Canonical Form for the Strength of Coupling Between Neighboring Spatial Cells for WDD Transport Methods in Homogeneous Configurations in 2D Geometry

摘要

Exact and asymptotic expressions for the matrix elements of the S_n-equivalent integral transport operator coupling a cell-average scalar flux with the fluxes in the neighboring cells have been obtained in 2D geometry for a WDD spatial discretization. Evaluation of these matrix elements for homogeneous configurations shows that elements pertaining to self-coupling and coupling with the first Cartesian neighbors are of the same order and dominate all other matrix elements in the optically thick cell limit. In particular, cross-derivative coupling with the first diagonal-neighbors is of higher-order in the cell optical thickness. These estimates also quantify the strength of cell-coupling vs separating distance. A recursive algorithm has been developed and coded for the numeric computation of the integral transport matrix in one mesh sweep. This permitted numerical verification of the theoretically predicted asymptotic behavior with increasing cell optical thickness. Overall our results provide quantitative understanding of the canonical behavior for the WDD class of spatial discretizations of the transport equation in the homogeneous optically thick cell limit. This provides further insight into the excellent convergence properties of diffusion-based acceleration schemes in homogeneous multi-dimensional problems. Future extension of the methodology presented here to the analysis of heterogeneous configurations in 2D geometry might shed light on the failure of diffusive preconditioners in multidimensional problems with sharp material heterogeneities. If this failure can be attributed to neglecting asymptotically significant coupling elements in diffusive preconditioners, then amending such elements will be attempted to restore acceleration robustness of the amended preconditioner.