Stability and accuracy of various difference schemes for the lattice Boltzmann method

Abstract

We test a second order central difference scheme and a first order upwind scheme for the advection of particles in the lattice Boltzmann method for fluid flow. A diffusion term is added to the Boltzmann equation in order to improve stability when using the second order scheme, this term is equivalent to the Lax-Wendroff scheme for a particular value of the diffusion constant. In contrast to the normal lattice Boltzmann method, we allow a particle Courant number less than one. We test the schemes for stability and accuracy using Taylor-Green vortex and channel flows in three dimensions, finding improved stability for some configurations and no loss in accuracy. Both modifications are expected to remove some spurious lattice invariants. The proposed particle diffusion term may also be used to improve the stability of other Boltzmann based methods that use higher order difference schemes. © Austral. Mathematical Soc. 2012.