American Institute of Mathematical Sciences

Advanced
February  2015, 8(1): 169-183. doi: 10.3934/dcdss.2015.8.169

An exponential integrator for a highly oscillatory vlasov equation

Emmanuel Frénod 1, , Sever A. Hirstoaga 2, and  Eric Sonnendrücker 3,

1. 

Université de Bretagne-Sud, UMR 6205, LMBA, F-56000 Vannes
2. 

Inria Nancy-Grand Est, CALVI Project, & IRMA (UMR CNRS 7501), Université de Strasbourg, 7, rue René-Descartes, 67084, Strasbourg, France
3. 

Max Planck Institute for Plasma Physics, EURATOM Association, Boltzmannstr. 2, 85748 Garching, Germany

Received  April 2013 Revised  May 2013 Published  July 2014

In the framework of a Particle-In-Cell scheme for some 1D Vlasov-Poisson system depending on a small parameter, we propose a time-stepping method which is numerically uniformly accurate when the parameter goes to zero. Based on an exponential time differencing approach, the scheme is able to use large time steps with respect to the typical size of the fast oscillations of the solution.
Keywords: highly oscillatory ODEs, exponential time differencing., Particle-in-Cell method, Vlasov-Poisson system.
Mathematics Subject Classification: Primary: 35Q83, 35J05, 65N75, 65L04, 37M0.
Citation: Emmanuel Frénod, Sever A. Hirstoaga, Eric Sonnendrücker. An exponential integrator for a highly oscillatory vlasov equation. Discrete & Continuous Dynamical Systems - S, 2015, 8 (1) : 169-183. doi: 10.3934/dcdss.2015.8.169
References:
[1]

C. K. Birdsall and A. B. Langdon, Plasma Physics Via Computer Simulation, Institute of Physics Publishing, Bristol and Philadelphia, 1991. Google Scholar

[2]

J. P. Boyd, Chebyshev and Fourier Spectral Methods, $2^{nd}$ edition, Dover, New York, 2001.  Google Scholar

[3]

S. M. Cox and P. C. Matthews, Exponential time differencing for stiff systems, J. Comput. Phys., 176 (2002), 430-455. doi: 10.1006/jcph.2002.6995.  Google Scholar

[4]

N. Crouseilles, M. Lemou and F. Méhats, Asymptotic preserving schemes for highly oscillatory Vlasov-Poisson equations, J. Comput. Phys., 248 (2013), 287-308. doi: 10.1016/j.jcp.2013.04.022.  Google Scholar

[5]

F. Filbet and E. Sonnendrücker, Modeling and numerical simulation of space charge dominated beams in the paraxial approximation, Math. Models Methods Appl. Sci., 16 (2006), 763-791. doi: 10.1142/S0218202506001340.  Google Scholar

[6]

E. Frénod, Application of the averaging method to the gyrokinetic plasma, Asymptot. Anal., 46 (2006), 1-28.  Google Scholar

[7]

E. Frénod, F. Salvarani and E. Sonnendrücker, Long time simulation of a beam in a periodic focusing channel via a two-scale PIC-method, Math. Models Methods Appl. Sci., 19 (2009), 175-197. doi: 10.1142/S0218202509003395.  Google Scholar

[8]

M. Hochbruck and A. Ostermann, Exponential integrators, Acta Numer., 19 (2010), 209-286. doi: 10.1017/S0962492910000048.  Google Scholar

[9]

A.-K. Kassam and L. N. Trefethen, Fourth-order time-stepping for stiff PDEs, SIAM J. Sci. Comput., 26 (2005), 1214-1233. doi: 10.1137/S1064827502410633.  Google Scholar

[10]

T. Tückmantel, A. Pukhov, J. Liljo and M. Hochbruck, Three-dimensional relativistic particle-in-cell hybrid code based on an exponential integrator, IEEE Trans. Plasma Sci., 38 (2010), 2383-2389. Google Scholar

show all references

References:
[1]

C. K. Birdsall and A. B. Langdon, Plasma Physics Via Computer Simulation, Institute of Physics Publishing, Bristol and Philadelphia, 1991. Google Scholar

[2]

J. P. Boyd, Chebyshev and Fourier Spectral Methods, $2^{nd}$ edition, Dover, New York, 2001.  Google Scholar

[3]

S. M. Cox and P. C. Matthews, Exponential time differencing for stiff systems, J. Comput. Phys., 176 (2002), 430-455. doi: 10.1006/jcph.2002.6995.  Google Scholar

[4]

N. Crouseilles, M. Lemou and F. Méhats, Asymptotic preserving schemes for highly oscillatory Vlasov-Poisson equations, J. Comput. Phys., 248 (2013), 287-308. doi: 10.1016/j.jcp.2013.04.022.  Google Scholar

[5]

F. Filbet and E. Sonnendrücker, Modeling and numerical simulation of space charge dominated beams in the paraxial approximation, Math. Models Methods Appl. Sci., 16 (2006), 763-791. doi: 10.1142/S0218202506001340.  Google Scholar

[6]

E. Frénod, Application of the averaging method to the gyrokinetic plasma, Asymptot. Anal., 46 (2006), 1-28.  Google Scholar

[7]

