Second-order discrete ordinate PL equations in multi-dimensional geometry

In this paper, discrete ordinate neutron transport equations which are equivalent to the P_L approximation are derived in order to eliminate the current problem in the numerical solutions for the transport equation. The number of the unknowns which remain to be solved in the discrete ordinate equations is consistent with that of the independent basic functions appearing in the P_L solution.

First, for even P_L approximation in one-dimensional slab geometry, a system of equations which are satisfied by the L unknowns is derived by eliminating the L-th Legendre moment.

For multi-dimensional geometry, discrete ordinate equations of the second order which are equivalent to the P_L approximation are proposed by following Davis' method for the derivation of the even-parity second-order form of the odd P_L equations. In this case, the total number of the quadrature points is just equal to that of the independent basic functions.

Finally, the boundary conditions of the P_L approximation in x-y geometry are transformed into those for the corresponding discrete ordinate equations. As for the boundary conditions, material interface, reflecting and vacuum boundary conditions are considered.