A Weighted Least-Squares Transport Equation Compatible with Source Iteration and Voids

TL;DR

This paper introduces a weighted least-squares transport equation compatible with source iteration and voids, improving causality and accuracy while maintaining a symmetric positive-definite system matrix for reactor physics calculations.

Contribution

A novel weighted least-squares transport formulation that enhances causality and aligns with the SAAF equation, suitable for iterative solution methods in void regions.

Findings

Weighted least-squares equation improves causality issues.

Method achieves acceptable accuracy compared to SAAFt.

Maintains symmetric positive-definite system matrix.

Abstract

Least-squares (LS) forms of the transport equation can circumvent the void problems of other second order forms, but are almost always non-conservative. Additionally, the standard LS form is not compatible with discrete ordinates method (Sn) iterative solution techniques such as source iteration. A new form of the least-squares transport equation has recently been developed that is compatible with voids and standard Sn iterative solution techniques. Performing Nonlinear Diffusion Acceleration (NDA) using an independently-differenced low-order equation enforces conservation for the whole system, and makes this equation suitable for reactor physics calculations. In this context independent means that both the transport and low-order solutions converge to the same scalar flux and current as the spatial mesh is refined, but for a given mesh, the solutions are not necessarily equal. In this paper we show that introducing a weight function to this least-squares equation improves issues with causality and can render our equation equal to the Self-Adjoint Angular Flux (SAAF) equation. Causality is a principle of the transport equation which states that information only travels downstream along characteristics. This principle can be violated numerically. We show how to limit the weight function in voids and demonstrate the effect of this limit on the accuracy. Using the C5G7 benchmark, we compare our method to the self-adjoint angular flux formulation with a void treatment (\SAAFt), which is not self-adjoint and has a non-symmetric coefficient matrix. We show that the weighted least-squares equation with NDA gives acceptable accuracy relative to the SAAFt equation while maintaining a symmetric positive-definite system matrix.