Numerical solution of the Boltzmann equation in cylindrical geometry

A numerical procedure that provides an accurate solution of the Boltzmann equation in cylindrical geometry with coordinates ( ρ,v→) is discussed. Statistical methods such as Monte Carlo can be used but suffer from statistical noise and thus do not resolve low density regions well. Furthermore, the slow speed of pure Monte Carlo methods makes self-consistent simulations quite difficult. A direct solution of the Boltzman equation avoids these difficulties but suffers from errors due to finite size mesh effects. In this work we examine a solution method, based on the convected scheme, that eliminates some specific sources of numerical diffusion in cylindrical geometry. The velocity is represented as (${\mathit{v}}_{\mathit{z}}$,${\mathit{v}}_{\mathrm{\perp }}$,scrM), where scrM is a moment arm or ‘‘reduced’’ angular momentum, scrM=ρsinφ, and φ is an azimuthal angle in velocity space (referenced to ρ^). The reason for all the coordinate choices are discussed. Propagator algorithm(s) for solving the kinetic equation are presented which remove certain numerical errors. Examples of the performance of the algorithm(s) under various conditions are presented and discussed. A self-consistent kinetic model of a dc positive column is described.