This paper is dedicated to the mathematical analysis of finite difference schemes for the angular diffusion operator present in the azimuth-independent Fokker-Planck equation. The study elucidates the reasons behind the lack of convergence in half range mode for certain widely recognized discrete ordinates methods, and establishes sets of sufficient conditions to ensure that the schemes achieve convergence of order $2$. In the process, interesting properties regarding Gaussian nodes and weights, which until now have remained unnoticed by mathematicians, naturally emerge.

翻译：本文致力于对与方位角无关的Fokker-Planck方程中角度扩散算子的有限差分格式进行数学分析。该研究阐明了某些广泛使用的离散纵标方法在半区间模式下缺乏收敛性的原因，并建立了一组充分条件，以确保这些格式达到二阶收敛。在此过程中，关于高斯节点与权重的有趣性质自然显现，而这些性质迄今为止尚未被数学家们注意到。