Connection between kinetic methods for fluid-dynamic equations and macroscopic finite-difference schemes

The lattice Boltzmann method (LBM) for the incompressible Navier–Stokes (NS) equations and the gas kinetic scheme for the compressible NS equations are based on the kinetic theory of gases. In the latter case, however, it is shown that the kinetic formulation is necessary only in the discontinuous reconstruction of fluid-dynamic variables for shock capturing. Analogously we will discuss the reduction of a kinetic method for the incompressible case, where the LBM scheme will be shown to shrink to an artificial compressibility type finite-difference scheme. We will prove first that a simple and compact LBM scheme cannot catch rarefied effects beyond Navier–Stokes and hence that it is worth the effort to develop kinetic-based FD alternatives. Finally we will propose two improvements to existing kinetic-based FD schemes: first of all, (a) the proposed scheme is formulated purely in terms of macroscopic quantities on a compact stencil; secondly (b) the semi-implicit formulation is proposed in order to increase the stability. We think that this work may be useful to others in realizing the actual possibilities of simple LBM schemes beyond Navier–Stokes and in adopting the suggested improvements in their actual FD codes.