# American Institute of Mathematical Sciences

April  2012, 5(2): 283-305. doi: 10.3934/dcdss.2012.5.283

## Analysis of a high order finite volume scheme for the 1D Vlasov-Poisson system

 1 Université Lyon, Université Lyon1, CNRS, UMR 5208 - Institut Camille Jordan, 43, Boulevard du 11 Novembre 1918, F-69622 Villeurbanne cedex, France 2 Université Bordeaux I, UMR CELIA CEA, CNRS et Institut de Mathmatiques de Bordeaux, 351, Cours de la Libération, F-33405 Talence cedex, France, France

Received  July 2009 Revised  December 2009 Published  September 2011

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 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.
Citation: Francis Filbet, Roland Duclous, Bruno Dubroca. Analysis of a high order finite volume scheme for the 1D Vlasov-Poisson system. Discrete and Continuous Dynamical Systems - S, 2012, 5 (2) : 283-305. doi: 10.3934/dcdss.2012.5.283
##### References:
 [1] T. D. Arber and R. G. L. Vann, A critical comparison of Eulerian-grid-based Vlasov solvers, J. Comput. Physics, 180 (2002), 339-357. doi: 10.1006/jcph.2002.7098. [2] N. Besse, Convergence of a semi-Lagrangian scheme for the one-dimensional Vlasov-Poisson system, SIAM J. Numer. Anal., 42 (2004), 350-382. doi: 10.1137/S0036142902410775. [3] J. P. Boris and D. L. Book, Solution of continuity equations by the method of flux-corrected transport, J. Comput. Phys., 20 (1976), 397-431. doi: 10.1016/0021-9991(76)90091-7. [4] J. A. Carrillo and F. Vecil, Nonoscillatory interpolation methods applied to Vlasov-based models, SIAM J. Sci. Comput., 29 (2007), 1179-1206. doi: 10.1137/050644549. [5] C. Z. Cheng and G. Knorr, The integration of the Vlasov equation in configuration space, J. Comput. Phys., 22 (1976), 330-351. doi: 10.1016/0021-9991(76)90053-X. [6] J. Cooper and A. J. Klimas, Boundary value problems for the Vlasov-Maxwell equation in one dimension, J. Math. Anal. Appl., 75 (1980), 306-329. doi: 10.1016/0022-247X(80)90082-7. [7] P. Degond and P.-A. Raviart, An asymptotic analysis of the Vlasov-Poisson system: The Child-Langmuir law, Asymptotic Anal., 4 (1991), 187-214. [8] R. J. DiPerna and P. L. Lions, Solutions globales d'équations du type Vlasov-Poisson, C. R. Acad. Sci. Paris Série I Math., 307 (1988), 306-329. [9] F. Filbet, Convergence d'un schéma de type volumes finis pour la resolution numérique du système de Vlasov-Poisson en dimension un, (French) [Convergence of a finite volume scheme for the numerical solution of the one-dimensional Vlasov-Poisson system], CRAS Paris, 330 (2000), 979-984. [10] F. Filbet, Convergence of a finite volume scheme for the Vlasov-Poisson system, SIAM J. Numer. Anal. 39 (2001), 1146-1169. doi: 10.1137/S003614290037321X. [11] F. Filbet, E. Sonnendrücker and P. Bertrand, Conservative numerical schemes for the Vlasov equation, J. Comput. Phys. 172 (2001), 166-187. doi: 10.1006/jcph.2001.6818. [12] F. Filbet and E. Sonnendrücker, Comparison of Eulerian Vlasov solvers, Computer Physics Communications, 150 (2003), 247-266. doi: 10.1016/S0010-4655(02)00694-X. [13] F. Filbet, Y. Guo and C.-W. Shu, Analysis of the relativistic Vlasov-Maxwell model in an interval, Quarterly Applied Math., 63 (2005), 691-714. [14] C. Greengard and P.-A. Raviart, A boundary-value problem for the stationary Vlasov-Poisson equations: The plane diode, Comm. Pure. Appl. Math., 43 (1990), 473-507. doi: 10.1002/cpa.3160430404. [15] Y. Guo, Global weak solutions of the Vlasov-Maxwell system with boundary conditions, Comm. Math. Phys., 154 (1993), 254-263. doi: 10.1007/BF02096997. [16] Y. Guo, C.-W. Shu and T. Zhou, The dynamics of a plane diode, SIAM J. Math. Anal., 35 (2004), 1617-1635. doi: 10.1137/S0036141003421133. [17] A. J. Klimas and W. M. Farrell, A splitting algorithm for Vlasov simulation with filamentation filtration, J. Comput. Phys., 110 (1994), 150-163. doi: 10.1006/jcph.1994.1011. [18] F. Poupaud, Boundary value problems for the stationary Vlasov-Maxwell system, Forum Math., 4 (1992), 499-527. doi: 10.1515/form.1992.4.499. [19] J. Schaeffer, Convergence of a difference scheme for the Vlasov-Poisson-Fokker-Planck system in one dimension, SIAM J. Numer. Anal., 35 (1998), 1149-1175. doi: 10.1137/S0036142996302554. [20] M. Shoucri and G. Knorr, Numerical integration of the Vlasov equation, J. Comput. Phys., 14 (1974), 84-92. doi: 10.1016/0021-9991(74)90006-0. [21] Y. Zheng and A. Majda, Existence of global weak solutions to one-component Vlasov-Poisson and Fokker-Planck-Poisson systems in one space dimension with measures as initial data, Comm. Pure Appl. Math., 47 (1994), 1365-1401. doi: 10.1002/cpa.3160471004. [22] T. Nakamura and T. Yabe, Cubic interpolated propagation scheme for solving the hyper-dimensional Vlasov-Poisson equation in phase space, Comput. Phys. Communications, 120 (1999), 122-154. doi: 10.1016/S0010-4655(99)00247-7.

