Stability and stabilisation of the lattice Boltzmann method: Magic steps and salvation operations

TL;DR

This paper analyzes the stability and accuracy of the lattice Boltzmann method, proposing conditions and stabilisation techniques to achieve second-order accuracy without artificial dissipation, validated through numerical simulations.

Contribution

It establishes necessary and sufficient conditions for second-order accuracy in LBM and introduces stabilisation recipes that maintain stability without artificial dissipation.

Findings

Identifies the invariant film as key to second-order accuracy.

Proposes stabilisation methods preserving positivity and accuracy.

Demonstrates stability up to Reynolds number 10000 in simulations.

Abstract

We revisit the classical stability versus accuracy dilemma for the lattice Boltzmann methods (LBM). Our goal is a stable method of second-order accuracy for fluid dynamics based on the lattice Bhatnager--Gross--Krook method (LBGK). The LBGK scheme can be recognised as a discrete dynamical system generated by free-flight and entropic involution. In this framework the stability and accuracy analysis are more natural. We find the necessary and sufficient conditions for second-order accurate fluid dynamics modelling. In particular, it is proven that in order to guarantee second-order accuracy the distribution should belong to a distinguished surface -- the invariant film (up to second-order in the time step). This surface is the trajectory of the (quasi)equilibrium distribution surface under free-flight. The main instability mechanisms are identified. The simplest recipes for stabilisation add no artificial dissipation (up to second-order) and provide second-order accuracy of the method. Two other prescriptions add some artificial dissipation locally and prevent the system from loss of positivity and local blow-up. Demonstration of the proposed stable LBGK schemes are provided by the numerical simulation of a 1D shock tube and the unsteady 2D-flow around a square-cylinder up to Reynolds number $\mathcal{O}(10000)$.