Abstract

To determine the angular discretization error asymptotic convergence rate of the uncollided scalar flux computed with the Discrete Ordinates (S${ }_N$) method, a comprehensive theory of the regularity order with respect to the azimuthal angle of the exact pointwise S_N 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 $S_N$ 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.

摘要

要确定使用离散纵坐标（S${ }_{ñ}$）方法，基于二维笛卡尔几何中的输运方程的积分形式，得出了关于精确的点向S _N非碰撞角通量的方位角的正则规律的综合理论。用这个理论，${\mathrm{小号}}_{ñ}$对于给定的问题配置，可以估计未碰撞的角通量。我们的新理论启发了一种新颖的Modified Simpson's（MS）正交函数，该函数通过避免在方位角上的不规则点之间进行积分，从而比任何传统正交函数更快地收敛了非碰撞标量通量。数值结果成功地验证了我们在测试配置的四个变体中的新理论，并且发现使用常规角正交类型和新正交类型计算的相应标量通量中的角离散误差收敛于不同阶次。传统正交类型获得的误差收敛速度受到精确角通量和正交积分间隔的规律性顺序的限制，而我们的新型MS正交类型的收敛速度是传统正交类型的2至4倍。通过引入不规则间隔的有效长度和相关的振荡函数，可以详细研究在某些正交误差中观察到的振荡。