Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-26T15:12:41.707Z Has data issue: false hasContentIssue false
  • English
  • Français

Long Time Behaviour of an Exponential Integrator for a Vlasov-Poisson System with Strong Magnetic Field

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Hyperbolic equations and systems Kinematics Numerical analysis: Ordinary differential equations

Published online by Cambridge University Press:  30 July 2015

Emmanuel Frénod ,
Sever A. Hirstoaga ,
Mathieu Lutz  and
Eric Sonnendrücker
Emmanuel Frénod
Affiliation:
Université Bretagne-Sud, UMR 6205, LMBA, F-56000 Vannes, France Inria Nancy-Grand Est, TONUS Project & IRMA (UMR CNRS 7501), Université de Strasbourg, France
Sever A. Hirstoaga*
Affiliation:
Inria Nancy-Grand Est, TONUS Project & IRMA (UMR CNRS 7501), Université de Strasbourg, France
Mathieu Lutz
Affiliation:
IRMA (UMR CNRS 7501) Université de Strasbourg, 7 rue René Descartes, F-67084 Strasbourg & Inria Nancy-Grand Est, TONUS Project, France
Eric Sonnendrücker
Affiliation:
Max Planck Institute for Plasma Physics, Boltzmannstr. 2, 85748 Garching, Germany & TU Munich, Zentrum Mathematik - M16, Boltzmannstr. 3, 85747 Garching, Germany
*
*Corresponding author. Email addresses: [email protected] (E. Frénod), [email protected] (S. A. Hirstoaga), [email protected] (M. Lutz), [email protected] (E. Sonnendrücker)
Get access

Abstract

With the aim of solving in a four dimensional phase space a multi-scale Vlasov-Poisson system, we propose in a Particle-In-Cell framework a robust time-stepping method that works uniformly when the small parameter vanishes. As an exponential integrator, the scheme is able to use large time steps with respect to the typical size of the solution’s fast oscillations. In addition, we show numerically that the method has accurate long time behaviour and that it is asymptotic preserving with respect to the limiting Guiding Center system.

Keywords

Vlasov-Poisson systemGuiding-CenterParticle-In-Cell methodhighly oscillatory ODEslong-time simulation

MSC classification

Secondary: 65L04: Stiff equations 65M99: None of the above, but in this section 35L03: Initial value problems for first-order hyperbolic equations 70B05: Kinematics of a particle
Type
Research Article
Information
Communications in Computational Physics , Volume 18 , Issue 2 , August 2015 , pp. 263 - 296
Copyright
Copyright © Global-Science Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Birdsall, C.K. and Langdon, A.B.. Plasma Physics via Computer Simulation. Institute of Physics, Bristol and Philadelphia, 1991.Google Scholar
[2]Bostan, M.. The Vlasov-Maxwell System with Strong Initial Magnetic Field: Guiding-Center Approximation. SIAM J. Multiscale Model. Simul., 6(3):1026–1058, 2007.Google Scholar
[3]Boyd, J. P.. Chebyshev and Fourier Spectral Methods. 2nd edition, Dover, New York, 2001.Google Scholar
[4]Cox, S. M. and Matthews, P. C.. Exponential time differencing for stiff systems. J. Comput. Phys., 176(2):430–455, 2002.Google Scholar
[5]Crouseilles, N., Frénod, E., Hirstoaga, S. A., and Mouton, A.. Two-scale macro-micro decomposition of the Vlasov equation with a strong magnetic field. Math. Models Methods Appl. Sci., 23(8):1527–1559, 2013.Google Scholar
[6]Crouseilles, N., Mehrenberger, M., and Sonnendrücker, E.. Conservative semi-Lagrangian schemes for Vlasov equations. J. Comput. Phys., 229(6):1927–1953, 2010.CrossRefGoogle Scholar
[7]Dubin, E. D. H., Krommes, J. A., Oberman, C., and Lee, W. W.. Nonlinear gyrokinetic equations. Phys. Fluids, 26(12):3524–3535, 1983.Google Scholar
[8]Frénod, E., Hirstoaga, S. A., and Lutz, M.. Long-time simulation of a highly oscillatory Vlasov equation with an exponential integrator. C. R. Mecanique, 342:595–609, 2014.Google Scholar
[9]Frénod, E., Hirstoaga, S. A., and Sonnendrücker, E.. An exponential integrator for a highly oscillatory Vlasov equation. Discrete Contin. Dyn. Syst. Ser. S, 8(1):169–183, 2015.Google Scholar
[10]Frénod, E. and Lutz, M.. On the geometrical gyrokinetic theory. Kinet. Relat. Models, to appear, 2015.Google Scholar
[11]Frénod, E., Raviart, P. A., and Sonnendrücker, E.. Asymptotic expansion of the Vlasov equation in a large external magnetic field. J. Math. Pures et Appl., 80(8):815–843, 2001.Google Scholar
[12]Frénod, E., Salvarani, F., and Sonnendrücker, E.. Long time simulation of a beam in a periodic focusing channel via a two-scale PIC-method. Math. Models Methods Appl. Sci., 19(2):175–197, 2009.Google Scholar
[13]Frénod, E. and Sonnendrücker, E.. Homogenization of the Vlasov equation and of the Vlasov-Poisson system with a strong external magnetic field. Asymptot. Anal., 18(3,4):193–214,1998.Google Scholar
[14]Frénod, E. and Sonnendrücker, E.. Long time behavior of the two dimensionnal Vlasov equation with a strong external magnetic field. Math. Models Methods Appl. Sci., 10(4):539–553, 2000.Google Scholar
[15]Golse, F. and Raymond, L. Saint. The Vlasov-Poisson system with strong magnetic field. J. Math. Pures. Appl., 78:791–817, 1999.CrossRefGoogle Scholar
[16]Hochbrück, M. and Ostermann, A.. Exponential integrators. Acta Numer., 19:209–286, 2010.Google Scholar
[17]Jin, S.. Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations. SIAM J. Sci. Comput., 21:441–454, 1999.Google Scholar
[18]Lee, W. W.. Gyrokinetic approach in particle simulation. Phys. Fluids, 26(2):556–562, 1983.Google Scholar
[19]Littlejohn, R. G.. A guiding center Hamiltonian: A new approach. J. Math. Phys., 20(12):2445–2458, 1979.Google Scholar
[20]Littlejohn, R. G.. Hamiltonian formulation of guiding center motion. Phys. Fluids, 24(9):1730–1749, 1981.CrossRefGoogle Scholar
[21]Shoucri, M.. A two-level implicit scheme for the numerical solution of the linearized vorticity equation. Internat. J. Numer. Methods Engrg., 17:1525–1538, 1981.Google Scholar
[22]Sonnendrücker, E.. Approximation numérique des équations de Vlasov-Maxwell, 2010. notes de cours de M2, Université de Strasbourg.Google Scholar