Solution of Standard Diamond Difference Equations for Discrete-Ordinate Neutron Transport Equations Equivalent to the P_L Approximation in x-y Geometry

Discrete-ordinate neutron transport equations in x-y geometry, which are equivalent to the P_L approximation, are developed for eliminating the ray effect in the usual discrete ordinate or S_N method. The standard diamond difference schemes for the discrete ordinate equations developed here are studied for vacuum and periodic boundary conditions. It is shown that the difference schemes, with an exception, lead to nonsingular systems of algebraic equations. The exception, which yields singular systems of difference equations, is the case where the following condition is satisfied: “In at least one of the x and y directions, the boundary conditions are periodic, and the number of mesh intervals is even.” It is also shown that the solutions yielded by these schemes with periodic boundary conditions converge in the L₂ norm to the solutions of the P_L equations.