This paper deals with the homogenization of a spectral equation posed in a periodic domain in linear transport theory. The particle density at equilibrium is given by the unique normalized positive eigenvector of this spectral equation. The corresponding eigenvalue indicates the amount of particle creation necessary to reach this equilibrium. When the physical parameters satisfy some symmetry conditions, it is known that the eigenvectors of this equation can be approximated by the product of two...
This paper deals with the homogenization of a spectral equation
posed in a periodic domain in linear transport theory. The particle
density at equilibrium is given by the unique normalized positive
eigenvector of this spectral equation. The corresponding eigenvalue
indicates the amount of particle creation necessary to reach this
equilibrium. When the physical parameters satisfy some symmetry
conditions, it is known that the eigenvectors of this equation can
be approximated by the product...
Numerical simulation of high frequency waves in highly heterogeneous media is a challenging problem. Resolving the fine structure of the wave field typically requires extremely small time steps and spatial meshes. We show that capturing macroscopic quantities of the wave field, such as the wave energy density, is achievable with much coarser discretizations. We obtain such a result using a time splitting algorithm that solves separately and successively propagation and scattering in the simplified...
This paper is concerned with the coupling of two models for the propagation of particles in scattering media. The first model is a linear transport equation of Boltzmann type posed in the phase space (position and velocity). It accurately describes the physics but is very expensive to solve. The second model is a diffusion equation posed in the physical space. It is only valid in areas of high scattering, weak absorption, and smooth physical coefficients, but its numerical solution is much cheaper...
This paper analyzes the random fluctuations obtained by a heterogeneous multi-scale first-order finite element method applied to solve elliptic equations with a random potential. Several multi-scale numerical algorithms have been shown to correctly capture the homogenized limit of solutions of elliptic equations with coefficients modeled as stationary and ergodic random fields. Because theoretical results are available in the continuum setting for such equations, we consider here the case of a second-order...
Numerical simulation of high frequency waves in highly heterogeneous
media is a challenging problem. Resolving the fine structure of the
wave field typically requires extremely small time steps and spatial
meshes. We show that capturing macroscopic quantities of the wave
field, such as the wave energy density, is achievable with much
coarser discretizations. We obtain such a result using a time
splitting algorithm that solves separately and successively
propagation and scattering in the...
We address the homogenization of an eigenvalue problem for the neutron transport
equation
in a periodic heterogeneous domain, modeling the criticality study of nuclear
reactor cores.
We prove that the neutron flux, corresponding to the first and unique positive
eigenvector,
can be factorized in the product of two terms, up to a remainder which goes
strongly to zero
with the period. One term is the first eigenvector of the transport equation in the
periodicity cell. The other term is the...
This paper is concerned with the coupling of two models for the
propagation of particles in scattering media. The first model is a
linear transport equation of Boltzmann type posed in the phase space
(position and velocity). It accurately describes the physics but is
very expensive to solve. The second model is a diffusion equation
posed in the physical space. It is only valid in areas of high
scattering, weak absorption, and smooth physical coefficients, but
its numerical solution is...
In this paper, we study the homogenization and localization of a spectral transport equation posed in a locally periodic heterogeneous domain. This equation models the equilibrium of particles interacting with an underlying medium in the presence of a creation mechanism such as, for instance, neutrons in nuclear reactors. The physical coefficients of the domain are -periodic functions modulated by a macroscopic variable, where is a small parameter. The mean free path of the particles is also...
In this paper, we study the homogenization and localization of a
spectral transport equation posed in a locally periodic
heterogeneous domain. This equation models the equilibrium of
particles interacting with an underlying medium in the presence of a
creation mechanism such as, for instance, neutrons in nuclear
reactors. The physical coefficients of the domain are
-periodic functions modulated by a macroscopic variable, where
is a small parameter. The mean free path of the particles is
...
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