Criterion of numerical instability of liquid state in LBE simulations

Abstract

The numerical stability of the lattice Boltzmann equation (LBE) method in simulations of a fluid described by an equation of state with possible vapor–liquid phase transitions is considered. The Courant–Friedrichs–Lewy number defined by the advection term in the Boltzmann equation is exactly equal to unity in classical LBE models. However, this condition does not ensure the numerical stability of LBE simulations with the equation of state. In our numerical LBE simulations, we find out that instability arises initially in the liquid phase, even if the vapor phase and, consequently, the vapor–liquid interface are absent. We demonstrate both in numerical tests and theoretically that the numerical stability of LBE simulations requires the criterion c ̃ ≤ c ̃ cr to be fulfilled for the liquid phase, where c ̃ = c s Δ t / h is the hydrodynamic Courant number. The hydrodynamic Courant number is proportional to the speed of sound c s , obtained from an equation of state of a fluid. This criterion is very similar to the well-known criteria of numerical stability of explicit finite difference schemes for a compressible fluid. The critical value of the Courant number c ̃ cr depends neither on the temperature T , nor on the fluid velocity, nor on the form of the equation of state. This critical value is equal to c ̃ cr = 1.1547 for the kinetic temperature of LBE pseudo-particles θ ̃ = 1 / 3 .