An exponential integrator for a highly oscillatory vlasov equation

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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

Abstract
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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 and 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.
[2]	J. P. Boyd, Chebyshev and Fourier Spectral Methods, $2^{nd}$ edition, Dover, New York, 2001.
[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.
[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.
[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.
[6]	E. Frénod, Application of the averaging method to the gyrokinetic plasma, Asymptot. Anal., 46 (2006), 1-28.
[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.
[8]	M. Hochbruck and A. Ostermann, Exponential integrators, Acta Numer., 19 (2010), 209-286. doi: 10.1017/S0962492910000048.
[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.
[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.

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.
[2]	J. P. Boyd, Chebyshev and Fourier Spectral Methods, $2^{nd}$ edition, Dover, New York, 2001.
[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.
[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.
[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.
[6]	E. Frénod, Application of the averaging method to the gyrokinetic plasma, Asymptot. Anal., 46 (2006), 1-28.
[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.
[8]	M. Hochbruck and A. Ostermann, Exponential integrators, Acta Numer., 19 (2010), 209-286. doi: 10.1017/S0962492910000048.
[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.
[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.

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