Lattice Boltzmann equation linear stability analysis: Thermal and athermal models
D. N. Siebert, L. A. Hegele Jr., and P. C. Philippi

Although several thermal lattice Boltzmann models have been proposed, this method has not yet been shown to be able to describe non-isothermal fully compressible flows in a satisfactory manner, mostly due to the presence of important deviations from the advection-diffusion macroscopic equations and also due to numerical instabilities. In this context, this paper presents a linear stability analysis for some Lattice Boltzmann models that were recently derived as discrete forms of the continuous Boltzmann equation (Philippi, P.C., Hegele Jr., L.A., dos Santos, L.O.E., Surmas, R., Physical Review E 63 (5), 056702, 2006), in order to investigate the sources of instability and to find, for each model, the upper and lower limits for the macroscopic variables, between which it is possible to ensure a stable behavior. The results for two-dimensional lattices with 9, 17, 25 and 37 velocities indicate that increasing the order of approximation of the lattice-Boltzmann equation enhances stability. Results are also presented for an athermal 2D nine-velocity model, the accuracy of which has been improved with respect to the standard D2Q9 model, by adding third-order terms in the LBE.

© 2008 The American Physical Society.