Asymptotic preserving methods for the low mach limit in discrete velocity models approximating kinetic equations

TL;DR

This paper introduces an asymptotic preserving numerical scheme for discrete velocity models in kinetic equations, ensuring accuracy and stability from kinetic to hydrodynamic scales, validated through numerical experiments.

Contribution

The authors develop a high-order, asymptotic preserving scheme combining IMEX Runge Kutta and WENO reconstructions for kinetic equations in low Mach regimes, bridging kinetic and fluid models.

Findings

Scheme remains stable and accurate across regimes

Reduces to incompressible Navier-Stokes in the limit

Numerical experiments confirm theoretical properties

Abstract

We consider a Lattice Boltzmann type discrete velocity model in the low Mach number scaling and develop a corresponding numerical scheme that remains uniformly valid across all regimes of the mean free path, from the kinetic to the hydrodynamic scale. The proposed framework ensures high order temporal accuracy through the use of Implicit Explicit Runge Kutta methods, which provide stability and efficiency in stiff regimes, while spatial resolution is enhanced by combining finite difference WENO reconstructions with high order central difference approximations. In the appropriate asymptotic limit, the scheme reduces to a high order finite difference formulation of the incompressible Navier Stokes equations, thereby guaranteeing physical consistency of the numerical approximation with the limit model. To corroborate the theoretical findings, a set of numerical experiments is performed on two dimensional benchmark problems, which confirm the accuracy, stability, and versatility of the method across different flow regimes.