Convergence of a certain monotone iteration in the reflection matrix for a nonmultiplying half-space

Abstract

A certain monotonically increasing nonlinear iterative procedure, first employed systematically by Shimizu and Aoki, has proven quite effective for computing the reflection matrix for a spatially homogeneous half-space, where the direction and energy variables are treated by suitable discrete approximations. In this paper it is shown that this procedure converges, provided only that the underlying half-space is nonmultiplying. (“Nonmultiplying” means that the maximum expected number of particles emerging from a collision does not exceed unity, where the maximum is taken over all energies and directions of the incident particles.) Furthermore, in this case it is shown that a certain norm of the approximate reflection matrices produced by the iterative process is bounded above by the maximum secondary scattering ratio.