Finite Boltzmann schemes

In the special case of relaxation parameter ω = 1 lattice Boltzmann schemes for (convection) diffusion and fluid flow are equivalent to finite difference/volume (FD) schemes, and are thus coined finite Boltzmann (FB) schemes. We show that the equivalence is inherent to the homology of the Maxwell–Boltzmann constraints for the equilibrium distribution, and the constraints for finite difference stencils as derived from Taylor series expansion. For convection–diffusion we analyse the equivalence between FB and the Lax–Wendroff FD scheme in detail. It follows that the Lax–Wendroff procedure is performed automatically in the finite Boltzmann schemes via the imposed Maxwell–Boltzmann constraints. Furthermore, we make some remarks on FB schemes for fluid flows, and show that an earlier related study can be extended to rectangular grids. Finally, our findings are briefly checked with simulations of natural convection in a differentially heated square cavity.