Convergence analysis of the coarse mesh finite difference method
The convergence rates of the nonlinear coarse mesh finite difference (CMFD) and the coarse mesh rebalance (CMR) methods are derived analytically for one- and two-dimensional geometries and one- and two-energy group solutions of the fixed source diffusion problem in a non-multiplying infinite homogeneous medium. The derivation is performed by linearizing the nonlinear algorithm and by applying Fourier error analysis to the linearized algorithm. The mesh size measured in units of the diffusion length was shown to be a dominant parameter for the convergence rate and for the stability of the iterative algorithms. For a small mesh size problem, CMFD is shown to be a more effective acceleration method than CMR. Both CMR and two-node CMFD algorithms are shown to be unconditionally stable. However, one-node CMFD becomes unstable for large mesh sizes. To remedy this instability, an under-relaxation of the current correction factor for the one-node CMFD method is successfully introduced and the domain of stability is significantly expanded. Furthermore, the optimum under-relaxation parameter is analytically derived and the one-node CMFD with the optimum relaxation is shown to be unconditionally stable. Additional numerical analysis was performed on the convergence of each algorithm with the U.S. NRC Neutron Kinetics code PARCS. It was confirmed that the insights about the CMFD and the CMR methods obtained for simple model problems in Chapters 2–4 are valid for the realistic three-dimensional two-group eigenvalue and transient fixed source problems.