Convergence Analysis of the Nonlinear Coarse Mesh Finite Difference Method

The convergence rates for the non-linear coarse mesh finite difference (CMFD) method and the coarse mesh rebalance (CMR) method are derived analytically for one-group, one-dimensional solutions of the fixed source diffusion problem in a non-multiplying infinite homogeneous medium. The derivation was performed by linearizing the non-linear algorithm and by applying Fourier error analysis to the linearized algorithm. The mesh size measured in units of the diffusion length is shown to be a dominant parameter for the convergence rate and for the stability of the iterative algorithms. Non-linear CMFD is shown to be a more effective acceleration method than CMR, especially for small mesh sizes. 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.