Time-eigenvalue calculations in multi-region Cartesian geometry using Green's functions

We have continued the progression of providing benchmark-quality eigenvalue calculations in multi-region Cartesian geometry by reformulating the effective multiplication equations to calculate time eigenvalues. As with effective multiplication eigenvalues, there are few benchmark-quality solutions for time-eigenvalue calculations in multi-region multiplying systems, especially for systems that have divergent temporal neutron populations. The purpose of this paper is to describe the reformulations required and to add benchmark-quality calculations for several test problems. Green's functions are used to model a multi-region, one-group, isotropically scattering, multiplying system in Cartesian geometry to obtain boundary flux values for a time-eigenvalue search and subsequent eigenfunction calculation. As usual with multi-region Cartesian systems, the solution is facilitated using (1) Placzek's lemma, which allows us to consider a multi-region system one region at a time as an infinite medium, and (2) the calculation of the Green's function solution for a nonphysical infinite multiplying medium.