The Even-Parity and Simplified Even-Parity Transport Equations in Two-Dimensional x-y Geometry

The finite element and lumped finite element methods for the spatial differencing of the even-parity discrete ordinates neutron transport equations (EPS_N) in two-dimensional x-y geometry are applied. In addition, the simplified even-parity discrete ordinates equations (SEPS_N) as an approximation to the EPS_N transport equations are developed. The SEPS_N equations are more efficient to solve than the EPS_N equations due to a reduction in angular domain of one-half, the applicability of a simple five-point diffusion operator, and directionally uncoupled reflective boundary conditions. Furthermore, the SEPS_N equations satisfy the same diffusion limits as EPS_N in an optically thick regime, appear to have no ray effect, and converge faster than EPS_N when using a diffusion synthetic acceleration (DSA). Also, unlike the case of EPS_N, the SEPS_N solutions are strictly positive, thus requiring no negative flux fixups. It is also demonstrated that SEPS_N is a generalization of the simplified P_N method. Most importantly, in these second-order approaches, an unconditionally effective DSA scheme can be achieved by simply integrating the differenced EPS_N and SEPS_N equations over the angles. It is difficult to obtain a consistent DSA scheme with the first-order S_N equations. This is because a second-order DSA equation must generally be derived directly from the differenced first-order S_N equations.