The Even-Parity and Simplified Even-Parity Transport Equations in Two-Dimensionalx-yGeometry

Abstract

The finite element and lumped finite element methods for the spatial differencing of the even-parity discrete ordinates neutron transport equations (EPS{sub N}) in two-dimensional x-y geometry are applied. In addition, the simplified even-parity discrete ordinates equations (SEPS{sub N}) as an approximation to the EPS{sub N} transport equations are developed. The SEPS{sub N} equations are more efficient to solve than the EPS{sub 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{sub N} equations satisfy the same diffusion limits as EPS{sub N} in an optically thick regime, appear to have no ray effect, and converge faster than EPS{sub N} when using a diffusion synthetic acceleration (DSA). Also, unlike the case of EPS{sub N}, the SEPS{sub N} solutions are strictly positive, thus requiring no negative flux fixups. It is also demonstrated that SEPS{sub N} is a generalization of the simplified P{sub N} method. Most importantly, in these second-order approaches, an unconditionally effective DSA scheme can be achieved by simply integrating the differenced EPS{sub N} and SEPS{sub N} equations over the angles. It is difficult to obtain a consistent DSA scheme with the first-order S{submore » N} equations. This is because a second-order DSA equation must generally be derived directly from the differenced first-order S{sub N} equations.« less