American Institute of Mathematical Sciences

Advanced
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

Francis Filbet 1, , Roland Duclous 2, and  Bruno Dubroca 2,

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.
Keywords: Finite Volume Schemes, Weak BV Estimate., Vlasov-Poisson System.
Mathematics Subject Classification: 65M12, 82D1.
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.

show all references

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.

[1]

Katherine Zhiyuan Zhang. Focusing solutions of the Vlasov-Poisson system. Kinetic and Related Models, 2019, 12 (6) : 1313-1327. doi: 10.3934/krm.2019051
[2]

Blanca Ayuso, José A. Carrillo, Chi-Wang Shu. Discontinuous Galerkin methods for the one-dimensional Vlasov-Poisson system. Kinetic and Related Models, 2011, 4 (4) : 955-989. doi: 10.3934/krm.2011.4.955
[3]

Jack Schaeffer. Global existence for the Vlasov-Poisson system with steady spatial asymptotic behavior. Kinetic and Related Models, 2012, 5 (1) : 129-153. doi: 10.3934/krm.2012.5.129
[4]

Gianluca Crippa, Silvia Ligabue, Chiara Saffirio. Lagrangian solutions to the Vlasov-Poisson system with a point charge. Kinetic and Related Models, 2018, 11 (6) : 1277-1299. doi: 10.3934/krm.2018050
[5]

Zili Chen, Xiuting Li, Xianwen Zhang. The two dimensional Vlasov-Poisson system with steady spatial asymptotics. Kinetic and Related Models, 2017, 10 (4) : 977-1009. doi: 10.3934/krm.2017039
[6]

Meixia Xiao, Xianwen Zhang. On global solutions to the Vlasov-Poisson system with radiation damping. Kinetic and Related Models, 2018, 11 (5) : 1183-1209. doi: 10.3934/krm.2018046
[7]

Jean Dolbeault. An introduction to kinetic equations: the Vlasov-Poisson system and the Boltzmann equation. Discrete and Continuous Dynamical Systems, 2002, 8 (2) : 361-380. doi: 10.3934/dcds.2002.8.361
[8]

Jack Schaeffer. On time decay for the spherically symmetric Vlasov-Poisson system. Kinetic and Related Models, 2022, 15 (4) : 721-727. doi: 10.3934/krm.2021021
[9]

Hao Wang. Uniform stability estimate for the Vlasov-Poisson-Boltzmann system. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 657-680. doi: 10.3934/dcds.2020292
[10]

Gerhard Rein, Christopher Straub. On the transport operators arising from linearizing the Vlasov-Poisson or Einstein-Vlasov system about isotropic steady states. Kinetic and Related Models, 2020, 13 (5) : 933-949. doi: 10.3934/krm.2020032
[11]

Hyung Ju Hwang, Jaewoo Jung, Juan J. L. Velázquez. On global existence of classical solutions for the Vlasov-Poisson system in convex bounded domains. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 723-737. doi: 10.3934/dcds.2013.33.723
[12]

Trinh T. Nguyen. Derivative estimates for screened Vlasov-Poisson system around Penrose-stable equilibria. Kinetic and Related Models, 2020, 13 (6) : 1193-1218. doi: 10.3934/krm.2020043
[13]

Silvia Caprino, Guido Cavallaro, Carlo Marchioro. Time evolution of a Vlasov-Poisson plasma with magnetic confinement. Kinetic and Related Models, 2012, 5 (4) : 729-742. doi: 10.3934/krm.2012.5.729
[14]

Gang Li, Xianwen Zhang. A Vlasov-Poisson plasma of infinite mass with a point charge. Kinetic and Related Models, 2018, 11 (2) : 303-336. doi: 10.3934/krm.2018015
[15]

Yulia O. Belyaeva, Björn Gebhard, Alexander L. Skubachevskii. A general way to confined stationary Vlasov-Poisson plasma configurations. Kinetic and Related Models, 2021, 14 (2) : 257-282. doi: 10.3934/krm.2021004
[16]

Xianglong Duan. Sharp decay estimates for the Vlasov-Poisson and Vlasov-Yukawa systems with small data. Kinetic and Related Models, 2022, 15 (1) : 119-146. doi: 10.3934/krm.2021049
[17]

Boris Andreianov, Mostafa Bendahmane, Kenneth H. Karlsen, Charles Pierre. Convergence of discrete duality finite volume schemes for the cardiac bidomain model. Networks and Heterogeneous Media, 2011, 6 (2) : 195-240. doi: 10.3934/nhm.2011.6.195
[18]

Silvia Caprino, Guido Cavallaro, Carlo Marchioro. A Vlasov-Poisson plasma with unbounded mass and velocities confined in a cylinder by a magnetic mirror. Kinetic and Related Models, 2016, 9 (4) : 657-686. doi: 10.3934/krm.2016011
[19]

Joackim Bernier, Michel Mehrenberger. Long-time behavior of second order linearized Vlasov-Poisson equations near a homogeneous equilibrium. Kinetic and Related Models, 2020, 13 (1) : 129-168. doi: 10.3934/krm.2020005
[20]

Dongming Wei. 1D Vlasov-Poisson equations with electron sheet initial data. Kinetic and Related Models, 2010, 3 (4) : 729-754. doi: 10.3934/krm.2010.3.729

2021 Impact Factor: 1.865

Tools

Metrics

  • PDF downloads (65)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy