Homogenization of a spectral equation with drift in linear transport

Department of Mathematics, University of Chicago, Chicago, IL 60637, U.S.A.; gbal@math.uchicago.edu.

Received: 4 December 2000
Revised: 22 May 2001

Abstract

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 term. The first one solves a local transport spectral equation posed in the periodicity cell and the second one a homogeneous spectral diffusion equation posed in the entire domain. This paper addresses the case where these symmetry conditions are not fulfilled. We show that the factorization remains valid with the diffusion equation replaced by a convection-diffusion equation with large drift. The asymptotic limit of the leading eigenvalue is also modified. The spectral equation treated in this paper can model the stability of nuclear reactor cores and describe the distribution of neutrons at equilibrium. The same techniques can also be applied to the time-dependent linear transport equation with drift, which appears in radiative transfer theory and which models the propagation of acoustic, electromagnetic, and elastic waves in heterogeneous media.