First-order quarter- and mixed-moment realizability theory and Kershaw closures for a Fokker-Planck equation in two space dimensions

TL;DR

This paper extends mixed-moment models to two dimensions for a Fokker-Planck equation, providing realizability theory and efficient Kershaw closures that improve upon existing models.

Contribution

It generalizes mixed-moment models to two space dimensions, develops a realizability theory, and derives Kershaw closures for improved efficiency.

Findings

Hyperbolic system with desirable properties

Realizability linked to quarter-moment theory

Effective closures tested on benchmark problems

Abstract

Mixed-moment models, introduced before for one space dimension, are a modification of the method of moments applied to a (linear) kinetic equation, by choosing mixtures of different partial moments. They are well-suited to handle such equations where collisions of particles are modelled with a Laplace-Beltrami operator. We generalize the concept of mixed moments to two dimension. The resulting hyperbolic system of equations has desirable properties, removing some drawbacks of the well-known $\MN[1]$ model. We furthermore provide a realizability theory for a first-order system of mixed moments by linking it to the corresponding quarter-moment theory. Additionally, we derive a type of Kershaw closures for mixed- and quarter-moment models, giving an efficient closure (compared to minimum-entropy models). The derived closures are investigated for different benchmark problems.