We present a new numerical method to solve the Vlasov-Darwin and Vlasov-Poisswell systems which are approximations of the Vlasov-Maxwell equation in the asymptotic limit of the infinite speed of light. These systems model low-frequency electromagnetic phenomena in plasmas, and thus "light waves" are somewhat supressed, which in turn allows thenumerical discretization to dispense with the Courant-Friedrichs-Lewy condition on the time step. We construct a numerical scheme based on semi-Lagrangian methods and time splitting techniques. We develop a four-dimensional phase space algorithm for the distribution function while the electromagnetic field is solved on a two-dimensional Cartesian grid. Finally, we present two nontrivial test cases: (a) the wave Landau damping and (b) the electromagnetic beam-plasma instability. For these cases our numerical scheme works very well and is in agreement with analytic kinetic theory.
@article{bwmeta1.element.bwnjournal-article-amcv17i3p361bwm,
author = {Besse, Nicolas and J. mauser, Norbert and Sonnendr\"ucker, Eric},
title = {Numerical approximation of self-consistent Vlasov models for low-frequency electromagnetic phenomena},
journal = {International Journal of Applied Mathematics and Computer Science},
volume = {17},
year = {2007},
pages = {361-374},
zbl = {1149.82028},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-amcv17i3p361bwm}
}
Besse, Nicolas; J. mauser, Norbert; Sonnendrücker, Eric. Numerical approximation of self-consistent Vlasov models for low-frequency electromagnetic phenomena. International Journal of Applied Mathematics and Computer Science, Tome 17 (2007) pp. 361-374. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-amcv17i3p361bwm/
[000] Bauer S. and Kunze M. (2005): The Darwin approximation of the relativistic Vlasov-Maxwell system. Annales Henri Poincaré, Vol.6, No.2, pp.283-308. | Zbl 1126.82041
[001] Bégué M. L., Ghizzo A. and Bertrand P. (1999): Two-dimensional Vlasov simulation of Raman scattering and plasma beatwave acceleration on parallel computers. Journal of Computational Physics, Vol.151, No.2, pp.458-478. | Zbl 0956.76068
[002] Benachour S., Filbet F., Laurencccot P. and Sonnendrucker E.(2003): Global existence for the Vlasov-Darwin system in R^3 for small initial data. Mathematical Methods in the Applied Sciences., Vol.26, No.4, pp.297-319. | Zbl 1024.35027
[003] Besse N. (2004): Convergence of a semi-Lagrangian scheme for the one-dimensional Vlasov-Poisson system. SIAM Journal on Numerical Analysis, Vol.42, No.1, pp.350-382. | Zbl 1071.82037
[004] Besse N. and Mehrenberger M. (2006): Convergence of classes of high-order semi-Lagrangian schemes for the Vlasov-Poisson system. Mathematics of Computation, (in print). | Zbl 1131.65080
[005] Besse N. and Sonnendrucker E. (2003): Semi-Lagrangian schemes for the Vlasov equation on an unstructured mesh of phase space. Journal of Computational Physics, Vol.191, No.2, pp.341-376. | Zbl 1030.82011
[006] Borodachev L. V. (2005): Elliptic formulation of discrete Vlasov-Darwin model with the implicit finite-difference representation of particle dynamics. Proceedings of 7-th International School/Symposium for Space Simulations, Kyoto, Japan. | Zbl 1089.78502
[007] Califano F., Prandi R., Pegoraro F. and Bulanov S.V. (1998): Nonlinear filamentation instability driven by an inhomogeneous current in a collisionless plasma. Physical Review E, Vol.58, No.6, pp.19-24.
[008] Cheng C. Z. and Knorr G. (1976): The integration of the Vlasov equation in configuration space. Journal of Computational Physics, Vol.22, No.3, pp.330-351.
[009] Coppi B., Laval G. and Pellat R. (1996): Dynamics of the geomagnetic tail. Physical Review Letters, Vol.16, No.26, pp.1207-1210.
