Stability of lattice Boltzmann methods in hydrodynamic regimes

Abstract

The Von Neumann linear stability theory as applied by Sterling and Chen [J. Comput. Chem. 123, 196 (1996)] to lattice Boltzmann numerical methods is revisited and extended. A simplifying assumption made by these authors (on the character of the most unstable mode) is abandoned and immediate refinements on their stability results are attained. The inadequacy of uniform background flow, as a general point of expansion, is evident from simulations of simple shear waves. The stability theory is consequently extended to address the destabilizing role of ``background'' shear. To this end, exact time-dependent solutions of the nine-velocity Bhatnagar-Gross-Krook (BGK) lattice Boltzmann model (LB9) are derived and used as expansion points for the stability theory. Calculations reveal both physical and nonphysical instabilities, the former being interpreted via classical inviscid stability theory and the latter forming an empirical instability criterion (fitting better at small values of the viscosity), $N<{R}^{0.56}$, where $N$ is the number of mesh points in the shearing direction and $R$ is the flow Reynolds number. This is interestingly close to the Kolmogorov-Batchelor-Kraichnan inertial range cutoff ${R}^{1/2}$ for two-dimensional isotropic turbulence. In this case, stability seems to require at least the spatial resolution required for accuracy. We also note that the particular class of solutions found above for the LB9 model can be compared directly to corresponding solutions of the Navier-Stokes equations. It is demonstrated that setting $\ensuremath{\tau}=1$, where $\ensuremath{\tau}$ is the relaxation time of the BGK collision operator, provides optimal accuracy in time. This observation may be relevant to current studies as letting $\ensuremath{\tau}\ensuremath{\rightarrow}1/2$ appears to be a common technique aimed at lowering the viscosity and thereby increasing the Reynolds number of LB simulations.