The Quadrature Discretization Method (QDM) in comparison with other numerical methods of solution of the Fokker–Planck equation for electron thermalization

The determination of the relaxation of electrons in atomic gases continues to be an important physical problem. The main interest is the determination of the time scale for the thermalization of electrons in different moderators and the nature of the time-dependent electron energy distribution. The theoretical basis for the study of electron thermalization is the determination of the electron distribution function from a solution of the Lorentz– Fokker–Planck equation. The present paper considers a detailed comparison of different numerical methods of solution of the Lorentz–Fokker–Planck equation for the electron distribution function. The methods include a pseudospectral method referred to as the Quadrature Discretization Method (QDM) which is based on non-standard polynomial basis sets, a finite-difference method, and a Lagrange interpolation method. The Fokker–Planck equation can be transformed to a Schrödinger equation, and methods developed for the solution of either equation apply to the other.