Effectiveness of a Consistently Formulated Diffusion-Synthetic Acceleration Differencing Approach

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Abstract
A consistently formulated differencing approach is applied to the diffusion-synthetic acceleration of discrete ordinates calculations based on various spatial differencing schemes. The diffusion “coupling” equations derived for each scheme are contrasted to conventional coupling relations and are shown to permit derivation of either point- or box-centered diffusion difference equations. The resulting difference equations are shown to be mathematically equivalent, in slab geometry, to equations derived by applying Larsen’s four-step procedure to the S2 equations. Fourier stability analysis of the acceleration method applied to slab model problems is used to demonstrate that, for any Sn differencing scheme (a) the upper bound on the spectral radius of the method occurs in the fine-mesh limit and equals that of the spatially continuous case (0.22466), and (b) the spectral radius decreases with increasing mesh size to an asymptotic value <0.13135. This model problem performance is somewhat superior to that of Larsen’s approach, for which the spectral radius is bounded by 0.25 in the wide-mesh limit. Numerical results of multidimensional, heterogeneous, scattering-dominated problems are also presented to demonstrate the rapid convergence of accelerated discrete ordinates calculations using various spatial differencing schemes.