On the Degradation of Cell-Centered Diffusive Preconditioners for Accelerating SN Transport Calculations in the Periodic Horizontal Interface Configuration
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We investigate the degraded effectiveness of diffusion-based acceleration schemes in terms of the adjacent-cell preconditioner (AP) in a periodically heterogeneous limit devised for the twodimensional (2-D) periodic horizontal interface (PHI) configuration. Specifically, we demonstrate that the diffusive low-order operator employed in the AP scheme lacks the structure of the integral transport operator in the above asymptotic limit since it (1) ignores cross-derivative coupling and (2) incorrectly estimates the strength of intra-layer coupling in the optically thin layers. In order to prove propositions 1 and 2, we derive expressions for the elements of the matrix representing a certain angular (SN) and spatially discretized form of the 2-D neutron transport integral operator. This is the transport operator that produces the full scalar flux solution if it is directly inverted on the once-collided particle source. The properties of this operator's elements are then investigated in the asymptotic limit for PHI. The results of the asymptotic analysis point to a sparse but nonlocal matrix structure due to long-range coupling of a cell's average flux with its neighboring cells, independent of the distance between the cells in the spatial mesh. In particular, for a cell in a thin layer, cross-derivative coupling of the cell's flux to its diagonal neighbors is of the same asymptotic order as self-coupling and coupling with its north/south Cartesian neighbors. Similarly, its coupling with the fluxes in the same thin layer is of the same order, independent of the distance between the cells in the layer, as the coupling with the east/west Cartesian neighbors. We also show that modifying the standard diffusion-based AP can lead to effective acceleration in PHI. Specifically, we devise three novel acceleration schemes, named APB, Optimized-AP (OAP), and Hybrid-AP (HAP), obtained by modifying the original AP formalism in 2-D. In the APB the five-point AP operator is extended to a nine-point stencil that accounts for cross-derivative coupling by including the matrix elements of the integral transport operator B, which couple a cell-averaged scalar flux to its first diagonal neighbors. In the OAP the five-point stencil of the original AP operator is retained while optimizing the value of the elements in the preconditioner that affect the coupling of a cell with its east/west Cartesian neighbors. Specifically, the optimum elements are obtained by minimizing the iteration's spectral radius and offer a more correct estimate of the strength of intra-layer coupling in a thin layer. Finally, the nine-point HAP operator represents a "hybrid" of the APB and OAP approaches, in the sense that the spectral properties of the optimized five-point OAP are further improved via the inclusion of crossderivative terms. Fourier analysis of the novel acceleration schemes indicates that robustness of the accelerated iterations can be recovered, in spite of sharp material discontinuities, by accounting for cross-derivative coupling and by optimizing the preconditioner elements. The new acceleration schemes have also been implemented in a 2-D transport code, and numerical tests successfully verify the predictions of the Fourier analysis. However, it is important to emphasize that the modifications attempted in this work are specific to the selected asymptotic limit for PHI and do not translate into new low-order operators for the general heterogeneous-material case. Rather, the above modified operators suggest that it may be possible to eventually derive such a general low-order unconditionally robust operator.
