This work is devoted to the numerical simulation of the Vlasov equation using a phase space grid. In contrast to Particle-In-Cell (PIC) methods, which are known to be noisy, we propose a semi-Lagrangian-type method to discretize the Vlasov equation in the two-dimensional phase space. As this kind of method requires a huge computational effort, one has to carry out the simulations on parallel machines. For this purpose, we present a method using patches decomposing the phase domain, each patch being devoted to a processor. Some Hermite boundary conditions allow for the reconstruction of a good approximation of the global solution. Several numerical results demonstrate the accuracy and the good scalability of the method with up to 64 processors. This work is a part of the CALVI project.
@article{bwmeta1.element.bwnjournal-article-amcv17i3p335bwm,
author = {Crouseilles, Nicolas and Latu, Guillaume and Sonnendr\"ucker, Eric},
title = {Hermite spline interpolation on patches for parallelly solving the Vlasov-Poisson equation},
journal = {International Journal of Applied Mathematics and Computer Science},
volume = {17},
year = {2007},
pages = {335-349},
zbl = {1159.65016},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-amcv17i3p335bwm}
}
Crouseilles, Nicolas; Latu, Guillaume; Sonnendrücker, Eric. Hermite spline interpolation on patches for parallelly solving the Vlasov-Poisson equation. International Journal of Applied Mathematics and Computer Science, Tome 17 (2007) pp. 335-349. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-amcv17i3p335bwm/
[000] Bermejo R. (1991): Analysis of an algorithm for the Galerkin-characteristic method. Numerische Mathematik, Vol. 60, pp. 163-194. | Zbl 0723.65073
[001] 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, pp. 341-376. | Zbl 1030.82011
[002] Birdsall C. K. and Langdon A. B.: Plasma Physics via Computer Simulation. Bristol: Institute of Physics Publishing.
[003] Bouchut F., Golse F. and Pulvirenti M. (2000): Kinetic Equations and Asymptotic Theory. Paris: Gauthier-Villars. | Zbl 0979.82048
[004] DeBoor C. (1978): A Practical Guide to Splines. New-York: Springer.
[005] Campos-Pinto M. and Merhenberger M. (2004): Adaptive Numerical Resolution of the Vlasov Equation.
[006] Cheng C. Z. and Knorr G. (1976): The integration of the Vlasov equation in configuration space. Journal of Computational Physics, Vol. 22, p. 330.
[007] Coulaud O., Sonnendrucker E., Dillon E., Bertrand P. and Ghizzo A. (1999): Parallelization of semi-Lagrangian Vlasov codes. Journal of Plasma Physics, Vol. 61, pp. 435-448.
[008] Feix M. R., Bertrand P. and Ghizzo A. (1994): Title? In: Kinetic Theory and Computing, (B. Perthame, Ed.).
[009] Filbet F., Sonnendrucker E. and Bertrand P. (2001): Conservative numerical schemes for the Vlasov equation. Journal of Computational Physics, Vol. 172, pp. 166-187. | Zbl 0998.65138
[010] Filbet F. and Sonnendrucker E. (2003): Comparison of Eulerian Vlasov solvers. Computer Physics Communications, Vol. 151, pp. 247-266. | Zbl 1196.82108
[011] Filbet F. and Violard E. (2002): Parallelization of a Vlasov Solver by Communication Overlapping. Proceedings PDPTA.
[012] Glassey R. T. (1996): The Cauchy Problem in Kinetic Theory. Philadelphia, PA: SIAM. | Zbl 0858.76001
[013] Ghizzo A., Bertrand P., Begue M. L., Johnston T. W. and Shoucri M. (1996): A Hilbert-Vlasov code for the study of high-frequency plasma beatwave accelerator. IEEE Transactions on Plasma Science, Vol. 24.
[014] Ghizzo A., Bertrand P., Shoucri M., Johnston T. W., Filjakow E. and Feix M. R. (1990): A Vlasov code for the numerical simulation of stimulated Raman scattering. Journal of Computational Physis, Vol. 90, pp. 431-457. | Zbl 0702.76080
[015] Grandgirard V., Brunetti M., Bertrand P., Besse N., Garbet N., Ghendrih P., Manfredi G., Sarrazin Y., Sauter O., Sonnendrucker E., Vaclavik J. and Villard L. (2006): A drift-kinetic semi-Lagrangian 4D code for ion turbulence simulation. Journal of Computational Physics, Vol. 217, pp. 395-423. | Zbl 1160.76385
[016] Gutnic M., Haefele M., Paun I. and Sonnendrucker E. (2004): Vlasov simulation on an adaptive phase space grid. Computer Physical Communications, Vol. 164, pp. 214-219. | Zbl 1196.76098
[017] Hammerlin G. and Hoffmann K. H. (1991): Numerical Mathematics, New-York: Springer. | Zbl 0713.65001
[018] Kim C. C. and Parker S. E. (2000): Massively parallel three-dimensional toroidal gyrokinetic flux-tube turbulence simulation. Journal of Computational Physics, Vol. 161, pp. 589-604. | Zbl 0980.76058
[019] McKinstrie C. J., Giacone R. E. and Startsev E. A. (1999): Accurate formulas for the Landau damping rates of electrostatic waves. Physics of Plasmas, Vol. 6, pp. 463-466.
[020] Manfredi G. (1997): Long time behaviour of strong linear Landau damping. Physical Review Letters, Vol. 79.
[021] Shoucri M. and Knorr G. (1974): Numerical integration of the Vlasov equation. Journal of Computational Physics, Vol. 14, pp. 84-92. | Zbl 0275.65029
[022] Sonnendrucker E., Filbet F., Friedman A., Oudet E. and Vay J. L. (2004): Vlasov simulation of beams on a moving phase space grid. Computer Physics Communications, Vol. 164, pp. 390-395.
[023] Sonnendrucker E., Roche J., Bertrand P. and Ghizzo A. (1999): The semi-Lagrangian method for the numerical resolution of the Vlasov equations. Journal of Computational Physics, Vol. 149, pp. 201-220. | Zbl 0934.76073
[024] Staniforth A. and Cote J. (1991): Semi-Lagrangian integration schemes for atmospheric models - A review. Monthly Weather Review, Vol. 119, pp. 2206-2223