Convergence Study of 2D Transport Method for Multigroup k-Eigenvalue Problems with High-Order Scattering Source Using Fourier Analysis

Abstract

The convergence behavior of a two-dimensional (2D) transport method has been derived by Fourier analysis for single-group problems with isotropic sources. However, in real calculation, to pursue precision, a high-order scattering source is a common option, and its influence on convergence performance is worth investigating. No theoretical convergence study of a 2D transport method for multigroup problems with high-order scattering sources was previously performed, but it is important work that would complement existing studies. This study presents a Fourier analysis for solving multigroup problems with high-order scattering. First, the influences of the number of inner iterations for the multigroup isotropic scattering problem are analyzed. It is found that with an increase of the number of inner iterations, the spectral radius decreases and finally reaches an asymptotic value. When the scattering ratio is increased, more inner iterations are required to reach the asymptotic value. Then, the influences of high-order scattering are analyzed. The Fourier analysis results show that for high-order scattering source problems, the influence of the number of inner iterations is different from the isotropic scattering case. The influences of first-order scattering and second-order scattering are not the same. With an increase of first-order scattering, the spectral radius first decreases in the small optical thickness region and then increases in the large optical thickness region, which may lead to the divergence of iterations. If second-order scattering is not too large, an increase of second-order scattering decreases the spectral radius for all optical thickness regions. First-order scattering and second-order scattering that are too large may result in an unpredictable slope of the spectral radius for optical thicknesses between 10⁻¹ and 1.

Keywords:

• Fourier analysis
• two-dimensional transport method
• k-eigenvalue problem
• high-order scattering, multigroup

Disclosure Statement

No potential conflict of interest was reported by the author(s).

Nomenclature

$\mathit{D}$ =	=	diffusive coefficient (m)
$\mathit{G}$ =	=	total number of energy groups
$\mathit{h}$ =	=	mesh size of fine mesh (m)
$\mathit{J}$ =	=	net current (m−2∙s−1)
$k_{\mathit{eff}}$ =	=	effective multiplication factor
$\mathit{M}$ =	=	total number of discrete angles
$\mathit{N}$ =	=	order of scattering order
$N_{\mathit{inner}}$ =	=	total number of inner iterations
$p^2$ =	=	number of fine mesh per coarse mesh
$\mathit{Q}$ =	=	total source term
$\mathit{w}$ =	=	quadrature weight
$Y_n^k$ =	=	spherical harmonic

Greek

$\mathrm{\Delta }$ =	=	coarse-mesh size (m)
$\mathit{\eta }$ =	=	sine of direction
$\mathit{\lambda }$ =	=	Fourier frequency
$\mathit{\mu }$ =	=	cosine of direction
$\mathit{v}$ =	=	average number of neutrons released per fission
${\mathrm{\Sigma }}_f$ =	=	fission cross section (m−1)
${\mathrm{\Sigma }}_s$ =	=	scattering cross section (m−1)
${\mathrm{\Sigma }}_t$ =	=	total cross section (m−1)
$\mathit{\phi }$ =	=	neutron angular flux (m–2∙s−1)
$\mathit{\chi }$ =	=	fission spectrum
$\mathit{\psi }$ =	=	scalar flux moment (m−2∙s−1)
$\mathrm{\Omega }$ =	=	angle direction vector

Subscript

$\mathit{g}$ =	=	discretized energy group
$\mathit{I}$ =	=	index of coarse mesh in X direction
$\mathit{I}\pm \frac{1}{2}$ =	=	coarse-mesh left/right boundary
$\mathit{i}$ =	=	index of fine mesh in X direction
$\mathit{i}\pm \frac{1}{2}$ =	=	fine-mesh left/right boundary in X direction
$\mathit{J}$ =	=	index of coarse mesh in Y direction
$\mathit{J}\pm \frac{1}{2}$ =	=	coarse-mesh behind/front boundary
$\mathit{j}$ =	=	index of fine mesh in Y direction
$\mathit{j}\pm \frac{1}{2}$ =	=	fine-mesh left/right boundary in Y direction
$\mathit{m}$ =	=	discretized angle
$\mathit{n}$ =	=	index of the high-order scattering

Superscript

$\mathit{a}$ =	=	index of the inner iteration
$\mathit{l}$ =	=	index of the outer iteration
$\mathit{l}+\frac{1}{2}$ =	=	medium step of l’th outer iteration

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [12005020]; Postdoctoral Fellowship Program of CPSF [GZC20233419].