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Coarse Mesh Iteration Approach for Analytical 1D Multigroup $S_N$ Eigenvalue Problems

Jilang Miao, Miaomiao Jin

TL;DR

This work tackles efficient eigenvalue solutions for analytical multigroup $S_N$ equations in 1D slab geometry by developing a coarse-mesh, analytically solvable fixed-source framework. It rewrites the transport problem as $ rac{d}{dx}oldsymbol{ m }(x) = A_0 oldsymbol{ m }(x) + Q(k_{eff},x)$ with an external source $Q$ derived from an eigensystem expansion on the coarse mesh and stabilizes the iteration using a Wielandt shift. The method demonstrates that coarse-mesh solutions achieve accuracy comparable to a fine-mesh fixed-source solver while delivering substantial speedups (up to roughly an order of magnitude depending on the SN order) and remains extendable to heterogeneous regions with proper interface handling. Results on a 35 cm slab show $k_{eff}$ differences shrinking from a few hundred to near-zero pcm as $S_N$ increases, and flux accuracies improving dramatically from percent-level errors to sub-percent levels. Overall, the coarse-mesh analytical SN approach provides a computationally efficient path for eigenvalue problems in slab geometries with strong potential for broader applicability in reactor-scale analyses.

Abstract

This paper extends the fixed source capability of analytical 1D multigroup $S_N$ equations to solve eigenvalue problems on coarse mesh.

Coarse Mesh Iteration Approach for Analytical 1D Multigroup $S_N$ Eigenvalue Problems

TL;DR

This work tackles efficient eigenvalue solutions for analytical multigroup equations in 1D slab geometry by developing a coarse-mesh, analytically solvable fixed-source framework. It rewrites the transport problem as with an external source derived from an eigensystem expansion on the coarse mesh and stabilizes the iteration using a Wielandt shift. The method demonstrates that coarse-mesh solutions achieve accuracy comparable to a fine-mesh fixed-source solver while delivering substantial speedups (up to roughly an order of magnitude depending on the SN order) and remains extendable to heterogeneous regions with proper interface handling. Results on a 35 cm slab show differences shrinking from a few hundred to near-zero pcm as increases, and flux accuracies improving dramatically from percent-level errors to sub-percent levels. Overall, the coarse-mesh analytical SN approach provides a computationally efficient path for eigenvalue problems in slab geometries with strong potential for broader applicability in reactor-scale analyses.

Abstract

This paper extends the fixed source capability of analytical 1D multigroup equations to solve eigenvalue problems on coarse mesh.
Paper Structure (14 sections, 30 equations, 1 figure, 2 tables, 1 algorithm)

This paper contains 14 sections, 30 equations, 1 figure, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Theory
  3. Formulation of Fixed Source Problem
  4. Coarse Mesh Iteration
  5. Representation of the source term
  6. Solution of the angular flux
  7. Evaluation of the eigenvalue $k_{eff}$
  8. Projection of angular flux on the new eigensystem
  9. Algorithm for coarse mesh iteration
  10. Results
  11. Accuracy of the coarse mesh iteration method
  12. Efficiency of the coarse mesh iteration method
  13. Conclusions
  14. Acknowledgments

Figures (1)

  • Figure 1: Convergence of fine and coarse mesh analytical S$_N$ methods. (a) Norm of flux change vs. iteration number. (b) Computation time compared between coarse mesh and fine mesh method. (c) Computation time compared between coarse mesh method with and without Wielandt's shift.