Uniformly accurate methods for Vlasov equations with non-homogeneous strong magnetic field

Chartier, Philippe; Crouseilles, Nicolas; Lemou, Mohammed; Méhats, Florian; Zhao, Xiaofei

doi:10.1090/mcom/3436

Uniformly accurate methods for Vlasov equations with non-homogeneous strong magnetic field
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by Philippe Chartier, Nicolas Crouseilles, Mohammed Lemou, Florian Méhats and Xiaofei Zhao HTML | PDF

Math. Comp. 88 (2019), 2697-2736 Request permission

Abstract:

In this paper, we consider uniformly accurate methods for solving the highly-oscillatory Vlasov equations with non-homogeneous magnetic field. The specific difficulty (and the resulting novelty of our approach) stems from the presence of a non-periodic oscillation, which necessitates a careful ad-hoc reformulation of the equations by using a confinement property of the solution. A two-scale numerical scheme is proposed where the error estimates show that our scheme not only remains insensitive to the stiffness of the problem in terms of both accuracy and computational cost, but also preserves the confinement property at the discrete level. Our results are illustrated numerically on several examples.

References

C. K. Birdsall and A. B. Langdon, Plasma Physics via Computer Simulation, Adam Hilger, 1991.
Mihai Bostan, The Vlasov-Maxwell system with strong initial magnetic field: guiding-center approximation, Multiscale Model. Simul. 6 (2007), no. 3, 1026–1058. MR 2368978, DOI 10.1137/070689383
M. Bostan, Asymptotic behavior for the Vlasov-Poisson equations with strong external magnetic field, Straight magnetic field lines. preprint (2018), hal-01683869.
Mihaï Bostan and Aurélie Finot, The effective Vlasov-Poisson system for the finite Larmor radius regime, Multiscale Model. Simul. 14 (2016), no. 4, 1238–1275. MR 3553917, DOI 10.1137/16M1060479
M. P. Calvo, Ph. Chartier, A. Murua, and J. M. Sanz-Serna, Numerical stroboscopic averaging for ODEs and DAEs, Appl. Numer. Math. 61 (2011), no. 10, 1077–1095. MR 2820967, DOI 10.1016/j.apnum.2011.06.007
Mari Paz Calvo, Philippe Chartier, Ander Murua, and Jesús María Sanz-Serna, A stroboscopic numerical method for highly oscillatory problems, Numerical analysis of multiscale computations, Lect. Notes Comput. Sci. Eng., vol. 82, Springer, Heidelberg, 2012, pp. 71–85. MR 3075792, DOI 10.1007/978-3-642-21943-6_{4}
Ph. Chartier, N. Crouseilles, and M. Lemou, An averaging technique for transport equations, arXiv:1609.09819v1, submitted, 2016.
Philippe Chartier, Nicolas Crouseilles, Mohammed Lemou, and Florian Méhats, Uniformly accurate numerical schemes for highly oscillatory Klein-Gordon and nonlinear Schrödinger equations, Numer. Math. 129 (2015), no. 2, 211–250. MR 3300419, DOI 10.1007/s00211-014-0638-9
Philippe Chartier, Mohammed Lemou, and Florian Méhats, Highly-oscillatory evolution equations with multiple frequencies: averaging and numerics, Numer. Math. 136 (2017), no. 4, 907–939. MR 3671592, DOI 10.1007/s00211-016-0864-4
Ph. Chartier, M. Lemou, F. Méhats, and G. Vilmart, A new class of uniformly accurate methods for highly oscillatory evolution equations, hal-01666472, 2017.
Philippe Chartier, Joseba Makazaga, Ander Murua, and Gilles Vilmart, Multi-revolution composition methods for highly oscillatory differential equations, Numer. Math. 128 (2014), no. 1, 167–192. MR 3248052, DOI 10.1007/s00211-013-0602-0