E. Frénod, F. Salvarani and E. Sonnendrücker, Long time simulation of a beam in a periodic focusing channel via a two-scale PIC-method, Math. Models Methods Appl. Sci., 19 (2009), 175-197. doi: 10.1142/S0218202509003395.  Google Scholar

[8]

M. Hochbruck and A. Ostermann, Exponential integrators, Acta Numer., 19 (2010), 209-286. doi: 10.1017/S0962492910000048.  Google Scholar

[9]

A.-K. Kassam and L. N. Trefethen, Fourth-order time-stepping for stiff PDEs, SIAM J. Sci. Comput., 26 (2005), 1214-1233. doi: 10.1137/S1064827502410633.  Google Scholar

[10]

T. Tückmantel, A. Pukhov, J. Liljo and M. Hochbruck, Three-dimensional relativistic particle-in-cell hybrid code based on an exponential integrator, IEEE Trans. Plasma Sci., 38 (2010), 2383-2389. Google Scholar

[1]

Jack Schaeffer. On time decay for the spherically symmetric Vlasov-Poisson system. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021021
[2]

Katherine Zhiyuan Zhang. Focusing solutions of the Vlasov-Poisson system. Kinetic & Related Models, 2019, 12 (6) : 1313-1327. doi: 10.3934/krm.2019051
[3]

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

Blanca Ayuso, José A. Carrillo, Chi-Wang Shu. Discontinuous Galerkin methods for the one-dimensional Vlasov-Poisson system. Kinetic & Related Models, 2011, 4 (4) : 955-989. doi: 10.3934/krm.2011.4.955
[5]

Jack Schaeffer. Global existence for the Vlasov-Poisson system with steady spatial asymptotic behavior. Kinetic & Related Models, 2012, 5 (1) : 129-153. doi: 10.3934/krm.2012.5.129
[6]

Gianluca Crippa, Silvia Ligabue, Chiara Saffirio. Lagrangian solutions to the Vlasov-Poisson system with a point charge. Kinetic & Related Models, 2018, 11 (6) : 1277-1299. doi: 10.3934/krm.2018050
[7]

Zili Chen, Xiuting Li, Xianwen Zhang. The two dimensional Vlasov-Poisson system with steady spatial asymptotics. Kinetic & Related Models, 2017, 10 (4) : 977-1009. doi: 10.3934/krm.2017039
[8]

Meixia Xiao, Xianwen Zhang. On global solutions to the Vlasov-Poisson system with radiation damping. Kinetic & Related Models, 2018, 11 (5) : 1183-1209. doi: 10.3934/krm.2018046
[9]

Jean Dolbeault. An introduction to kinetic equations: the Vlasov-Poisson system and the Boltzmann equation. Discrete & Continuous Dynamical Systems, 2002, 8 (2) : 361-380. doi: 10.3934/dcds.2002.8.361
[10]

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

Joackim Bernier, Michel Mehrenberger. Long-time behavior of second order linearized Vlasov-Poisson equations near a homogeneous equilibrium. Kinetic & Related Models, 2020, 13 (1) : 129-168. doi: 10.3934/krm.2020005
[12]

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 & Continuous Dynamical Systems, 2013, 33 (2) : 723-737. doi: 10.3934/dcds.2013.33.723
[13]

Francis Filbet, Roland Duclous, Bruno Dubroca. Analysis of a high order finite volume scheme for the 1D Vlasov-Poisson system. Discrete & Continuous Dynamical Systems - S, 2012, 5 (2) : 283-305. doi: 10.3934/dcdss.2012.5.283
[14]

Trinh T. Nguyen. Derivative estimates for screened Vlasov-Poisson system around Penrose-stable equilibria. Kinetic & Related Models, 2020, 13 (6) : 1193-1218. doi: 10.3934/krm.2020043
[15]

Gang Li, Xianwen Zhang. A Vlasov-Poisson plasma of infinite mass with a point charge. Kinetic & Related Models, 2018, 11 (2) : 303-336. doi: 10.3934/krm.2018015
[16]

Yulia O. Belyaeva, Björn Gebhard, Alexander L. Skubachevskii. A general way to confined stationary Vlasov-Poisson plasma configurations. Kinetic & Related Models, 2021, 14 (2) : 257-282. doi: 10.3934/krm.2021004
[17]

Silvia Caprino, Guido Cavallaro, Carlo Marchioro. A Vlasov-Poisson plasma with unbounded mass and velocities confined in a cylinder by a magnetic mirror. Kinetic & Related Models, 2016, 9 (4) : 657-686. doi: 10.3934/krm.2016011
[18]

Dongming Wei. 1D Vlasov-Poisson equations with electron sheet initial data. Kinetic & Related Models, 2010, 3 (4) : 729-754. doi: 10.3934/krm.2010.3.729
[19]

Mihaï Bostan. Asymptotic behavior for the Vlasov-Poisson equations with strong uniform magnetic field and general initial conditions. Kinetic & Related Models, 2020, 13 (3) : 531-548. doi: 10.3934/krm.2020018
[20]

Megan Griffin-Pickering, Mikaela Iacobelli. Global strong solutions in $ {\mathbb{R}}^3 $ for ionic Vlasov-Poisson systems. Kinetic & Related Models, 2021, 14 (4) : 571-597. doi: 10.3934/krm.2021016

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (88)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy