Optimal Stability of Advection-Diffusion Lattice Boltzmann Models with Two Relaxation Times for Positive/Negative Equilibrium

Despite the growing popularity of Lattice Boltzmann schemes for describing multi-dimensional flow and transport governed by non-linear (anisotropic) advectiondiffusion equations, there are very few analytical results on their stability, even for the isotropic linear equation. In this paper, the optimal two-relaxation-time (OTRT) model is defined, along with necessary and sufficient (easy to use) von Neumann stability conditions for a very general anisotropic advection-diffusion equilibrium, in one to three dimensions, with or without numerical diffusion. Quite remarkably, the OTRT stability bounds are the same for any Peclet number and they are defined by the adjustable equilibrium parameters. Such optimal stability is reached owing to the free (“kinetic”) relaxation parameter. Furthermore, the sufficient stability bounds tolerate negative equilibrium functions (the distribution divided by the local mass), often labeled as “unphysical”. We prove that the non-negativity condition is (i) a sufficient stability condition of the TRT model with any eigenvalues for the pure diffusion equation, (ii) a sufficient stability condition of its OTRT and BGK/SRT subclasses, for any linear anisotropic advection-diffusion equation, and (iii) unnecessarily more restrictive for any Peclet number than the optimal sufficient conditions. Adequate choices of the two relaxation rates and the free-tunable equilibrium parameters make the OTRT subclass more efficient than the BGK one, at least in the advection-dominant regime, and allow larger time steps than known criteria of the forward time central finite-difference schemes (FTCS/MFTCS) for both, advection and diffusion dominant regimes. I. Ginzburg ( ) Cemagref, Antony Regional Centre, HBAN, Parc de Tourvoie BP 44, 92163 Antony Cedex, France e-mail: [email protected] D. d’Humières Laboratoire de Physique Statistique, École Normale Supérieure, Associated to CNRS, Pierre and Marie Curie and D. Diderot Universities, 24 Rue Lhomond, 75231 Paris Cedex 05, France e-mail: [email protected] A. Kuzmin Mechanical and Manufacturing Engineering, Schulich School of Engineering, University of Calgary, Calgary, T2N 1N4 AB, Canada e-mail: [email protected] Optimal Stability of Advection-Diffusion Lattice Boltzmann Models 1091