Numerical approximation of self-consistent Vlasov models for low-frequency electromagnetic phenomena

• Autorzy: Nicolas Besse , Norbert J. Mauser , Eric Sonnendrücker
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

low-frequency electromagnetic phenomena; semi-Lagrangian methods; Vlasov-Poisswell model; Vlasov-Darwin model

• Wydawca: University of Zielona Gora Press
• Czasopismo: International Journal of Applied Mathematics and Computer Science
• Rocznik: 2007
• Tom: 17
• Numer: 3
• Strony: 361-374

Daty

wydano

2007

Twórcy

autor

Nicolas Besse

• Laboratoire de Physique des Milieux Ionisés et Applications & Institut de Mathématiques Elie Cartan, Université Henri Poincaré Nancy-I, BP 239 54506 Vandoeuvre-lès-Nancy Cedex, France

autor

Norbert J. Mauser

• Wolfgang Pauli Institute c/o Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, A-1090 Wien, Austria

autor

Eric Sonnendrücker

• Institut de Recherche Mathématique Avancée, UMR 7501 CNRS/ULP, rue René Descartes, 67084 Strasbourg Cedex, France

Bibliografia

• 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.
• 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.
• 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.
• 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.
• 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).
• 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.
• 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.
• 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.
• 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.
• Coppi B., Laval G. and Pellat R. (1996): Dynamics of the geomagnetic tail. Physical Review Letters, Vol.16, No.26, pp.1207-1210.
• 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.
• Fijalkow E. (1999): A numerical solution to the Vlasov equation. Computer Physics Communications, Vol.116, No.2, pp.319-328.
• 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.
• 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.
• 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.
• Kulsrud R. M. (1998): Magnetic reconnection in a magnetohydrodynamic plasma. Physics of Plasmas, Vol.5, No.5, pp.1599-1606.
• 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.
• Masmoudi N. and Mauser N.J. (2001): The selfconsistent Pauli equation. Monatshefte fur Mathematik, Vol.132, No.1, pp.19-24.
• 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.
• Ottaviani M. and Porcelli F. (1993): Nonlinear collisionless magnetic reconnection. Physical Review Letters, Vol.71, No.23, pp.3802-3805.
• 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.
• 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.
• Schmz H. and Grauer R. (2006): newblock Darwin-Vlasov simulations of magnetised plasmas. Journal of Computational Physics, Vol.214, No.2, pp.738-756.
• 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.
• 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.
• 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