Convergence of inner/outer source iterations with finite termination of the inner iterations

Abstract

A two-stage (nested) iteration strategy, in which the outer iteration is analogous to a block Gauss-Seidel method and the inner iteration to a Jacobi method for each of these blocks, often are used in the numerical solution of discretized approximations to the neutron transport equation. This paper is concerned with the effect, within a continuous space model, of errors from finite termination of the inner iterations upon the overall convergence. The main result is that convergence occurs, to the solution of the original problem, in the limit that both the outer iteration index and the minimum over all outer indices and all groups ("blocks") jointly become large. Positivity properties (in the sense of cone preserving) are used extensively.