Iterative solutions of the Boltzmann transport equation are computationally intensive. Spatial multigrid methods have led to efficient iterative algorithms for solving a variety of partial differential equations; thus, it is natural to explore their application to transport equations. Manteuffel et al. conducted such an exploration in one spatial dimension, using two-cell inversions as the relaxation or smoothing operation, and reported excellent results. In this dissertation we extensively test Manteuffel's one-dimensional method and our modified versions thereof. We demonstrate that the performance of such spatial multigrid methods can degrade significantly given strong heterogeneities. We also extend Manteuffel's basic approach to two-dimensional problems, employing four-cell inversions for the relaxation operation. We find that for uniform homogeneous problems the two-dimensional multigrid method is not as rapidly convergent as the one-dimensional method. For strongly heterogeneous problems the performance of the two-dimensional method is much like that of the one-dimensional method, which means it can be slow to converge. We conclude that this approach to spatial multigrid produces a method that converges rapidly for many problems but not for others. That is, this spatial multigrid method is not unconditionally rapidly convergent. However, our analysis of the distribution of eigenvalues of the iteration operators indicates that this spatial multigrid method may work very well as a preconditioner within a Krylov iteration algorithm, because its eigenvalues tend to be relatively well clustered. Further exploration of this promising result appears to be a fruitful area of further research.
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