme. A Vlasov-Poisson problem modeling a heavy ion beam in an axisymmetric configu-ration 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 semi-Lagrangian 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 Semi-Lagrangien. Le cas test d’un faisceau axisymétrique d’ions lourds est étudie ́ dans le cadre du système Vlasov-Poisson. 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 Semi-Lagrangien 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
Project-Team Calvi Scientific Computing and Visualization
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The semi-Lagrangian method on curvilinear grids
, 2015
"... We study the semi-Lagrangian method on curvilinear grids. The clas-sical backward semi-Lagrangian method [1] preserves constant states but is not mass conservative. Natural reconstruction of the field permits nev-ertheless to have at least first order in time conservation of mass, even if the spatia ..."
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We study the semi-Lagrangian method on curvilinear grids. The clas-sical backward semi-Lagrangian method [1] preserves constant states but is not mass conservative. Natural reconstruction of the field permits nev-ertheless 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 semi-Lagrangian scheme along the lines of [2] is then described; here conservation of mass is au-tomatically satisfied and constant states are shown to be preserved up to first order in time.1 1
HIGH-ORDER HAMILTONIAN SPLITTING FOR VLASOV–POISSON EQUATIONS
, 2015
"... HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished 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 multi-disciplinary open access archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished 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.
