On the Convergence of the Eigenvalue Response Matrix Method

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A study of the convergence behavior of the eigenvalue response matrix method (ERMM) for nuclear reactor eigenvalue problems is presented. The eigenvalue response matrix equations are traditionally solved by a two-level iterative scheme in which an inner eigenproblem yields particle balance across node boundaries and an outer fixed-point iteration updates the global k-eigenvalue. Past work has shown the method converges rapidly, but the properties of its convergence have not been studied in detail. To perform a formal assessment of these properties, the one-dimensional, one-group diffusion approximation is used to derive the asymptotic error constant of the fixed-point iteration. Several problems are solved numerically, and the observed convergence behavior is compared to the analytic model based on buckling and nodal dimensions (in mean free paths). The results confirm the method converges quickly, with no degradation in the convergence rate for small nodes, which is an observation that suggests ERMM can be used for large-scale, parallel computations with no penalty from the decomposition of a domain into smaller nodes. In addition, results from multigroup problems show that convergence depends strongly on the heterogeneity and the energy representation of a model. In particular, the convergence for two-group and heterogeneous, one-group models is substantially slower than for the homogeneous, one-group model.