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Non-negativity and stability analyses of lattice Boltzmann method for advection-diffusion equation. (English) Zbl 1245.76119
Summary: Stability is one of the main concerns in lattice Boltzmann method (LBM). The objectives of this study are to investigate the linear stability of lattice Boltzmann equation with the Bhatnagar-Gross-Krook collision operator (LBGK) for the advection-diffusion equation, and to understand the relationship between the stability of LBGK and non-negativity of equilibrium distribution functions (EDFs). This study conducted linear stability analysis on the LBGK, whose stability depends on the lattice Peclet number, the Courant number, the single relaxation time, and the flow direction. The von Neumann analysis was applied to delineate the stability domains by systematically varying these parameters. Moreover, the dimensionless EDFs were analyzed to identify non-negative domains of dimensionless EDFs. As a result, this study obtained linear stability and non-negativity domains for three different lattices with linear and second-order EDFs. It was found that the second-order EDFs have larger stability and non-negativity domains than the linear EDFs, and outperform linear EDFs in terms of stability and numerical dispersion. Furthermore, the non-negativity of the EDFs is a sufficient condition for linear stability and becomes a necessary condition when the relaxation time is very close to 0.5. The stability and non-negativity domains provide useful information to guide the selection of dimensionless parameters to obtain stable LBM solutions. We use mass transport problems to demonstrate the consistence between theoretical findings and LBM solutions.

##### MSC:
 76M28 Particle methods and lattice-gas methods 76R99 Diffusion and convection 80A20 Heat and mass transfer, heat flow (MSC2010)
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