On the Ronen Method in Simple 1-D Geometries for Neutron Transport Theory Solutions

Abstract

In this work, we apply the Ronen method to obtain highly-accurate approximations to the solution of the neutron transport equation in simple homogeneous problems. Slab, cylindrical, and spherical geometries are studied. This method demands successive resolutions of the diffusion equation, where the local diffusion constants are modified in order to reproduce new estimates of the currents by a transport operator. The diffusion solver employs here finite differences and the transport-corrected currents are forced in the numerical scheme by means of drift terms, like in the CMFD scheme. Boundary conditions are discussed introducing proper approximations to save the particle balance in case of reflection in the slab. The solution from the Ronen iterations is compared against reference results provided by the collision probability method. More accurate estimates of the currents are provided by integral transport equations using first flight escape probabilities. Slow convergence on the scalar flux is analyzed, although the results match the reference solutions in the limit of fine meshes and far from the bare boundary.