Stabilisation of the lattice-Boltzmann method using the Ehrenfests' coarse-graining

TL;DR

This paper introduces a novel stabilization technique for the lattice-Boltzmann method using Ehrenfests' coarse-graining, effectively suppressing post-shock oscillations in fluid simulations.

Contribution

A simple stabilization method for LBM using entropy-based monitoring and Ehrenfests' steps, improving numerical stability near shocks.

Findings

Ehrenfests' steps suppress post-shock oscillations.

Method improves stability in shock tube simulations.

Outperforms other LBM variants in stability.

Abstract

The lattice-Boltzmann method (LBM) and its variants have emerged as promising, computationally efficient and increasingly popular numerical methods for modelling complex fluid flow. However, it is acknowledged that the method can demonstrate numerical instabilities, e.g., in the vicinity of shocks. We propose a simple and novel technique to stabilise the lattice-Boltzmann method by monitoring the difference between microscopic and macroscopic entropy. Populations are returned to their equilibrium states if a threshold value is exceeded. We coin the name Ehrenfests' steps for this procedure in homage to the vehicle that we use to introduce the procedure, namely, the Ehrenfests' idea of coarse-graining. The one-dimensional shock tube for a compressible isothermal fluid is a standard benchmark test for hydrodynamic codes. We observe that, of all the LBMs considered in the numerical experiment with the one-dimensional shock tube, only the method which includes Ehrenfests' steps is capable of suppressing spurious post-shock oscillations.