Higher order mixed moment approximations for the Fokker-Planck equation in one space dimension

TL;DR

This paper introduces mixed-moment minimum-entropy models and Kershaw closures for the one-dimensional Fokker-Planck equation, improving the modeling of distribution functions and overcoming zero net-flux issues.

Contribution

The paper develops realizability theory and a new analytical closure called Kershaw closures for mixed moments in Fokker-Planck models, enhancing model accuracy and positivity.

Findings

Kershaw closures provide non-negative distribution functions.

Numerical tests show improved performance over standard schemes.

Models overcome the zero net-flux problem of full-moment approaches.

Abstract

We study mixed-moment models (full zeroth moment, half higher moments) for a Fokker-Planck equation in one space dimension. Mixed-moment minimum-entropy models are known to overcome the zero net-flux problem of full-moment minimum entropy Mn models. Realizability theory for these mixed moments of arbitrary order is derived, as well as a new closure, which we refer to as Kershaw closures. They provide non-negative distribution functions combined with an analytical closure. Numerical tests are performed with standard first-order finite volume schemes and compared with a finite-difference Fokker-Planck scheme.