Article Contents
Article Contents

# An exponential integrator for a highly oscillatory vlasov equation

• 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.
Mathematics Subject Classification: Primary: 35Q83, 35J05, 65N75, 65L04, 37M05.

 Citation:

•  [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.