Nonlinear Diffusion Acceleration of the Least-Squares Transport Equation in Geometries with Voids

TL;DR

This paper extends the Nonlinear-Diffusion Acceleration method to handle geometries with voids by modifying the closure term and diffusion coefficient, ensuring stability and efficiency in both optically thick and thin regions.

Contribution

The paper introduces modifications to NDA for void regions using a weighted least-squares equation, enabling stable and efficient transport calculations in complex geometries.

Findings

The modified NDA algorithm is stable and efficient in void-containing geometries.

Numerical tests show NDA WLS performs comparably or slightly worse than existing methods.

The approach maintains a symmetric discretization matrix, beneficial for computational stability.

Abstract

In this paper we show the extension of the Nonlinear-Diffusion Acceleration (NDA) to geometries containing small voids using a weighted least-squares (WLS) high order equation. Even though the WLS equation is well defined in voids, the low-order drift diffusion equation was not defined in materials with a zero cross section. This paper derives the necessary modifications to the NDA algorithm. We show that a small change to the NDA closure term and a non-local definition of the diffusion coefficient solve the problems for voids regions. These changes do not affect the algorithm for optical thick material regions, while making the algorithm well defined in optically thin ones. We use a Fourier analysis to perform an iterative analysis to confirm that the modifications result in a stable and efficient algorithm. Numerical results of our method will be presented in the second part of the paper. We test this formulation with a small, one-dimensional test problem. Additionally we present results for a modified version of the C5G7 benchmark containing voids as a more complex, reactor like problem. We compared our results to PDT, Texas A\&M's transport code, utilizing a first order discontinuous formulation as reference and the self-adjoint angular flux equation with void treatment (SAAFt), a different second order form. The results indicate, that the NDA WLS performed comparably or slightly worse then the asymmetric SAAFt, while maintaining a symmetric discretization matrix.