Properties of the S(N)-equivalent integral transport operator and the iterative acceleration of neutral particle transport methods

We have derived expressions for the elements of the matrix representing a certain angular (SN) and spatial discretized form of the neutron integral transport operator. This is the transport operator that if directly inverted on the once-collided fixed particle source produces, without the need for an iterative procedure, the converged limit of the scalar fluxes for the iterative procedure. The asymptotic properties of this operator's elements have then been investigated in homogeneous and periodically heterogeneous limits in one-dimensional and two-dimensional geometries. The thesis covers the results obtained from this asymptotic study of the matrix structure of the discrete integral transport operator and illustrates how they relate to the iterative acceleration of neutral particle transport methods. Specifically, it will be shown that in one-dimensional problems (both homogeneous and periodically heterogeneous) and homogeneous two-dimensional problems, containing optically thick cells, the discrete integral transport operator acquires a sparse matrix structure, implying a strong local coupling of a cell-averaged scalar flux only with its nearest Cartesian neighbors. These results provide further insight into the excellent convergence properties of diffusion-based acceleration schemes for this broad class of transport problems. In contrast, the results of the asymptotic analysis for two-dimensional periodically heterogeneous problems point to a sparse but non-local 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. The latter results indicate that cross-derivative coupling, namely coupling of a cell's average flux to its diagonal neighbors, is of the same order as self-coupling and coupling with its first Cartesian neighbors. Hence they substantiate the conjecture that the loss of robustness of diffusion-based acceleration schemes, in particular of the Adjacent-cell Preconditioner (AP) considered in this work, in the presence of sharp material discontinuities in periodically heterogeneous multi-dimensional problems, is due to a structural deficiency of such low-order operators since they ignore cross-derivative coupling. This conjecture has been successfully verified by amending the AP formalism to account for cross-derivative coupling by the inclusion of matrix elements that account for the coupling of a cell's average flux to its first diagonal neighbors. Preliminary results of the Fourier analysis for the novel acceleration scheme indicate that robustness of the accelerated iterations can be recovered by accounting for cross-derivative coupling. The new acceleration scheme has also been implemented in a two-dimensional transport code and numerical results from the code have successfully verified the predictions of the Fourier analysis.