Theoretical Convergence Rate Lower Bounds for Variants of Coarse Mesh Finite Difference to Accelerate Neutron Transport Calculations

The lower bounds for the theoretical convergence rate of variants of the Coarse Mesh Finite Difference (CMFD) method for neutron transport acceleration are studied in this paper by generalization of the method into three categories: artificially diffusive CMFD, flux relaxation, and higher-order spatial prolongation operators. A Fourier analysis of the methods demonstrates that artificial diffusion and flux relaxation are mathematically equivalent and arbitrarily scale the coarse mesh to fine mesh projection (CMP) vector. The high-order spatial prolongation method is shown to affect the shape of the CMP vector. As a result, any of the CMFD variants based on these three sets of modifications correspond to a specific CMP vector. The optimization process is performed for the multidimensional vector, and the minimum spectral radius among all possible CMP vectors is shown to be the theoretical lower bound for the CMFD convergence rate. The spectral radius associated with the CMFD convergence rate lower bound is found to be slightly smaller (less than 0.04) than optimally diffusive CMFD(odCMFD), and the difference between odCMFD to the CMFD lower bound is much smaller than the difference between both standard CMFD and partial current-based CMFD to the CMFD lower bound. In addition, the odCMFD method has a distinct advantage in ease of implementation and minimal overhead. Conversely, the implementation necessary to achieve the CMFD lower bound would be very complicated, especially for two-and three-dimensional problems.