Lattice Bhatnagar–Gross–Krook

Abstract

Abstract In the previous chapter, we have learned how to cut down the complexity of the LBE collision process by formulating a discrete scattering version of the collision operator. This scattering version is de-facto a multi-relaxation model equation in which, owing to the discreteness of velocity space, it becomes possible to analyze the spectrum of the scattering operator in full depth. This analysis delivers an exact expression for the fluid viscosity via the single eigenvalue associated with the slowest non-conserved quantity, namely the momentum flux tensor. In light of this, it is natural to wonder whether the scattering operator can be simplified further and brought down to a single-time relaxation form. The idea is again patterned after continuum kinetic theory, in particular after the celebrated Bhatnagar–Gross–Krook (BGK) model Boltzmann introduced as early as 1954. Its transcription to the discrete lattice world proves exceedingly fruitful, and it leads to a hydrodynamic-compliant kinetic equation of great simplicity and efficiency. In this chapter we shall describe the Lattice Bhatnagar–Gross–Krook scheme, LBGK for short, the ultimate LBE model in terms of simplicity and effectiveness. The main lesson taught by the self-standing LBE of the previous chapter is that the scattering matrix and local equilibria can be regarded as free parameters of the theory, to be tuned to our best purposes within the limits set by the conservation laws and numerical stability. Within this picture, the viscosity of the LB fluid is entirely controlled by a single parameter, namely the leading nonzero eigenvalue of the scattering matrix Aij . The remaining eigenvalues are then set so as to minimize the interference of non-hydrodynamic modes (ghosts) with the dynamics of macroscopic observables. This spawns a very natural question: since transport is related to a single nonzero eigenvalue, why not simplify things further by choosing a one parameter scattering matrix? The point, raised almost simultaneously by a number of authors [54], is indeed well taken.