Geodesic
Numerical approximation of self-consistent Vlasov models for low-frequency electromagnetic phenomena
International Journal of Applied Mathematics and Computer Science, Tome 17 (2007) no. 3, pp. 361-374.

Voir la notice de l'article provenant de la source Library of Science

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 the numerical 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.
Keywords: Vlasov-Darwin model, Vlasov-Poisswell model, semi-Lagrangian method, low-frequency electromagnetic
Mots-clés : model Vlasova-Darwina, model Vlasova-Poiswell'a, metoda Lagrangiana, niskoczęstotliwościowe zjawisko elektromagnetyczne 
@article{IJAMCS_2007_17_3_a6,
     author = {Besse, N. and Mauser, N. J. and Sonnendr\"ucker, E.},
     title = {Numerical approximation of self-consistent {Vlasov} models for low-frequency electromagnetic phenomena},
     journal = {International Journal of Applied Mathematics and Computer Science},
     pages = {361--374},
     publisher = {mathdoc},
     volume = {17},
     number = {3},
     year = {2007},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IJAMCS_2007_17_3_a6/}
}
TY  - JOUR
AU  - Besse, N.
AU  - Mauser, N. J.
AU  - Sonnendrücker, E.
TI  - Numerical approximation of self-consistent Vlasov models for low-frequency electromagnetic phenomena
JO  - International Journal of Applied Mathematics and Computer Science
PY  - 2007
SP  - 361
EP  - 374
VL  - 17
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IJAMCS_2007_17_3_a6/
LA  - en
ID  - IJAMCS_2007_17_3_a6
ER  - 
%0 Journal Article
%A Besse, N.
%A Mauser, N. J.
%A Sonnendrücker, E.
%T Numerical approximation of self-consistent Vlasov models for low-frequency electromagnetic phenomena
%J International Journal of Applied Mathematics and Computer Science
%D 2007
%P 361-374
%V 17
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IJAMCS_2007_17_3_a6/
%G en
%F IJAMCS_2007_17_3_a6
Besse, N.; Mauser, N. J.; Sonnendrücker, E. Numerical approximation of self-consistent Vlasov models for low-frequency electromagnetic phenomena. International Journal of Applied Mathematics and Computer Science, Tome 17 (2007) no. 3, pp. 361-374. http://geodesic.mathdoc.fr/item/IJAMCS_2007_17_3_a6/

[1] 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.

[2] Bégué M.L., Ghizzo A. and Bertrand P. (1999): Twodimensional Vlasov simulation of Raman scattering and plasma beatwave acceleration on parallel computers. Journal of Computational Physics, Vol. 151, No. 2, pp. 458-478.

[3] Benachour S., Filbet F., Laurençcot P. and Sonnendrücker E.(2003): Global existence for the Vlasov-Darwin system in R3 for small initial data. Mathematical Methods in the Applied Sciences., Vol. 26, No. 4, pp. 297-319.

[4] 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.

[5] 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).

[6] Besse N. and Sonnendrücker 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.

[7] 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.

[8] 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.

[9] 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.

[10] Coppi B., Laval G. and Pellat R. (1996): Dynamics of the geomagnetic tail. Physical Review Letters, Vol. 16, No. 26, pp. 1207-1210.

[11] Degond P. and Raviart P.-A. (1992): An analysis of Darwin model of approximation to Maxwell's eqautions. ForumMathematics, Vol. 4, pp. 13-44.

[12] Fijalkow E. (1999): A numerical solution to the Vlasov equation. Computer Physics Communications, Vol. 116, No. 2, pp. 319-328.

[13] Filbet F., Sonnendrücker E. and Bertrand P. (2000): Conservative numerical schemes for the Vlasov equation. Journal of Computational Physics, Vol. 172, No. 1, pp. 166-187.

[14] 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.

[15] 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.

[16] Kulsrud R.M. (1998): Magnetic reconnection in a magnetohydrodynamic plasma. Physics of Plasmas, Vol. 5, No. 5, pp. 1599-1606.

[17] Lee W.W.L., Startsev E., Hong Q. and Davidson R.C. (2001): Electromagnetic (Darwin) model for threedimensional perturbative particle simulation of high intensity beams. Proceedings of the Particle Accelerator Conference, PACS'2001, Chicago, USA, IEEE Part, Vol. 3, p. 1906.

[18] Masmoudi N. and Mauser N.J. (2001): The selfconsistent Pauli equation. Monatshefte für Mathematik, Vol. 132, No. 1, pp. 19-24.

[19] 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.

[20] Ottaviani M. and Porcelli F. (1993): Nonlinear collisionless magnetic reconnection. Physical Review Letters, Vol. 71, No. 23, pp. 3802-3805.

[21] 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.

[22] Raviart P.-A. and Sonnendrücker E. (1996): A Hierarchy of Approximate Models for the Maxwell Equations. NumerischeMathematik, Vol. 73, No. 3, pp. 329-372.

[23] Schmitz H. and Grauer R. (2006): Darwin-Vlasov simulations of magnetised plasmas. Journal of Computational Physics, Vol. 214, No. 2, pp. 738-756.

[24] 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.

[25] Sonnendrücker 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.

[26] 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.