Singular Difference Approximations for the Discrete Ordinate Equations in x-y Geometry

Two well-known finite difference approximations to the discrete ordinate equations in x-y geometry are shown to lead to a singular system of equations for the case of reflecting boundary conditions. These difference schemes are the diamond approximation of Carlson, and the central difference approximation. Despite this singularity it is shown for the diamond scheme that a solution always exists and is, in some sense, unique. For the central difference scheme, however, it is shown that a solution need not, and in most cases will not, exist.