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