Hybrid approaches for accelerated convergence of block-Jacobi iterative methods for solution of the neutron transport equation

Abstract

The Parallel Block Jacobi (PBJ) [1,2] scheme is an iterative method with application in neutron transport solution which requires asynchronous sub-domain interfaces, where asynchronous here means flux continuity is not imposed across sub-domain interfaces. In this work, we present the development, analysis, and testing of hybrid methods which utilize both PBJ-type methods and synchronous sweep-based methods in a single iterative step. These methods are developed to reduce the penalty to iterative performance of the former incurred in problems containing optically thin cells, while maintaining the PBJ method's applicability to massively parallel solution on unstructured grids. As a prelude to this ultimate goal, this study is conducted in serial operation on 2-D Cartesian grids to facilitate implementation of the various tested iterative methods, to enable the reported spectral analysis, and for shorter computational execution times allowing the large amounts of performance data to be collected. The analysis consists of two Fourier analyses, demonstrating the sequential execution of a PBJ method and SI to be iteratively robust with respect to optical thickness. The results of these analyses and preliminary computational experimentation demonstrate the effectiveness of methods that execute PBJ and SI only in optically thick and thin regions of a problem, respectively. Further experimentation, including a parametric study, then demonstrates this approach to be effective when the thin region is solved using Inexact Parallel Block Jacobi (IPBJ), with sub-domains of a size such that the applicability to massively parallel solution on unstructured grids is not diminished. Finally, testing on realistic test problems containing large void regions confirms the viability of this approach. Performance comparisons include the traditional iterative acceleration methods Diffusion Synthetic Acceleration, Partial-current Nonlinear Diffusion Acceleration, and Adjacent-cell Preconditioner. The results show that while the hybrid PBJ methods generally consume more iterations than these traditional methods, our hybrid approach maintains an iteration count on the same order of magnitude as these acceleration methods, which are not fully developed for massively parallel solution on unstructured grids.