[010] Degond P. and Raviart P.-A. (1992): An analysis of Darwin model of approximation to Maxwell's eqautions. Forum Mathematics, Vol.4, pp.13-44. | Zbl 0755.35137
[011] Fijalkow E. (1999): A numerical solution to the Vlasov equation. Computer Physics Communications, Vol.116, No.2, pp.319-328. | Zbl 1019.76035
[012] Filbet F., Sonnendrucker E. and Bertrand P. (2000): Conservative numerical schemes for the Vlasov equation. Journal of Computational Physics, Vol.172, No.1, pp.166-187. | Zbl 0998.65138
[013] Gibbons M. R. and Hewett D.W. (1995): The Darwin direct implicit particle-on-cell (DADIPIC) method for simulation of low frequency plasma phenomena. Journal of Computational Physics, Vol.120, No.2, pp.231-247. | Zbl 0841.76066
[014] Gibbons M. R. and Hewett D.W. (1997): Characterization of the Darwin direct implicit particle-in-cell method an resulting guidelines for operation. Journal of Computational Physics, Vol.130, No.1, pp.54-66. | Zbl 0870.76056
[015] Kulsrud R. M. (1998): Magnetic reconnection in a magnetohydrodynamic plasma. Physics of Plasmas, Vol.5, No.5, pp.1599-1606.
[016] Lee W. W.L., Startsev E., Hong Q. and Davidson R.C. (2001): Electromagnetic (Darwin) model for three-dimensional perturbative particle simulation of high intensy beams. Proceedings of the Particle Accelerator Conference, PACS'2001, Chicago, USA, IEEE Part, Vol.3, p.1906.
[017] Masmoudi N. and Mauser N.J. (2001): The selfconsistent Pauli equation. Monatshefte fur Mathematik, Vol.132, No.1, pp.19-24. | Zbl 0973.35160
[018] Omelchenko Y. A. and Sudan R.N. (1997): A 3-D Darwin-EM Hybrid PIC code for ion ring studies. Journal of Computational Physics, Vol.133, No.1, pp.146-159. | Zbl 0883.76067
[019] Ottaviani M. and Porcelli F. (1993): Nonlinear collisionless magnetic reconnection. Physical Review Letters, Vol.71, No.23, pp.3802-3805.
[020] Pallard C. (2006): The initial value problem for the relativistic Vlasov-Darwin system. International Mathematics Research Notices, Vol.2006, Article ID 57191, available at: DOI:100.1155/IMRN/2006/57191.
[021] Raviart P. -A. and Sonnendrucker E. (1996): A Hierarchy of Approximate Models for the Maxwell Equations. Numerische Mathematik, Vol.73, No.3, pp.329-372. | Zbl 0862.35122
[022] Schmz H. and Grauer R. (2006): newblock Darwin-Vlasov simulations of magnetised plasmas. Journal of Computational Physics, Vol.214, No.2, pp.738-756. | Zbl 1136.82374
[023] Sabatier M., Such N., Mineau P., Feix M., Shoucri M., Bertrand P. and Fijalkow E. (1990): Numerical simulations of the Vlasov equation using a flux conservation scheme; comparaison with the cubic spline interpolation code. Technical Report No.330e, Centre Canadien de Fusion Magnetique, Varennes, Canada.
[024] Sonnendrucker E., Ambrosiano J.J. and Brandon S.T. (1995): A finite element formulation of the Darwin PIC model for use on unstructured grids. Journal of Computational Physics, Vol.121, No.2, pp.281-297. | Zbl 0834.76052
[025] Taguchi T., Antonsen T.M. Jr., Liu C.S. and Mima K. (2001): Structure formation and tearing of an MeV cylindrical electron beam in a laser-produced plasma. Physical Review Letters, Vol.86, No.2, pp.5055-5058