Fast Algorithms for Numerical, Conservative, and Entropy Approximations of the Fokker–Planck–Landau Equation

Buet, Christophe; Cordier, Stéphane; Degond, Pierre; Lemou, Mohammed

doi:10.1006/jcph.1997.5669

Cited by 72 publications

(89 citation statements)

References 12 publications

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“…We use the "double-peak" non-equilibrium initial condition (4.4) with the macroscopic variables (4.5), where 6]. We take N x = 100, while N v = 32, 64, 128 respectively.…”

Section: The Conservation Of Moments In Solving the Linear System (33)mentioning

confidence: 99%

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A class of asymptotic-preserving schemes for the Fokker–Planck–Landau equation

Jin

Yan

2011

Journal of Computational Physics

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We present a class of asymptotic-preserving (AP) schemes for the nonhomogeneous Fokker-Planck-Landau (nFPL) equation. Filbet and Jin [14] designed a class of AP schemes for the classical Boltzmann equation, by penalization with the BGK operator, so they become efficient in the fluid dynamic regime. We generalize their idea to the nFPL equation, with a different penalization operator, the Fokker-Planck operator that can be inverted by the conjugate-gradient method. We compare the effects of different penalization operators, and conclude that the FokkerPlanck (FP) operator is a good choice. Such schemes overcome the stiffness of the collision operator in the fluid regime, and can capture the fluid dynamic limit without numerically resolving the small Knudsen number. Numerical experiments demonstrate that the schemes possess the AP property for general initial data, with numerical accuracy uniformly in the Knudsen number.

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“…We use the "double-peak" non-equilibrium initial condition (4.4) with the macroscopic variables (4.5), where 6]. We take N x = 100, while N v = 32, 64, 128 respectively.…”

Section: The Conservation Of Moments In Solving the Linear System (33)mentioning

confidence: 99%

“…In [31], [2], [10], [4], [5], [6], conservative finite difference type discretizations were applied to the space homogeneous equation was derived. To reduce the computational cost caused by the high dimensional integral in the collision operator, spectral schemes were derived in [25], [26], [15].…”

Section: Introductionmentioning

confidence: 99%

A class of asymptotic-preserving schemes for the Fokker–Planck–Landau equation

Jin

Yan

2011

Journal of Computational Physics

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“…This model is a simplified one which allows us to make the derivation of the hybrid model easier. Admittedly, the Fokker-Planck-Landau (FPL) collision operator is more adapted to describe binary collisions between charged particles, for which the interaction potential is the long-range Coulomb interaction; but its direct numerical implementation is very expensive, in spite of several approaches for reducing its computing cost (see [8]). Consequently, we propose to adopt in this paper a BGK type model which expresses the relaxation towards a local equilibrium.…”

Section: Derivation Of the Hybrid Modelmentioning

confidence: 99%

A hybrid kinetic–fluid model for solving the Vlasov–BGK equation

Crouseilles¹,

Degond²,

Lemou³

2005

Journal of Computational Physics

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Our purpose is to derive a model for charged particles which combines a kinetic description of the fast particles with a fluid description of the slow ones. In a previous paper, a similar model was derived from a kinetic BGK equation that uses a constant relaxation time and does not include the effect of an electric field. In this paper, we consider a more general kinetic equation including an electric field and a varying relaxation time. Fast particles will be described through a collisional kinetic equation of Vlasov-BGK type while thermal particles will be modeled by a hydrodynamic model. Then, we construct a numerical scheme for this model and validate the approach by presenting various numerical tests.

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“…Finally, ν is a nonnegative constant related to the collision frequency, Q α,β (f α , f β ) describes α-β collisions and reads Many different deterministic numerical schemes have been considered to Fokker-Planck type equations [2,3,8,13,16]. Due to the computational complexity of the equation (essentially caused by the large number of variables and the three-fold collision integral), many papers have been devoted to treat simpler space homogeneous situations (the distribution function f α does not depend on x) in the isotropic case or for cylindrically symmetric problems.…”

Section: Introductionmentioning

confidence: 99%

“…Even if this scheme gives interesting properties, its direct implementation is very expensive in high dimensions. Thus, we adopt the multigrid method used in [3] to reduce the computational cost.…”

Section: Introductionmentioning

confidence: 99%

Conservative and entropic numerical schemes for the non homogeneous Fokker-Planck-Landau equation

Crouseilles

2005

ESAIM: Proc.

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Abstract. In this paper, we investigate the approximation of the solution to the Vlasov equation coupled with the Fokker-Planck-Landau collision operator using a phase space grid. On the one hand, the transport is treated by a flux conservative method and the distribution function is reconstructed to to preserve energy and to control numerical oscillations. On the other hand, the collision operators are approximated to preserve the main properties (conservations and decrease of the entropy). Several numerical results are presented in one dimension in space and three dimensions in velocity.

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Fast Algorithms for Numerical, Conservative, and Entropy Approximations of the Fokker–Planck–Landau Equation

Cited by 72 publications

References 12 publications

A class of asymptotic-preserving schemes for the Fokker–Planck–Landau equation

A class of asymptotic-preserving schemes for the Fokker–Planck–Landau equation

A hybrid kinetic–fluid model for solving the Vlasov–BGK equation

Conservative and entropic numerical schemes for the non homogeneous Fokker-Planck-Landau equation

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