Truncation Errors, Exact and Heuristic Stability Analysis of Two-Relaxation-Times Lattice Boltzmann Schemes for Anisotropic Advection-Diffusion Equation (Communications in Computational Physics, 2012)
This paper establishes relations between the stability and the high-ordertruncated corrections for modeling of the mass conservation equation with the two-relaxation-times (TRT) collision operator. First we propose a simple method to derivethe truncation errors from the exact, central-difference type, recurrence equations ofthe TRT scheme. They also supply its equivalent three-time-level discretization form.Two different relationships of the two relaxation rates nullify the third (advection) andfourth (pure diffusion) truncation errors, for any linear equilibrium and any velocity set. However, the two relaxation times alone cannot remove the leading-orderadvection-diffusion error, because of the intrinsic fourth-order numerical diffusion.The truncation analysis is carefully verified for the evolution of concentration waveswith the anisotropic diffusion tensors. The anisotropic equilibrium functions are presented in a simple but general form, suitable for the minimal velocity sets and thed2Q9, d3Q13, d3Q15 and d3Q19 velocity sets. All anisotropic schemes are complemented by their exact necessary von Neumann stability conditions and equivalentfinite-difference stencils. The sufficient stability conditions are proposed for the moststable (OTRT) family, which enables modeling at any Peclet numbers with the samevelocity amplitude. The heuristic stability analysis of the fourth-order truncated corrections extends the optimal stability to larger relationships of the two relaxation rates,in agreement with the exact (one-dimensional) and numerical (multi-dimensional) stability analysis. A special attention is put on the choice of the equilibrium weights. Bycombining accuracy and stability predictions, several strategies for selecting the relaxation and free-tunable equilibrium parameters are suggested and applied to theevolution of the Gaussian hill.