A Fourier analysis is conducted for the discrete ordinates, or SN, approximation of the neutron transport problem solved with Richardson iteration (source iteration) and Richardson iteration preconditioned with transport synthetic acceleration (TSA), using the inexact parallel block-Jacobi (IPBJ) algorithm both in slab and two-dimensional Cartesian geometry. Both traditional, or “beta,” TSA (TTSA) and a modified TSA (MTSA), in which only the scattering in the low-order equations is reduced by some nonnegative factor < 1, are considered.
The results for the unaccelerated algorithm show that convergence of IPBJ can degrade, leading in particular to stagnation of the generalized minimum residual method with restart parameter m, GMRES(m), in problems containing optically thin subdomains. The IPBJ algorithm preconditioned with TTSA can be effective, provided the parameter is properly tuned for a given scattering ratio c, but is potentially unstable. Compared to TTSA, MTSA is less sensitive to the choice of , more effective for the same computational effort, measured in terms of the effective scattering ratio c′, and it is unconditionally stable.