An analysis of over-relaxation in kinetic approximation

TL;DR

This paper analyzes the over-relaxation kinetic scheme, providing insights into its equivalent equations to improve stability and accuracy in kinetic methods like Lattice-Boltzmann.

Contribution

It offers a detailed analysis of the over-relaxation scheme and derives its equivalent equation to aid in developing stable boundary conditions.

Findings

Derived the equivalent equation of the over-relaxation scheme

Provided stability insights for boundary condition design

Enhanced understanding of kinetic approximation accuracy

Abstract

The over-relaxation approach is an alternative to the Jin-Xin relaxation method (Jin and Xin [1]) in order to apply the equilibrium source term in a more precise way (Coulette et al. [2, 3]). This is also a key ingredient of the Lattice-Boltzmann method for achieving second order accuracy (Dellar [4]). In this work we provide an analysis of the over-relaxation kinetic scheme. We compute its equivalent equation, which is particularly useful for devising stable boundary conditions for the hidden kinetic variables.