@article{hu_azmy_2021, title={On the Regularity Order of the Pointwise Uncollided Angular Flux and Asymptotic Convergence of the Discrete Ordinates Approximation of the Scalar Flux}, volume={195}, ISSN={["1943-748X"]}, DOI={10.1080/00295639.2020.1860634}, abstractNote={Abstract To determine the angular discretization error asymptotic convergence rate of the uncollided scalar flux computed with the discrete ordinates (S ) method, a comprehensive theory of the regularity order with respect to the azimuthal angle of the exact pointwise SN uncollided angular flux is derived based on the integral form of the transport equation in two-dimensional Cartesian geometry. With this theory, the regularity order of the pointwise uncollided angular flux can be estimated for a given problem configuration. Our new theory inspired a novel Modified Simpson’s (MS) quadrature that converges the uncollided scalar flux faster than any of the traditional quadratures by avoiding integration across points of irregularity in the azimuthal angle. Numerical results successfully verify our new theory in four variants of a test configuration, and the angular discretization errors in the corresponding scalar flux computed with conventional angular quadrature types and with our new quadrature types are found to converge with different orders. The error convergence rates obtained with traditional quadrature types are limited by the regularity order of the exact angular flux and the quadrature’s integration intervals while our new MS quadrature types converge with order two to four times higher than traditional quadratures. A detailed study of oscillations observed in certain quadrature errors is provided by introducing the effective length of the irregular interval and the associated oscillating function.}, number={6}, journal={NUCLEAR SCIENCE AND ENGINEERING}, author={Hu, Xiaoyu and Azmy, Yousry Y.}, year={2021}, month={Jun}, pages={598–613} }
 @article{hu_azmy_2020, title={Asymptotic convergence of the angular discretization error in the scalar flux computed from the particle transport equation with the method of discrete ordinates}, volume={138}, ISSN={["0306-4549"]}, DOI={10.1016/j.anucene.2019.107199}, abstractNote={The asymptotic convergence of the angular discretization error in the scalar flux solution of the particle transport equation computed with the Discrete Ordinates (SN) method with increasing quadrature order is examined. Five selected angular quadrature types are considered: Level Symmetric (LS), Legendre-Chebyshev Quadrangular (LCQ), Legendre-Chebyshev Triangular(LCT), Quadruple Range (QR) and Quadruple Range Spence-type (QRS) quadrature sets. We relate the SN angular discretization error to the quadrature error, and split the total flux into the uncollided flux and the fully collided flux, and then we verify the uncollided scalar flux error and the fully collided scalar flux error separately. After developing the theoretical basis for the relationship between true solution regularity and quadrature rule error, we employ a two-dimensional problem to verify the observed and theoretical convergence orders for the region-averaged uncollided and fully collided scalar flux errors. Numerical results show that the angular discretization errors in the region-averaged scalar flux obtained by different quadrature types converge with different rates, that are commensurate with the regularity order of the exact angular flux within the quadrature-designed integration interval. The angular discretization error in the uncollided region-averaged scalar flux obtained by LC class quadratures converges linearly, and the error obtained by QR class quadrature sets converges quadratically. The angular discretization error in the fully collided region-averaged scalar flux converges linearly for LC class quadratures, and faster-than-second order for QR class quadrature sets.}, journal={ANNALS OF NUCLEAR ENERGY}, author={Hu, Xiaoyu and Azmy, Yousry Y.}, year={2020}, month={Apr} }