Exact Lattice Boltzmann Equation

The lattice Boltzmann equation is derived from the Bhatnagar-Gross-Krook kinetic equation using the Euler-Maclaurin integration formula. Unlike previous attempts to connect the lattice Boltzmann method with the kinetic theory, the result is free of any relaxation-type approximation. It shows that the conventional lattice Bhatnagar-Gross-Krook simulations of hydrodynamics belong to a parameter domain which is disconnected from the kinetic theory domain.

Figure 1

Lattice Boltzmann equations as a map of the kinetic-theory parameter $\lambda =\Delta t/\tau$ into the LB interval $\beta \in [0,1]$ (2). Curve 1, exact (21); curve 2, HCD extension (11); curves 3–6, polynomial extensions ${\phi }_n$ at various $n$: 3, ${\phi }_1$ (${\lambda }_1=2$); 4, ${\phi }_2$; 5, ${\phi }_3$ (${\lambda }_1\approx 2.51$); 6, ${\phi }_9$ (${\lambda }_1\approx 4.70$). $\beta =0$ corresponds to populations $g_i$; $\beta =1/2$ corresponds to the equilibrium populations $g_i^{\mathrm{eq}}$; $\beta =1$ corresponds to the mirror populations $g_i^{*}=2g_i^{\mathrm{eq}}-g_i$; overrelaxation range is $1/2<\beta <1$; shadowed strip $\beta \in [0.99,1]$ indicates typical operation domain of LB simulations. Curves 2, 3, 5, and 6 extend into the overrelaxation subinterval $\beta \in [1/2,1]$.