The Extended Diamond Difference - Constant Nodal Method with Decoupled Cell Iteration Scheme in Two-Dimensional Discrete Ordinate Transport Problems

Abstract

Analytical solutions for the Boltzmann transport equation are typically limited to specific scenarios, prompting the research transport community to extensively explore numerical approaches with special attention to the discrete ordinates (SN) formulation. The spectral nodal methods are widely recognized for their capability to preserve the analytical solution of the SN mathematical model. However, these formulations exhibit notable complexity with high-order polynomial approximations for the transverse leakage terms, motivating researchers to pursue simpler hybrid approaches by partially preserving the analytical solution. Both spectral nodal and hybrid spectral nodal methods introduce nonstandard auxiliary equations not amenable to standard iterative schemes due to a highly coupling in angular directions. Following these ideas, we present a finite element – spectral nodal hybrid method referred to as Extended Diamond Difference-Constant Nodal (EDD-CN) method for two-dimensional fixed-source discrete ordinate transport problems with constant nodal approximation for the leakage terms. The proposed method seeks to preserve eigenfunctions associated with the dominant pair of eigenvalues in the transverse integrated SN equations. In addition, we propose a novel iterative scheme called Decoupled Cell Iteration (DCI) scheme which decouples the angular directions in the nonstandard auxiliary equations and follows conventional iteration transport sweeps for solving SN problems.