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Spectral Method for 1-D Neutron Transport Equation

Haonan Zhang, Huiyuan Li, Zhimin Zhang

TL;DR

This work tackles the 1D steady-state neutron transport equation by introducing a fully spectral discretization that couples a spectral-(Petrov-)Galerkin scheme in the spatial variable with a Legendre-Gauss collocation approach in the angular variable, and uses Legendre-Gauss quadrature for angular integrals. The authors establish solvability of the discretization and derive an $L^2$-error estimate, showing optimal angular convergence and suboptimal but sharp spatial convergence due to boundary effects, under suitable regularity assumptions on the cross sections and source. They validate the method through extensive numerical experiments, including smooth and discontinuous coefficient cases, and demonstrate spectral accuracy, robustness, and competitiveness against HWENO and low-order DG methods; they also introduce a multi-domain strategy to handle discontinuities. The results indicate that the fully spectral approach yields high accuracy with favorable computational efficiency and offer a clear path toward extensions to higher dimensions via spectral-element or DG frameworks, enabling precise neutron transport simulations on complex geometries.

Abstract

In this paper, we present an efficient fully spectral approximation scheme for exploring the one-dimensional steady-state neutron transport equation. Our methodology integrates the spectral-(Petrov-)Galerkin scheme in the spatial dimension with the Legendre-Gauss collocation scheme in the directional dimension. The directional integral in the original problem is discretized with Legendre-Gauss quadrature. We furnish a rigorous proof of the solvability of this scheme and, to our best knowledge, conduct a comprehensive error analysis for the first time. Notably, the order of convergence is optimal in the directional dimension, while in the spatial dimension, it is suboptimal and, importantly, non-improvable. Finally, we verify the computational efficiency and error characteristics of the scheme through several numerical examples.

Spectral Method for 1-D Neutron Transport Equation

TL;DR

This work tackles the 1D steady-state neutron transport equation by introducing a fully spectral discretization that couples a spectral-(Petrov-)Galerkin scheme in the spatial variable with a Legendre-Gauss collocation approach in the angular variable, and uses Legendre-Gauss quadrature for angular integrals. The authors establish solvability of the discretization and derive an -error estimate, showing optimal angular convergence and suboptimal but sharp spatial convergence due to boundary effects, under suitable regularity assumptions on the cross sections and source. They validate the method through extensive numerical experiments, including smooth and discontinuous coefficient cases, and demonstrate spectral accuracy, robustness, and competitiveness against HWENO and low-order DG methods; they also introduce a multi-domain strategy to handle discontinuities. The results indicate that the fully spectral approach yields high accuracy with favorable computational efficiency and offer a clear path toward extensions to higher dimensions via spectral-element or DG frameworks, enabling precise neutron transport simulations on complex geometries.

Abstract

In this paper, we present an efficient fully spectral approximation scheme for exploring the one-dimensional steady-state neutron transport equation. Our methodology integrates the spectral-(Petrov-)Galerkin scheme in the spatial dimension with the Legendre-Gauss collocation scheme in the directional dimension. The directional integral in the original problem is discretized with Legendre-Gauss quadrature. We furnish a rigorous proof of the solvability of this scheme and, to our best knowledge, conduct a comprehensive error analysis for the first time. Notably, the order of convergence is optimal in the directional dimension, while in the spatial dimension, it is suboptimal and, importantly, non-improvable. Finally, we verify the computational efficiency and error characteristics of the scheme through several numerical examples.

Paper Structure

This paper contains 18 sections, 5 theorems, 86 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Jacobi polynomials
  5. Useful lemmas
  6. Spectral approximation scheme and implementation
  7. Approximation scheme
  8. Implementation
  9. Error estimate
  10. Numerical results
  11. Example 1
  12. Example 2
  13. Example 3
  14. Example 4
  15. Example 5
  16. ...and 3 more sections

Key Result

Lemma 2.1

Assuming $\varphi \in L_{\omega^{0,-1}}^{2}(D)$ and $\partial_x\varphi \in L_{\omega^{1,0}}^{2}(D)$, we have following relation: Similarly, when $\varphi \in L_{\omega^{-1,0}}^{2}(D)$ and $\partial_x\varphi \in L_{\omega^{0,1}}^{2}(D)$, we have Hereafter, the notation $A \lesssim B$ means that there exists a generic positive constant $C$, independent of $N$ and any function, such that $A\leq CB$

Figures (14)

  • Figure 1: Global indexing of the unknowns and the mask matrices. the inflow boundary condition; $\bullet$: interior and the outflow boundary.
  • Figure 2: The flux of Example 1 for spectral method and $L^1$-errors of flux evaluated by the HWENO method.
  • Figure 3: The $L^{2}$-errors versus $N$ and $M+1$ of Example 1 using the spectral method.
  • Figure 4: The flux of Example 2 obtained using the spectral method and $L^2$-errors of $P_{1}$ DG scheme with the positivity-preserving limiter.
  • Figure 5: The $L^{2}$-errors versus $N$ and $M+1$ of Example 2 for the spectral method.
  • ...and 9 more figures

Theorems & Definitions (6)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3: canuto2007spectral
  • Theorem 3.1
  • Theorem 4.1
  • Remark 4.1