The Error Bounds for Three-Dimensional Nodal LTSN Method

Abstract

In this paper we present a proof about the convergence of the 3D Nodal-LTSN Method in order to solve the transport problem in a parallelepiped domain. For that, we define functions associated to the errors, one in the approximated flux, another in the quadrature formula and establish a relation between them. We present a Nodal-LTSN method to generate an analytical solution for discrete ordinates problems in three-dimensional cartesian geometry. 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. 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 the particles leave behind. We present numerical results generated by the offered method applied to typical shielding model problems.

Integral Methods in Science and Engineering

“…In this section we extend the mathematical analysis of the error bound estimate and convergence to the mono-energetic nodal-LT S N solution in a rectangle proposed in [HaEtAl08]. We discuss the conditions for the convergence of the discrete ordinates nodal method, LT S N , mentioned in [Ze90], [KaLeHe82], [Kh97], [PaVi99], [KhSb05a], [KhSb05b], [HaEtAl08].…”

“…We discuss the conditions for the convergence of the discrete ordinates nodal method, LT S N , mentioned in [Ze90], [KaLeHe82], [Kh97], [PaVi99], [KhSb05a], [KhSb05b], [HaEtAl08]. We define the so-called errors of the approximated flux and the error in the quadrature formula and then establish a relationship between both errors in order to give a global estimate of the approximated flux.…”

“…At this point, based on the physics of shielding problems, we assume that the neutron flux attenuates exponentially with increasing distance from the source, a hypothesis also assumed in [BaVi91], [HaViBa09], [HaEtAl08], [HaPaVi05], where the only approximation involved is in the transverse leakage terms. For instance, in (13.4) we use…”