Fourier Analysis of Inexact Parallel Block Jacobi with Source Iteration Preconditioning and Computational Experimentation with Parallel Block Jacobi / Source Iteration Hybrid Methods

摘要

Parallel Block Jacobi (PBJ) iterative methods used in transport calculations experienceiterative slowdown in problems with optically thin spatial cells due to asynchronicity.Previously we tested hybrid methods which execute both Parallel Block Jacobi – IntegralTransport Matrix Method (PBJ-ITMM) and Source Iterations (SI) interchangeably and foundthis to effectively accelerate PBJ-ITMM. This acceleration was conjectured to be a result ofthe most prevalent phenomena in optically thin and thick cells being streaming and collision,respectively, with SI able to model the streaming and PBJ-ITMM able to resolve localcollisions. In this work, we support this claim and further investigate its implications byperforming Fourier analysis on Inexact Parallel Block Jacobi (IPBJ) executed in series withSI. Our analysis establishes that lagging the scattering source in IPBJ causes the spectral radiusof the P-IP-SI method (IPBJ preconditioned with SI) to asymptote to c~2 with increasing celloptical thickness, where c is the scattering ratio, far exceeding the spectral radius of P-PI-SI(PBJ-ITMM preconditioned with SI) in thick cells. Upon theoretically demonstrating theadvantage of using P-PI-SI over P-IP-SI, we perform computational experiments on bothmethods. These experiments show that asynchronous combination of PBJ-ITMM and SI (AHPI-SI) is the most effective PBJ / SI hybrid method among considered options.