STABILITY OF LATTICE BOLTZMANN METHODS IN HYDRODYNAMIC REGIMES

The Von Neumann linear stability theory as applied by Sterling and Che n [J. Comput. Chem. 123, 196 (1996)] to lattice Boltzmann numerical me thods is revisited and extended. A simplifying assumption made by thes e authors (on the character of the most unstable mode) is abandoned an d immediate refinements on their stability results are attained. The i nadequacy of uniform background flow, as a general point of expansion, is evident from simulations of simple shear waves. The stability theo ry is consequently extended to address the destabilizing role of ''bac kground'' shear. To this end, exact time-dependent solutions of the ni ne-velocity Bhatnagar-Gross-Krook (BGK) lattice Boltzmann model (LB9) are derived and used as expansion points for the stability theory. Cal culations reveal both physical and nonphysical instabilities, the form er being interpreted via classical inviscid stability theory and the l atter forming an empirical instability criterion (fitting better at sm all values of the viscosity), N < R-0.56, where N is the number of mes h points in the shearing direction and R is the dow Reynolds number. T his is interestingly close to the Kolmogorov-Batchelor-Kraichnan inert ial range cutoff R-1/2 for two-dimensional isotropic turbulence. In th is case, stability seems to require at least the spatial resolution re quired for accuracy. We also note that the particular class of solutio ns found above for the LB9 model can be compared directly to correspon ding solutions of the Navier-Stokes equations. It is demonstrated that setting tau = 1, where tau is the relaxation time of the BGK collisio n operator, provides optimal accuracy in time. This observation may be relevant to current studies as letting tau --> 1/2 appears to be a co mmon technique aimed at lowering the viscosity and thereby increasing the Reynolds number of LB simulations.