The first part of this introduction is devoted to the known derivation of the lattice
Boltzmann method (LBM): We track two different derivations, a historical one (via lattice
gas automata) and a theoretical version (via a discretization of the Boltzmann equation).
Thereby the collision term is approximated with a single relaxation time model (BGK)
and we motivate the introduction of this common approximation. By applying a multiscale
expansion (Chapman-Enskog), the solution of the numerical method is verified as a
meaningful approximation of the solution of the Navier-Stokes equations. To state a well
posed problem, common boundary conditions are introduced and their realization within a
LBM is discussed.
In the second part, the LBM is extended to handle coupled problems. Four cases are investigated:
(i) multiphase and multicomponent flow, (ii) additional forces, (iii) the coupling to
heat transport, (iv) coupling of electric circuits with power dissipation (as heat) and heat
transport.
Keywords: BBGKY hierarchy, BGK approximation, boundary conditions, Chapman-Enskog expansion,
circuit coupling, D3Q19, discrete velocity space, Gauß-Hermite quadrature, lattice gas automata, Navier-
Stokes equations, thermal coupling.
