A note on the Lattice Boltzmann Method beyond the Chapman-Enskog limits

– A non-perturbative analysis of the Bhatnagar-Gross-Krook (BGK) model kinetic equation for finite values of the Knudsen number is presented. This analysis indicates why discrete kinetic versions of the BGK equation, and notably the Lattice Boltzmann method, can provide semi-quantitative results also in the non-hydrodynamic, finite-Knudsen regime, up to Kn ∼ O(1). This may help the interpretation of recent Lattice Boltzmann simulations of microflows, which show satisfactory agreement with continuum kinetic theory in the moderateKnudsen regime. In the last decade, the lattice Boltzmann Method (LB) has developed into a very flexible and effective numerical method for the simulation of a large variety of complex flows, mostly in the macroscopic domain [1–3]. Fueled by relentless progress in micro, nano and bio-sciences, the recent years have witnessed a growing interest in exploring the possibility to enrich LB in the direction of describing micro-structured flows [4–6]. The LB method is based on a stylized stream-and-collide microscopic dynamics of fictitious particles, located on the nodes of discrete lattices and interacting according to local collision rules that drive the system towards a local equilibrium [2,3]. Mathematically: fi(~x + ~ci∆t, t+∆t)− fi(~x, t) = −ω∆t [ fi(~x, t)− f (eq) i (~x, t) ] where fi(~x, t) is the probability to find a particle at position ~x and time t, moving along the lattice direction defined by the discrete speeds ~ci (i = 1, ..., b). The second term at the rhs of the above equations denotes relaxation towards a local equilibrium, the lattice analogue of a Maxwellian distribution in continuum kinetic theory: f i (~x, t) = nwi(1 + β~ci · ~u+ β 2 [(~ci · ~u) 2 − u]. In the above, n(~x, t) is the fluid density, ~u the flow speed, β = 1/cs is the inverse square speed of sound, and wi a discrete set of weights normalized to unity. Finally, ω, indicates a typical