The aim of this paper is to investigate a deterministic particle method for a model containing a Fokker–Planck collision operator in velocity and strong oscillations (characterized by a small parameter ) induced by a space and velocity transport operator. First, we investigate the properties (collisional invariants and equilibrium) of the asymptotic model obtained when . Second a numerical method is developed to approximate the solution of the multiscale Fokker–Planck model. To do so, a deterministic particle method (recently introduced for the Landau equation in Carrillo et al. 2020) is proposed for Fokker–Planck type operators. This particle method consists in reformulating the collision operator in an advective form and in regularizing the advection field in such a way that it conserves the geometric bracket structure. In the Fokker–Planck homogeneous case, the properties of the resulting method are analysed. In the non homogeneous case, the particle method is coupled with a uniformly accurate time discretization in that enables to capture numerically the solution of the asymptotic model. Numerous numerical results are displayed, illustrating the behavior of the method.
Keywords: Vlasov equation, Fokker–Planck collision operator, highly oscillatory systems, multiscale numerical schemes, Particle method.
Anaïs Crestetto 1; Nicolas Crouseilles 2; Damien Prel 1
CC-BY 4.0
@article{SMAI-JCM_2024__10__141_0,
author = {Ana{\"\i}s Crestetto and Nicolas Crouseilles and Damien Prel},
title = {Deterministic particle method for {Fokker{\textendash}Planck} equation with strong oscillations},
journal = {The SMAI Journal of computational mathematics},
pages = {141--173},
publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
volume = {10},
year = {2024},
doi = {10.5802/smai-jcm.109},
language = {en},
url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.109/}
}
TY - JOUR AU - Anaïs Crestetto AU - Nicolas Crouseilles AU - Damien Prel TI - Deterministic particle method for Fokker–Planck equation with strong oscillations JO - The SMAI Journal of computational mathematics PY - 2024 SP - 141 EP - 173 VL - 10 PB - Société de Mathématiques Appliquées et Industrielles UR - https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.109/ DO - 10.5802/smai-jcm.109 LA - en ID - SMAI-JCM_2024__10__141_0 ER -
%0 Journal Article %A Anaïs Crestetto %A Nicolas Crouseilles %A Damien Prel %T Deterministic particle method for Fokker–Planck equation with strong oscillations %J The SMAI Journal of computational mathematics %D 2024 %P 141-173 %V 10 %I Société de Mathématiques Appliquées et Industrielles %U https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.109/ %R 10.5802/smai-jcm.109 %G en %F SMAI-JCM_2024__10__141_0
Anaïs Crestetto; Nicolas Crouseilles; Damien Prel. Deterministic particle method for Fokker–Planck equation with strong oscillations. The SMAI Journal of computational mathematics, Volume 10 (2024), pp. 141-173. doi : 10.5802/smai-jcm.109. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.109/
[1] Uncertainty Quantification for the Homogeneous Landau–Fokker–Planck Equation via Deterministic Particle Galerkin methods (2023) | arXiv
[2] Diffusion and guiding center approximation for particle transport in strong magnetic fields, Kinet. Relat. Models, Volume 1 (2008) no. 3, pp. 331-354 | DOI | Zbl
[3] Plasma Physics via Computer Simulation, Series in Plasma Physics and Fluid Dynamics, Taylor & Francis, 2004 | DOI
[4] Asymptotic analysis of parabolic equations with stiff transport terms by a multi-scale approach, Discrete Contin. Dyn. Syst., Ser. A, Volume 37 (2017) no. 9, pp. 4637-4676 | DOI | Zbl
[5] Finite Larmor radius approximation for collisional magnetic confinement. Part I: The linear Boltzmann equation, Quart. J., Volume LXXII (2014) no. 2, pp. 323-345 | DOI | Zbl
[6] Finite Larmor radius approximation for collisional magnetic confinement. Part II: The Fokker–Planck–Landau equation, Quart. J., Volume LXXII (2014) no. 2, pp. 513-548 | Zbl
[7] Finite Larmor radius regime: Collisional setting and fluid models, Commun. Contemp. Math., Volume 22 (2020) no. 06, 1950047 | DOI | Zbl
[8] A blob method for diffusion, Calc. Var. Partial Differ. Equ., Volume 58 (2019) no. 2, 53 | Zbl
[9] A particle method for the homogeneous Landau equation, J. Comput. Phys.: X, Volume 7 (2020), 100066, 24 pages | DOI | Zbl
[10] Uniformly accurate numerical schemes for highly oscillatory Klein–Gordon and nonlinear Schrödinger equations, Numer. Math., Volume 129 (2015) no. 2, pp. 211-250 | DOI | Zbl
