Perfect entropy functions of the Lattice Boltzmann method

– In this letter, we derive entropy functions whose local equilibria are suitable to recover the Navier-Stokes equations in the framework of the Lattice Boltzmann method. For the two-dimensional nine-velocity lattice we demonstrate that such an entropy function is unique, and that the expansion of the corresponding local equilibrium is the well-known local equilibrium of Y. H. Qian et al. (Europhys. Lett., 17 (1992) 479). Based on the knowledge of entropy functions, we introduce a new version of the Lattice Boltzmann method with an H-theorem built in. Since the pioneering work on the Lattice Gas [1], the lattice-based approach to simulation of hydrodynamics received considerable attention over the past decade. In the widely used Lattice Boltzmann method [2], one considers populations of fictitious particles, Ni(r, t), where i = 1, . . . , b labels discrete velocities ci. The set of discrete velocities, which can also include a zero vector (“rest population”), is associated with outgoing links at each site r of a regular isotropic lattice. Populations are updated at discrete time steps t according to an equation, Ni(r + ci, t+ 1)−Ni(r, t) = ∆i . (1) In the following, we restrict our attention to the isothermal Navier-Stokes equation. For that case, the collision integral ∆i must obey only the local conservation laws ∑b i=1{1, ciα}∆i = 0 for the local hydrodynamic fields, i.e., the density ρ = ∑b i=1Ni(r, t), and momentum ρuα = ∑b i=1 ciαNi(r, t). Here α = 1, . . . , d label the Cartesian components of d-dimensional vectors. If the long-time large-scale limit of eq. (1) recovers the Navier-Stokes equation, then hydrodynamics is implemented in a fairly simple, fully discrete kinetic picture. In the following, we denote N as the b-dimensional vector of populations. An important part of any realization of the Lattice Boltzmann method is the problem of the local equilibrium N. From the perspective of classical kinetic theory, local equilibria are found as minimum points of a convex function H(N), subject to constraints fixed by the hydrodynamic fields,