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

The lattice Boltzmann method (LBM) is known to have stability deficiencies. For example, local blow-ups and spurious oscillations are readily observed when the method is used to model high-Reynolds fluid flow. Beginning from thermodynamic considerations, the LBM can be recognised as a discrete dynamical system generated by entropic involution and free-flight and the stability analysis is more natural. In this paper we solve the stability problem of the LBM on the basis of this thermodynamic point of view. The main instability mechanisms are identified. The simplest and most effective receipt for stabilisation adds no artificial dissipation, preserves the second-order accuracy of the method, and prescribes coupled steps: to start from a local equilibrium, then, after free-flight, perform the overrelaxation collision, and after a second free-flight step go to new local equilibrium. 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 LBMs are provided by the numerical simulation of a 1D shock tube and the unsteady 2D-flow around a square-cylinder up to Reynolds number O(10000)