Analysis of difference schemes for the Fokker-Planck angular diffusion operator

TL;DR

This paper analyzes finite difference schemes for the Fokker-Planck angular diffusion operator, explaining convergence issues and establishing conditions for second-order accuracy, while revealing new properties of Gaussian nodes and weights.

Contribution

It provides a mathematical analysis of convergence issues in discrete ordinates methods and introduces conditions for achieving second-order accuracy in angular diffusion schemes.

Findings

Identifies reasons for lack of convergence in half range mode

Establishes sufficient conditions for second-order convergence

Discovers new properties of Gaussian nodes and weights

Abstract

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.