Convergence analysis of MOC inner iterations with large negative self-scattering cross-section

Abstract

When the transport correction is applied to the total cross-section, the self-scattering cross-section could have a negative value in order to preserve the balance of partial cross-sections. The negative self-scattering cross-section may lead to a negative impact on the convergence behavior for the method of characteristics (MOC), especially in a problem with large moderator regions containing hydrogen. In order to address this issue, the spectral radius of the inner iteration of MOC is theoretically estimated for various self-scattering cross-sections. It is found that the spectral radius of the inner iteration of MOC could exceed unity for a large negative self-scattering cross-section, which results in numerical divergence. A countermeasure for the divergence using the successive over relaxation method is also discussed in this paper.

Improvement of transport-corrected scattering stability and performance using a Jacobi inscatter algorithm for 2D-MOC

“…Estimating the spectral radius from the fission source residual yields 0.946 with Jacobi vs 0.919 with Gauss Seidel. This is similar to what would be observed if a relaxation factor were applied, as in the approach proposed by Tabuchi and Yamamoto [8]. In a sense, because Jacobi lags the scattering source, the solution is effectively relaxed, providing additional stability.…”

“…This type of approach is more consistent with CASMO [10] and OpenMOC [11], but the impact on convergence and stability has not been evaluated in these codes. Furthermore, since the scattering source is lagged with the Jacobi approach, it indirectly relaxes the solution and likely falls somewhat in line with the relaxation approach proposed by Tabuchi and Yamamoto [8].…”

“…Convergence issues with transport-corrected scattering are a long-standing issue and are not specific to MPACT [7,8]. While it is possible that cross section library generation can lead to unphysical/negative solutions, particularly where there is a large variation in the energy group size, in many instances the convergence issues seem to be related to the path to the solution more than the final solution itself.…”

“…While it is possible that cross section library generation can lead to unphysical/negative solutions, particularly where there is a large variation in the energy group size, in many instances the convergence issues seem to be related to the path to the solution more than the final solution itself. One proposed solution for instabilities caused by the transport correction from Tabuchi and Yamamoto [8] has been to apply a relaxation factor during the MOC inner iterations, which has proven to be moderately successful, but it requires several inner iterations for MOC, which can be fairly costly. Other potential solutions, such as solving several iterations with positive-definite isotropic cross sections then incorporating the negative, transport-corrected scattering terms, have also been evaluated with mixed success.…”

Recent activities in the field of reactor physics

“…The generalized equivalence theory and the superhomogeneisation (SPH) method, which have been widely used in current core calculations with the advanced nodal method to mitigate cell/assembly homogenization errors, have been applied to the integro-differential transport equation to further improve the accuracy of whole-core calculations [8,9]. Improvements on robustness, efficiency and accuracy of transport calculations based on the method of characteristics [10][11][12] and calculations of the space-dependent kinetic equation [13] are expected to enlarge the range of applicability of those advanced methods in realistic problems.…”