Asymptotic solution of neutron transport problems for small mean free paths

Larsen, Edward W.; Keller, Joseph B.

doi:10.1063/1.1666510

Cited by 348 publications

(240 citation statements)

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“…[51], [19], [11]) and radiative transfer [9], [10]. Its first application to semiconductors and the rigorous derivation of the Classical DriftDiffusion model is found in [63], [44].…”

Section: Introductionmentioning

confidence: 99%

Quantum Energy-Transport and Drift-Diffusion Models

2005

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We show that Quantum Energy-Transport and Quantum Drift-Diffusion models can be derived through diffusion limits of a collisional Wigner equation. The collision operator relaxes to an equilibrium defined through the entropy minimization principle. Both models are shown to be entropic and exhibit fluxes which are related with the state variables through spatially non-local relations. Thanks to an expansion of these models, 2 perturbations of the Classical Energy-Transport and Drift-Diffusion models are found. In the DriftDiffusion case, the quantum correction is the Bohm potential and the model is still entropic. In the Energy-Transport case however, the quantum correction is a rather complex expression and the model cannot be proven entropic.

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“…[51], [19], [11]) and radiative transfer [9], [10]. Its first application to semiconductors and the rigorous derivation of the Classical DriftDiffusion model is found in [63], [44].…”

Section: Introductionmentioning

confidence: 99%

Quantum Energy-Transport and Drift-Diffusion Models

2005

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“…These kinds of models are called in the literature Fokker-Planck models or "SHE model" (for spherical harmonics expansion) because of its earlier derivation in [27]. The diffusive limits have been extensively studied; see, for example, [3], [4], [7], [5], or [21] in different contexts: radiative transfer, semiconductors, neutron transport.…”

Section: Introductionmentioning

confidence: 99%

A Boltzmann Model for Trapped Particles in a Surface Potential

Degond¹,

Parzani²,

Vignal³

2006

Multiscale Model. Simul.

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Abstract. In this article, we propose a model describing the transport of trapped particles in a surface potential. The potential confines particles close to the surface, increasing the number of surface collisions. First, we consider the case of noncharged particles. From a kinetic description, we rigorously derive a two dimensional Boltzmann equation. In the case of charged particles we introduce the coupling with the Poisson equation. We perform a formal asymptotic analysis which leads to a two dimensional Boltzmann equation coupled with a three dimensional Poisson equation. We illustrate the charged particle model with some numerical simulations of a gas discharge on a satellite solar array. We use a particle in cell (P.I.C.) scheme that is a particle discretization for the Boltzmann equation and a Fourier approximation for the Poisson equation.Key words. Boltzmann equation, Poisson equation, surface collisions, asymptotic model, simulation of gas discharge, particle in cell scheme, particle method AMS subject classifications. 82C70, 82C40, 82C21, 82B40, 82B21, 35J05, 76M28 DOI. 10.1137/050642897 1. Introduction. In this article, we are interested in particles subject to a given external potential in a half space. We propose a mathematical model to describe their transport, and we perform numerical simulations. The starting point of our model is a kinetic model constituted of the Vlasov equation set in the half space. It is completed by some boundary conditions for the description of the surface collisions.In order to reduce the cost of the numerical simulations, it is classical to derive some asymptotic models with a smaller number of variables than the kinetic description. It is possible when the physical conditions allow one to do it, for example when particles are subject to many collisions. The resulting model depends on the considered physical process (see [6] and the references given there).Here we want to establish an asymptotic model when the applied potential confines the particles close to the surface, increasing the number of collisions with it. Furthermore, we assume that the dominant surface collision process is specular; the other collisions are supposed to be only a perturbation. Although it is not true in practical situations we assume that the error is of the same order as the error we make with the asymptotic limit.This work is a continuation of [15], in which the considered problem is the same as here but the dominant surface collisions are supposed diffusive. In [15], the resulting model is a diffusive model in two space dimensions; its variables are the time, the position on the surface, and the total energy of the particles. These kinds of models are called in the literature Fokker-Planck models or "SHE model" (for spherical harmonics expansion) because of its earlier derivation in [27]

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“…This regime may be obtained by the introduction of the following scaling in the transport equation [16,17,23]:…”

Section: The P N Equationsmentioning

confidence: 99%

Resolution of the time dependentP_nequations by a Godunov type scheme having the diffusion limit

Cargo

Samba

2010

ESAIM: M2AN

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Abstract.We consider the Pn model to approximate the time dependent transport equation in one dimension of space. In a diffusive regime, the solution of this system is solution of a diffusion equation. We are looking for a numerical scheme having the diffusion limit property: in a diffusive regime, it has to give the solution of the limiting diffusion equation on a mesh at the diffusion scale. The numerical scheme proposed is an extension of the Godunov type scheme proposed by Gosse to solve the P1 model without absorption term. It requires the computation of the solution of the steady state Pn equations. This is made by one Monte-Carlo simulation performed outside the time loop. Using formal expansions with respect to a small parameter representing the inverse of the number of mean free path in each cell, the resulting scheme is proved to have the diffusion limit. In order to avoid the CFL constraint on the time step, we give an implicit version of the scheme which preserves the positivity of the zeroth moment.Mathematics Subject Classification. 82C70, 35B40, 74S10.

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Asymptotic solution of neutron transport problems for small mean free paths

Cited by 348 publications

References 1 publication

Quantum Energy-Transport and Drift-Diffusion Models

Quantum Energy-Transport and Drift-Diffusion Models

A Boltzmann Model for Trapped Particles in a Surface Potential

Resolution of the time dependentP_nequations by a Godunov type scheme having the diffusion limit

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Asymptotic solution of neutron transport problems for small mean free paths

Cited by 348 publications

References 1 publication

Quantum Energy-Transport and Drift-Diffusion Models

Quantum Energy-Transport and Drift-Diffusion Models

A Boltzmann Model for Trapped Particles in a Surface Potential

Resolution of the time dependentPnequations by a Godunov type scheme having the diffusion limit

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Resolution of the time dependentP_nequations by a Godunov type scheme having the diffusion limit