  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.
