Lattice-Boltzmann Part 2: LB as Solver for Partial Differential Equations and Multiple Relaxation Time Models

Since its introduction about two decades ago the lattice-Boltzmann (LB) method has been developed into a viable numerical tool for computational fluid dynamics and beyond. With its roots in kinetic theory this special version of a cellular automaton concept allows to obtain continuum flow quantities from simple and local update rules based on particle interactions. Instead of solving the Navier-Stokes equation directly (DNS), the discrete Boltzmann equation is solved by using simple models for the collision operator. The simplest such operator is the single relaxation time Bhatnagar-Gross-Krook (BGK) model, whereas more general models feature multiple relaxation times (MRT) which allow different shear and bulk viscosities.
In this second part of my talk on LB methods I will demonstrate how LB schemes can be used as alternative solvers for a variety of partial differential equations, a feature that summarises best the spirit of the method. I will also show how the simple BGK collision operator can be generalised to permit different relaxation times for mass, momentum and energy density.