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

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Abstract

Despite the growing popularity of Lattice Boltzmann schemes for describing multi-dimensional flow and transport governed by non-linear (anisotropic) advection-diffusion 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 sub-classes, 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 sub-class 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.

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Authors and Affiliations

  1. Cemagref, Antony Regional Centre, HBAN, Parc de Tourvoie BP 44, 92163, Antony Cedex, France

    Irina Ginzburg

  2. 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

    Dominique d’Humières

  3. Mechanical and Manufacturing Engineering, Schulich School of Engineering, University of Calgary, Calgary, T2N 1N4, AB, Canada

    Alexander Kuzmin

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  1. Irina Ginzburg
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  2. Dominique d’Humières
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  3. Alexander Kuzmin
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Correspondence to Irina Ginzburg.

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Ginzburg, I., d’Humières, D. & Kuzmin, A. Optimal Stability of Advection-Diffusion Lattice Boltzmann Models with Two Relaxation Times for Positive/Negative Equilibrium. J Stat Phys 139, 1090–1143 (2010). https://doi.org/10.1007/s10955-010-9969-9

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