Spectral Analysis of the Three-Dimensional Laplace Transform Nodal Method for Two-Groups Discrete Ordinates Problems in Cartesian Geometry

Abstract

In this work, we describe a spectrum of the three-dimensional Laplace Transform Nodal method (LTSN) in order to solve the transport problem in a parallelepiped domain with two energy groups.
We present the LTSN nodal method to generate an analytical solution for discrete ordinates (SN) problems in three-dimensional cartesian geometry and two energy groups. We first transverse integrate the SN equations and then we apply the Laplace transform. The essence of this method is the diagonalization of the LTSN transport matrices and the spectral analysis garantees this, because the eigenvalues can have multiplicity greater than one and corresponding linearly independent eigenvectors.
The transverse leakage terms that appear in the transverse integrated SN equations are represented by exponential functions with decay constants that depend on the characteristics of the material of the medium of the particles leave behind. We use continuity conditions across the region interfaces, in order to obtain the approximated problem solution. The only approximation we use in the derivation of the present method is the exponential approximation for the transverse leakage terms.