Philippe Chartier, Norbert J. Mauser, Florian Méhats, and Yong Zhang, Solving highly-oscillatory NLS with SAM: numerical efficiency and long-time behavior, Discrete Contin. Dyn. Syst. Ser. S 9 (2016), no. 5, 1327–1349. MR 3591947, DOI 10.3934/dcdss.2016053
Nicolas Crouseilles, Mohammed Lemou, and Florian Méhats, Asymptotic preserving schemes for highly oscillatory Vlasov-Poisson equations, J. Comput. Phys. 248 (2013), 287–308. MR 3066153, DOI 10.1016/j.jcp.2013.04.022
Nicolas Crouseilles, Mohammed Lemou, Florian Méhats, and Xiaofei Zhao, Uniformly accurate forward semi-Lagrangian methods for highly oscillatory Vlasov-Poisson equations, Multiscale Model. Simul. 15 (2017), no. 2, 723–744. MR 3635455, DOI 10.1137/16M1059497
Nicolas Crouseilles, Mohammed Lemou, Florian Méhats, and Xiaofei Zhao, Uniformly accurate particle-in-cell method for the long time solution of the two-dimensional Vlasov-Poisson equation with uniform strong magnetic field, J. Comput. Phys. 346 (2017), 172–190. MR 3670633, DOI 10.1016/j.jcp.2017.06.011
Pierre Degond and Francis Filbet, On the asymptotic limit of the three dimensional Vlasov-Poisson system for large magnetic field: formal derivation, J. Stat. Phys. 165 (2016), no. 4, 765–784. MR 3568167, DOI 10.1007/s10955-016-1645-2
Francis Filbet, Tao Xiong, and Eric Sonnendrücker, On the Vlasov-Maxwell system with a strong magnetic field, SIAM J. Appl. Math. 78 (2018), no. 2, 1030–1055. MR 3782409, DOI 10.1137/17M1112030
Francis Filbet and Luis Miguel Rodrigues, Asymptotically stable particle-in-cell methods for the Vlasov-Poisson system with a strong external magnetic field, SIAM J. Numer. Anal. 54 (2016), no. 2, 1120–1146. MR 3485969, DOI 10.1137/15M104952X
Francis Filbet and Luis Miguel Rodrigues, Asymptotically preserving particle-in-cell methods for inhomogeneous strongly magnetized plasmas, SIAM J. Numer. Anal. 55 (2017), no. 5, 2416–2443. MR 3709891, DOI 10.1137/17M1113229
Emmanuel Frénod, Francesco Salvarani, and Eric 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), no. 2, 175–197. MR 2498432, DOI 10.1142/S0218202509003395
E. Frénod and E. Sonnendrücker, Long time behavior of the two-dimensional Vlasov equation with a strong external magnetic field, Math. Models Methods Appl. Sci. 10 (2000), no. 4, 539–553. MR 1758722, DOI 10.1142/S021820250000029X
Emmanuel Frénod, Sever A. Hirstoaga, Mathieu Lutz, and Eric Sonnendrücker, Long time behaviour of an exponential integrator for a Vlasov-Poisson system with strong magnetic field, Commun. Comput. Phys. 18 (2015), no. 2, 263–296. MR 3377254, DOI 10.4208/cicp.070214.160115a
G. B. Jacobs and J. S. Hesthaven, Implicit-explicit time integration of a high-order particle-in-cell method with hyperbolic divergence cleaning, Comput. Phys. Comm. 180 (2009), no. 10, 1760–1767. MR 2678449, DOI 10.1016/j.cpc.2009.05.020
Shi Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations, SIAM J. Sci. Comput. 21 (1999), no. 2, 441–454. MR 1718639, DOI 10.1137/S1064827598334599
W. W. Lee, Gyrokinetic approach in particle simulation, Phys. Fluids 26 (1983).
L. Saint-Raymond, The gyrokinetic approximation for the Vlasov-Poisson system, Math. Models Methods Appl. Sci. 10 (2000), no. 9, 1305–1332. MR 1796566, DOI 10.1142/S0218202500000641
E. Sonnendrücker, Numerical methods for Vlasov equations, Lecture notes, 2016.