A Necessary Condition for the Convergence of Iterative Solutions to the Neutron Diffusion Equation

Linear boundary value problems of neutron diffusion theory in a symmetric region are analyzed in the context of the theory of point groups. If the symmetries of the region commute with the operator of the equation to be solved, the symmetries generate a classification of the solution space. This technique is applied to study the numerical behavior of numerical solution methods. It is shown that approximations on the boundary and inside the region must be compatible with regard to their symmetry components for stable iterative solutions. The necessary compatibility conditions are given. In the case of a regular hexagon linear approximations on the six faces require at least a sixth order approximation inside the hexagon. The predicted convergence problems if the necessary conditions are not met have been observed in the neutron physics code VARIANT, which is a routinely used neutron diffusion equation solver at ANL. (author)