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##### References:
 [1] T. D. Arber and R. G. L. Vann, A critical comparison of Eulerian-grid-based Vlasov solvers, J. Comput. Physics, 180 (2002), 339-357. doi: 10.1006/jcph.2002.7098. [2] N. Besse, Convergence of a semi-Lagrangian scheme for the one-dimensional Vlasov-Poisson system, SIAM J. Numer. Anal., 42 (2004), 350-382. doi: 10.1137/S0036142902410775. [3] J. P. Boris and D. L. Book, Solution of continuity equations by the method of flux-corrected transport, J. Comput. Phys., 20 (1976), 397-431. doi: 10.1016/0021-9991(76)90091-7. [4] J. A. Carrillo and F. Vecil, Nonoscillatory interpolation methods applied to Vlasov-based models, SIAM J. Sci. Comput., 29 (2007), 1179-1206. doi: 10.1137/050644549. [5] C. Z. Cheng and G. Knorr, The integration of the Vlasov equation in configuration space, J. Comput. Phys., 22 (1976), 330-351. doi: 10.1016/0021-9991(76)90053-X. [6] J. Cooper and A. J. Klimas, Boundary value problems for the Vlasov-Maxwell equation in one dimension, J. Math. Anal. Appl., 75 (1980), 306-329. doi: 10.1016/0022-247X(80)90082-7. [7] P. Degond and P.-A. Raviart, An asymptotic analysis of the Vlasov-Poisson system: The Child-Langmuir law, Asymptotic Anal., 4 (1991), 187-214. [8] R. J. DiPerna and P. L. Lions, Solutions globales d'équations du type Vlasov-Poisson, C. R. Acad. Sci. Paris Série I Math., 307 (1988), 306-329. [9] F. Filbet, Convergence d'un schéma de type volumes finis pour la resolution numérique du système de Vlasov-Poisson en dimension un, (French) [Convergence of a finite volume scheme for the numerical solution of the one-dimensional Vlasov-Poisson system], CRAS Paris, 330 (2000), 979-984. [10] F. Filbet, Convergence of a finite volume scheme for the Vlasov-Poisson system, SIAM J. Numer. Anal. 39 (2001), 1146-1169. doi: 10.1137/S003614290037321X. [11] F. Filbet, E. Sonnendrücker and P. Bertrand, Conservative numerical schemes for the Vlasov equation, J. Comput. Phys. 172 (2001), 166-187. doi: 10.1006/jcph.2001.6818. [12] F. Filbet and E. Sonnendrücker, Comparison of Eulerian Vlasov solvers, Computer Physics Communications, 150 (2003), 247-266. doi: 10.1016/S0010-4655(02)00694-X. [13] F. Filbet, Y. Guo and C.-W. Shu, Analysis of the relativistic Vlasov-Maxwell model in an interval, Quarterly Applied Math., 63 (2005), 691-714. [14] C. Greengard and P.-A. Raviart, A boundary-value problem for the stationary Vlasov-Poisson equations: The plane diode, Comm. Pure. Appl. Math., 43 (1990), 473-507. doi: 10.1002/cpa.3160430404. [15] Y. Guo, Global weak solutions of the Vlasov-Maxwell system with boundary conditions, Comm. Math. Phys., 154 (1993), 254-263. doi: 10.1007/BF02096997. [16] Y. Guo, C.-W. Shu and T. Zhou, The dynamics of a plane diode, SIAM J. Math. Anal., 35 (2004), 1617-1635. doi: 10.1137/S0036141003421133. [17] A. J. Klimas and W. M. Farrell, A splitting algorithm for Vlasov simulation with filamentation filtration, J. Comput. Phys., 110 (1994), 150-163. doi: 10.1006/jcph.1994.1011. [18] F. Poupaud, Boundary value problems for the stationary Vlasov-Maxwell system, Forum Math., 4 (1992), 499-527. doi: 10.1515/form.1992.4.499. [19] J. Schaeffer, Convergence of a difference scheme for the Vlasov-Poisson-Fokker-Planck system in one dimension, SIAM J. Numer. Anal., 35 (1998), 1149-1175. doi: 10.1137/S0036142996302554. [20] M. Shoucri and G. Knorr, Numerical integration of the Vlasov equation, J. Comput. Phys., 14 (1974), 84-92. doi: 10.1016/0021-9991(74)90006-0. [21] Y. Zheng and A. Majda, Existence of global weak solutions to one-component Vlasov-Poisson and Fokker-Planck-Poisson systems in one space dimension with measures as initial data, Comm. Pure Appl. Math., 47 (1994), 1365-1401. doi: 10.1002/cpa.3160471004. [22] T. Nakamura and T. Yabe, Cubic interpolated propagation scheme for solving the hyper-dimensional Vlasov-Poisson equation in phase space, Comput. Phys. Communications, 120 (1999), 122-154. doi: 10.1016/S0010-4655(99)00247-7.
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