Multilevel source iteration accelerators for the linear transport equation in slab geometry

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Abstract
In this paper we show how classical error estimates for various discretizations of the source iteration map in slab geometry can be used to construct accurate approximate inverses in the context of fast multilevel methods. For discretizations that give strongly convergent collectively compact sequences of approximate source iteration maps, the Atkinson-Brakhage approximate inverse can be applied. For discretizations that give rise to norm convergent sequences, a more direct approach can be used. Our implementation of these ideas, based on use of GMRES iteration to solve the coarse mesh problems, gives the solution to an accuracy of fine mesh truncation error at a cost proportional to that of an evaluation of the fine mesh source iteration map. These methods require only the source iteration map and are hence easier to adapt to multiprocessor computers than methods that require solution of diffusion equations. We illustrate our results with a report on numerical experiments with both strongly and norm convergent source iteration maps using the Kendall Square KSR1 computer.