Fourier Convergence Analysis of the Rebalance Methods for Discrete Ordinates Transport Equations in Eigenvalue Problems

This paper analyzes the convergence of the rebalance iteration methods for accelerating the power iteration method of the discrete ordinates transport equation in the eigenvalue problem. The rebalance iteration methods include the coarse mesh rebalance (CMR), the coarse mesh finite difference (CMFD), and the partial current-based CMFD methods. The convergence analysis is performed with the well-known Fourier analysis through linearization. In the linearized form, these rebalance methods are formulated in a unified way where the rebalance methods are different only in a parameter. The analyses are applied for both one- and two-group problems in a homogeneous infinite medium and a finite medium having periodic boundary conditions. The theoretical analysis shows that the convergences of the rebalance methods for the eigenvalue problems are closely related with the ones for the fixed source problems and that the convergences for the eigenvalue problems can be analyzed with the formula for the fixed source problem after transforming the scattering cross sections into a different cross-section set. The numerical tests show that the Fourier convergence analysis provides a reasonable estimate for the numerical spectral radii for the model problems.