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

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.