Behavior of the Diamond Difference and Low-Order Nodal Numerical Transport Methods in the Thick Diffusion Limit for Slab Geometry

Abstract

The objective of this work is to investigate the thick diffusion limit of various spatial discretizations of the one-dimensional, steady-state, monoenergetic, discrete ordinates neutron transport equation. This work specifically addresses the two lowest order nodal methods, AHOT-N0 and AHOT-N1, as well as reconsiders the asymptotic limit of the Diamond Difference method. The asymptotic analyses of the AHOT-N0 and AHOT-N1 nodal methods show that AHOT-N0 does not possess the thick diffusion limit for cell edge or cell average fluxes except under very limiting conditions, which is to be expected considering the AHOT-N0 method limits to the Step method in the thick diffusion limit. The AHOT-N1 method, which uses a linear in-cell representation of the flux, was shown to possess the thick diffusion limit for both cell average and cell edge fluxes. The thick diffusion limit of the DD method, including the boundary conditions, was derived entirely in terms of cell average scalar fluxes. It was shown that, for vacuum boundaries, only when {sigma}{sub t}, h, and Q are constant and {sigma}{sub a} = 0 is the asymptotic limit of the DD method close to the finite-differenced diffusion equation in the system interior, and that the boundary conditions between the systems will only agree in the absence of an external source. For a homogeneous medium an effective diffusion coefficient was shown to be present, which was responsible for causing numeric diffusion in certain cases. A technique was presented to correct the numeric diffusion in the interior by altering certain problem parameters. Numerical errors introduced by the boundary conditions and material interfaces were also explored for a two-region problem using the Diamond Difference method. A discrete diffusion solution which exactly solves the one-dimensional diffusion equation in a homogeneous region with constant cross sections and a uniform external source was also developed and shown to be equal to the finite-differenced diffusion discretization for c = 1 in the system interior, where again the boundary conditions again only agree in the absence of an external source. It was also shown that for c < 1 the exact discrete diffusion solution is written in terms of hyperbolic functions, with expressions which limit to the exact solution for the c = 1 case as c {yields} 1. Finally, a transport discretization is developed which reproduces the exact S2 solution for the case of a purely scattering homogeneous region with vacuum boundary conditions, and an extension to the discretization for the case of c < 1 is found by considering a Taylor series expansion of the exact answer centered at c = 0.