The Lattice Boltzmann method was originally devised as a computational alternative for the simulation of macroscopic flows, as described by the Navier–Stokes equations of continuum mechanics. In many respects, this still is the main place where it belongs today. Yet, in the past decade, LB has made proof of a largely unanticipated versatility across a broad spectrum of scales, from fully developed turbulence, to microfluidics, all the way down to nanoscale flows. Even though no systematic analogue of the Chapman–Enskog asymptotics is available in this beyond-hydro region (no guarantee), the fact remains that, with due extensions of the basic scheme, the LB has proven capable of providing several valuable insights into the physics of flows at micro- and nano-scales. This does not mean that LBE can solve the actual Boltzmann equation or replace Molecular Dynamics, but simply that it can provide useful insights into some flow problems which cannot be described within the realm of the Navier–Stokes equations of continuum mechanics. This Chapter provides a cursory view of this fast-growing front of modern LB research.
Decoupled scheme based on the Hermite expansion to construct lattice Boltzmann models for the compressible Navier-Stokes equations with arbitrary specific heat ratio