Homogenization and Diffusion Asymptotics of the Linear Boltzmann Equation

Goudon, Thierry; Mellet, Antoine

doi:10.1051/cocv:2003018

Cited by 25 publications

(37 citation statements)

References 32 publications

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“…This notion has been introduced in [35] and further developed in [1] and is an alternative approach to the energy method of Tartar [45] (see also [3,4,23,24,36]). This situation holds in particular for superlattices which are obtained by growing periodically successive slices of two materials resulting in a periodic electrostatic potential [15,20,21,30,49]. We shall here consider that transport is classical and that thickness of the potential barriers is of the order of magnitude of the collision mean free path.…”

Section: Introductionmentioning

confidence: 99%

Diffusion Approximation and Homogenization of the Semiconductor Boltzmann Equation

Abdallah¹,

Tayeb²

2005

Multiscale Model. Simul.

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This paper deals with the diffusion approximation of the Boltzmann equation for semiconductors in the presence of spatially oscillating electrostatic potential. When the oscillation period is of the same order of magnitude as the mean free path, the asymptotics leads to the driftdiffusion equation with a homogenized electrostatic potential and a diffusion matrix involving the small-scale information. The convergence is proven rigorously for Boltzmann statistics, while it is incomplete for Fermi-Dirac statistics. Introduction.The drift-diffusion equation is a standard model of particle transport in many applications such as neutron transport [17,18], plasmas [14,22], semiconductors [25,26,33,34,41,42,40,44], gas discharges [39], etc. The driftdiffusion model, very well suited for numerical simulations, is designed to describe the macroscopic behavior of the device. It consists of a mass balance equation relying on the particle and current densities,

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Section: Introductionmentioning

confidence: 99%

Diffusion Approximation and Homogenization of the Semiconductor Boltzmann Equation

Abdallah¹,

Tayeb²

2005

Multiscale Model. Simul.

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“…A Hybrid-Hilbert expansion is used to analyze the diffusion limit of the BGK model (coupled to Poisson) with inflow boundary data [9]. The same expansion is also used to study the behavior of the Fermi-Dirac statistics without detailed balance principle assumption [17,23]. In these previous examples, a correction of the linear Hilbert expansion is introduced to approximate the singularity due to the Poisson coupling and to control the time derivative of electrostatic field [23].…”

Section: In the Linear Multi-dimensional Case Let F ε Be A Weak Solumentioning

confidence: 99%

“…Progressively, different fluid models have been obtained as a hydrodynamic limit of kinetic equations for prescribed macroscopic potential [25,26]. Nowadays, more attention is paid to the variations of the potential by coupling the distribution with the Poisson equation [4,5,11,15,16,18] and recently by adding a potential with microscopic variations in the same order of the mean free path of the diffusive operator, leading to phenomena of homogenization [10,17,19,22,30]. The concept of relative entropy dissipation is useful to approximate the solution of the kinetic model to its corresponding equilibrium state [6,7,12,21,24,28,29].…”

Section: Introductionmentioning

confidence: 99%

Homogenized Diffusion Limit of a Vlasov–Poisson–Fokker–Planck Model

Tayeb

2016

Ann. Henri Poincaré

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The approximation by diffusion and homogenization of the initial-boundary value problem of the Vlasov-Poisson-Fokker-Planck model is studied for a given velocity field with spatial macroscopic and microscopic variations. The L 1 -contraction property of the Fokker-Planck operator and a two-scale Hybrid-Hilbert expansion are used to prove the convergence towards a homogenized Drift-Diffusion equation and to exhibit a rate of convergence.

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“…where σ δ (x) = σ(x, x σ ) to explicit reflects the fast variable x σ dependence. On the theoretical level, to our best knowledge, except a few cases [22], the theory is largely in lack, except a few cases [22], and to a large extend, we do not yet know the resonance of the two parameters, and how they contribute in the asymptotic limits of the equation. And on the computation level, the only numerical study aware to the authors is presented in [25] where the limits are taken in order: δ 1.…”

Section: Introductionmentioning

confidence: 97%

Generalized Multiscale Finite Element Method for the Steady State Linear Boltzmann Equation

Chung¹,

Efendiev²,

Li³

et al. 2020

Multiscale Model. Simul.

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The Boltzmann equation, as a model equation in statistical mechanics, is used to describe the statistical behavior of a large number of particles driven by the same physics laws. Depending on the media and the particles to be modeled, the equation has slightly different forms. In this article, we investigate a model Boltzmann equation with highly oscillatory media in the small Knudsen number regime, and study the numerical behavior of the Generalized Multi-scale Finite Element Method (GMsFEM) in the fluid regime when high oscillation in the media presents. The Generalized Multi-scale Finite Element Method (GMsFEM) is a general approach [7] to numerically treat equations with multi-scale structures. The method is divided into the offline and online steps. In the offline step, basis functions are prepared from a snapshot space via a well-designed generalized eigenvalue problem (GEP), and these basis functions are then utilized to patch up for a solution through DG formulation in the online step to incorporate specific boundary and source information. We prove the wellposedness of the method on the Boltzmann equation, and show that the GEP formulation provides a set of optimal basis functions that achieve spectral convergence. Such convergence is independent of the oscillation in the media, or the smallness of the Knudsen number, making it one of the few methods that simultaneously achieve numerical homogenization and asymptotic preserving properties across all scales of oscillations and the Knudsen number.

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Homogenization and Diffusion Asymptotics of the Linear Boltzmann Equation

Cited by 25 publications

References 32 publications

Diffusion Approximation and Homogenization of the Semiconductor Boltzmann Equation

Diffusion Approximation and Homogenization of the Semiconductor Boltzmann Equation

Homogenized Diffusion Limit of a Vlasov–Poisson–Fokker–Planck Model

Generalized Multiscale Finite Element Method for the Steady State Linear Boltzmann Equation

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