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

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 $\mathcal{O}(10000)$.

Significantly extended version with new LBM accuracy analysis and extended bibliography

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• Entropy Balance and Dispersive Oscillations in Lattice Boltzmann Models
• Efficient supersonic flows through high-order guided equilibrium with lattice Boltzmann
• A hybrid multi-time-step framework for pore-scale and continuum-scale modeling of solute transport in porous media
• Stable simulation of fluid flow with high-Reynolds number using Ehrenfests' steps