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Developing an Analytical Fixed Source Solver for the 1D Multigroup $S_N$ Equations

Jilang Miao, Miaomiao Jin

TL;DR

The paper addresses efficient 1D fixed-source solutions within the multigroup $S_N$ transport framework by developing an analytic fixed-source solver that leverages block-diagonalization of the transport operator and regional interface conditions. It embeds this solver in power iteration to tackle 1D eigenvalue problems, handling homogeneous and heterogeneous slabs with boundary and continuity constraints and enabling piecewise-constant sources on a fine mesh. Key contributions include explicit construction of region-wise coefficients via a global $NGR\times NGR$ linear system, demonstrated accuracy against Monte Carlo references (e.g., $k_{eff}$ differences shrinking from hundreds to a few pcm as $S_N$ order increases) and substantial computational speedups over traditional sweeping methods (up to ~31x per-iteration and >100x overall, enhanced further by Wielandt's shift). The results suggest the method’s potential as a fast axial solver for 2D-1D and 3D nodal $S_N$ schemes, enabling efficient and accurate reactor transport simulations.

Abstract

We extend the analytical multigroup $S_N$ method to solve 1D fixed source problems. The fixed source solver is applied in power iteration of eigenvalue problems.

Developing an Analytical Fixed Source Solver for the 1D Multigroup $S_N$ Equations

TL;DR

The paper addresses efficient 1D fixed-source solutions within the multigroup transport framework by developing an analytic fixed-source solver that leverages block-diagonalization of the transport operator and regional interface conditions. It embeds this solver in power iteration to tackle 1D eigenvalue problems, handling homogeneous and heterogeneous slabs with boundary and continuity constraints and enabling piecewise-constant sources on a fine mesh. Key contributions include explicit construction of region-wise coefficients via a global linear system, demonstrated accuracy against Monte Carlo references (e.g., differences shrinking from hundreds to a few pcm as order increases) and substantial computational speedups over traditional sweeping methods (up to ~31x per-iteration and >100x overall, enhanced further by Wielandt's shift). The results suggest the method’s potential as a fast axial solver for 2D-1D and 3D nodal schemes, enabling efficient and accurate reactor transport simulations.

Abstract

We extend the analytical multigroup method to solve 1D fixed source problems. The fixed source solver is applied in power iteration of eigenvalue problems.
Paper Structure (13 sections, 16 equations, 1 figure, 2 tables, 1 algorithm)

This paper contains 13 sections, 16 equations, 1 figure, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Theory
  3. S$_N$ equation in a homogeneous slab
  4. S$_N$ solution in a heterogeneous slab
  5. Incoming flux boundary condition
  6. Angular flux continuity condition
  7. Solution of the coefficients
  8. Application of the fixed source solution
  9. Results
  10. Accuracy of the eigenvalue problem
  11. Efficiency of the fixed source solver
  12. Conclusions
  13. Acknowledgments

Figures (1)

  • Figure 1: (a--h) Scalar flux from S$_N$ compared with MC. (i--p) Angular flux ($\omega_n \psi_n(x)$) from S$_N$ compared with MC. (a),(e),(i),(m) show both flux value and relative error (%) from MC. The uncertainty of each tally $T$ from MC is shown with the shading area between $\pm 100\times \frac{\sigma_{\bar{T}}}{\bar{T}}$. (q--s) Convergence rate compared with sweeping method. (q) $L^2$ norm of scalar flux change as function of iteration number. (r--s) $L^2$ norm of scalar flux change as function of computation time.