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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

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

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