Asymptotic solution of neutron transport problems for small mean free paths

TLDR

A method is presented to solve initial and boundary value problems for energy‑dependent and one‑speed neutron transport equations. The method constructs an asymptotic expansion of the neutron density in a small mean‑free‑path parameter ε, decomposing it into interior, boundary‑layer, and initial‑layer components, with the interior part expanded in powers of ε and the layers decaying exponentially with distance or time. For near‑critical reactors the leading interior term satisfies a diffusion equation, the boundary layer satisfies a half‑space problem, and the method’s applicability is demonstrated in the one‑speed case, indicating it can handle more realistic and complex problems than existing methods.

Abstract

A method is presented for solving initial and boundary value problems for the energy dependent and one speed neutron transport equations. It consists in constructing an asymptotic expansion of the neutron density ψ(r, v, τ) with respect to a small parameter ε, which is the ratio of a typical mean free path of a neutron to a typical dimension of the domain under consideration. The density ψ is expressed as the sum of an interior part ψi, a boundary layer part ψb, and an initial layer part ψ0. Then ψi is sought as a power series in ε, while ψb decays exponentially with distance from a boundary or interface at a rate proportional to ε−1. Similarly ψ0 decays at a rate proportional to ε−1 with time after the initial time. For a near critical reactor, the leading term in ψi is determined by a diffusion equation. The leading term in ψb is determined by a half-space problem with a plane boundary. The initial and boundary conditions for the diffusion equation are obtained by requiring ψ0 and ψb to decay away from the initial instant and from the boundary, respectively. The results are illustrated by specializing them to the one speed case. The method may make it possible to treat more realistic and more complex problems than can be handled by other methods.