[11] Uniformly accurate methods for Vlasov equations with non-homogeneous strong magnetic field, Math. Comput., Volume 88 (2019) no. 320, pp. 2697-2736 | DOI | Zbl
[12] Uniformly Accurate Methods for Three Dimensional Vlasov Equations under Strong Magnetic Field with Varying Direction, SIAM J. Sci. Comput., Volume 42 (2020) no. 2, p. B520-B547 | DOI | Zbl
[13] Numerical methods for the two-dimensional Vlasov–Poisson equation in the finite Larmor radius approximation regime, J. Comput. Phys., Volume 375 (2018), pp. 619-640 | DOI | Zbl
[14] Derivative-free high-order uniformly accurate schemes for highly oscillatory systems, IMA J. Numer. Anal., Volume 42 (2021) no. 2, pp. 1623-1644 | DOI | Zbl
[15] High order averaging, formal series and numerical integrations I: B-series, FOCM, Volume 10 (2010) no. 6
[16] A formal series approach to averaging: exponentially small error estimates, Discrete Contin. Dyn. Syst., Volume 32 (2012) no. 9 | Zbl
[17] Multiscale Numerical Schemes for the Collisional Vlasov Equation in the Finite Larmor Radius Approximation Regime, Multiscale Model. Simul., Volume 21 (2023) no. 3, pp. 1210-1236 | DOI | Zbl
[18] Asymptotic Preserving schemes for highly oscillatory Vlasov–Poisson equations, J. Comput. Phys., Volume 248 (2013), pp. 287-308 | DOI | Zbl
[19] Uniformly accurate forward semi-Lagrangian methods for highly oscillatory Vlasov–Poisson equations, Multiscale Model. Simul., Volume 15 (2017) no. 2, pp. 723-744 | DOI | Zbl
[20] Uniformly accurate Particle-In-Cell method for the long time two-dimensional Vlasov–Poisson equation with strong magnetic field, J. Comput. Phys., Volume 346 (2017), pp. 172-190 | DOI | Zbl
[21] The weighted particle method for convection-diffusion equations. I. The case of an isotropic viscosity, Math. Comput., Volume 93 (1989) no. 345, pp. 485-507 | Zbl
[22] A Deterministic Approximation of Diffusion Equations Using Particles, SIAM J. Sci. Stat. Comput., Volume 11 (1990) no. 2, pp. 293-310 | DOI | Zbl
[23] The paraxial approximation of the Vlasov–Maxwell equations, Math. Models Methods Appl. Sci., Volume 3 (1993), pp. 513-562 | DOI
[24] Nonlinear gyrokinetic equations, Phys. Fluids, Volume 26 (1983) no. 12, pp. 3524-3535 | DOI | Zbl
[25] Long time simulation of a beam in a periodic focusing channel via a two-scale pic-method, Math. Models Methods Appl. Sci., Volume 19 (2009) no. 02, pp. 175-197 | DOI | Zbl
[26] Long time behavior of the two-dimensional Vlasov equation with a strong external magnetic field, Math. Models Methods Appl. Sci., Volume 10 (2000) no. 4, pp. 539-553 | DOI
[27] Large-Time Behavior of Solutions to Vlasov–Poisson–Fokker–Planck Equations: From Evanescent Collisions to Diffusive Limit, J. Stat. Phys., Volume 170 (2018) no. 5, pp. 895-931 | DOI | Zbl
[28] Anisotropic Boltzmann–Gibbs dynamics of strongly magnetized Vlasov–Fokker–Planck equations, Kinet. Relat. Models, Volume 12 (2019) no. 3, pp. 593-636 | DOI | Zbl
[29] Structure-preserving marker-particle discretizations of Coulomb collisions for particle-in-cell codes, Plasma Phys. Control. Fusion, Volume 63 (2021) no. 4, 044003 | DOI
[30] Exponential integrators, Acta Numer., Volume 19 (2010), pp. 209-286 | DOI | Zbl
[31] A structure-preserving particle discretisation for the Lenard–Bernstein operator (2023) | arXiv
[32] Asymptotic state of the finite-Larmor-radius guiding-centre plasma, J. Plasma Phys., Volume 41 (1989) no. 1, pp. 157-170 | DOI
[33] Metriplectic Integrators for the Landau Collision Operator, Phys. Plasmas, Volume 24 (2017) no. 10, 102311
[34] Variational integrators for stochastic dissipative Hamiltonian systems (2020) | arXiv
[35] Gyrokinetic particle simulation model, J. Comput. Phys., Volume 72 (1987) no. 1, pp. 243-269 | DOI | Zbl
[36] High-order stochastic integration schemes for the Rosenbluth–Trubnikov collision operator in particle simulations (2024) | arXiv
[37] Two-scale semi-Lagrangian simulation of a charged particle beam in a periodic focusing channel, Kinet. Relat. Models, Volume 2 (2009) no. 2, pp. 251-274 | DOI | Zbl
[38] Numerical Methods for the Vlasov–Maxwell equations, Lecture notes, 2015
[39] Stochastic variational principles for the collisional Vlasov–Maxwell and Vlasov–Poisson equations, Proc. R. Soc. A: Math. Phys. Eng. Sci., Volume 477 (2021) no. 2252 | DOI
[40] Multispecies structure-preserving particle discretization of the Landau collision operator, Phys. Plasmas, Volume 29 (2022) no. 12, 123906 | DOI
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