Hypocoercivity and diffusion limit of a finite volume scheme for linear kinetic equations

Bessemoulin-Chatard, Marianne; Herda, Maxime; Rey, Thomas

doi:10.1090/mcom/3490

Cited by 21 publications

(7 citation statements)

References 57 publications

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“…This limit discretization is in the same fashion as the famous Chang-Cooper [12,9], Il'In [28] and Scharfetter-Gummel discretizations [36]. It is also close to the discretization adpoted in [6] and where the counterpart of this paper's results were proved for classical Fokker-Planck equations. 2) in the previous section.…”

Section: Remark 23 (Additional Properties)supporting

confidence: 63%

“…For the same model, hypocoercivity properties for other types of schemes were studied in [19,21]. Concerning the classical kinetic Fokker-Planck equation, there has been to our knowledge two main contributions [18,6] dealing respectively with H 1 and L 2 hypocoercivity. In the present contribution, the range of models is extended to the fractional-Fokker-Planck case.…”

Section: Introductionmentioning

confidence: 99%

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On a structure-preserving numerical method for fractional Fokker-Planck equations

Ayi¹,

Herda²,

Hivert³

et al. 2021

Preprint

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In this paper, we introduce and analyse numerical schemes for the homogeneous and the kinetic Lévy-Fokker-Planck equation. The discretizations are designed to preserve the main features of the continuous model such as conservation of mass, heavy-tailed equilibrium and (hypo)coercivity properties. We perform a thorough analysis of the numerical scheme and show exponential stability. Along the way, we introduce new tools of discrete functional analysis, such as discrete nonlocal Poincaré and interpolation inequalities adapted to fractional diffusion. Our theoretical findings are illustrated and complemented with numerical simulations.

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Section: Remark 23 (Additional Properties)supporting

confidence: 63%

Section: Introductionmentioning

confidence: 99%

On a structure-preserving numerical method for fractional Fokker-Planck equations

Ayi¹,

Herda²,

Hivert³

et al. 2021

Preprint

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“…The diffusion in θ terms in (1.1a)-(1.1c) describe the rotational diffusion of cells while the y−derivative term models the free transport with velocity V sin θ in y direction when cells are at FSP. Notice the degeneracy of the system (the missing diffusion in y) which is standard in kinetic theory [25,8,3,16]. The terms on the right hand side of (1.1b)-(1.1c) describe the transitions from FSP to BCP.…”

Section: The Fokker-planck Systemmentioning

confidence: 99%

Fokker–Plank System for Movement of Micro-organism Population in Confined Environment

Perthame

Tang

2021

J Stat Phys

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We consider self-propelled particles confined between two parallel plates, moving with a constant velocity while their moving direction changes by rotational diffusion. The probability distribution of such micro-organisms in confined environment is singular because particles accumulate at the boundaries. This leads us to distinguish between the probability distribution densities in the bulk and in the boundaries. They satisfy a degenerate Fokker-Planck system and we propose boundary conditions that take into account the switching between free-moving and boundary-contacting particles. Relative entropy property, a priori estimates and the convergence to an unique steady state are established. The steady states of both the PDE and individual based stochastic models are compared numerically.

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“…We emphasize that in Table 1 we used the exact Fourier expansion of the moment vector given by (3.5). One other possibility would be to use a Discrete Fourier Transform on the original moment vector, by sampling uniformly its value (and using the discrete version of Parseval identity, see identity (6.1) in [3]). This, however, will introduce an additional spectrally small error on the moment of the constrained function f c N which may reduce accuracy when using a small number of nodes.…”

Section: Numerical Examplesmentioning

confidence: 99%

Moment preserving Fourier-Galerkin spectral methods and application to the Boltzmann equation

Pareschi¹,

Rey²

2021

Preprint

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Spectral methods, thanks to the high accuracy and the possibility of using fast algorithms, represent an effective way to approximate collisional kinetic equations in kinetic theory. On the other hand, the loss of some local invariants can lead to the wrong long time behavior of the numerical solution. We introduce in this paper a novel Fourier-Galerkin spectral method that improves the classical spectral method by making it conservative on the moments of the approximated distribution, without sacrificing its spectral accuracy or the possibility of using fast algorithms. The method is derived directly using a constrained best approximation in the space of trigonometric polynomials and can be applied to a wide class of problems where preservation of moments is essential. We then apply the new spectral method to the evaluation of the Boltzmann collision term, and prove spectral consistency and stability of the resulting Fourier-Galerkin approximation scheme. Various numerical experiments illustrate the theoretical findings.

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Hypocoercivity and diffusion limit of a finite volume scheme for linear kinetic equations

Cited by 21 publications

References 57 publications

On a structure-preserving numerical method for fractional Fokker-Planck equations

On a structure-preserving numerical method for fractional Fokker-Planck equations

Fokker–Plank System for Movement of Micro-organism Population in Confined Environment

Moment preserving Fourier-Galerkin spectral methods and application to the Boltzmann equation

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