Accuracy of discrete-velocity BGK models for the simulation of the incompressible Navier-Stokes equations

Abstract

The lattice Boltzmann (LB) method has been used as a Navier-Stokes CFD method since its introduction in 1988. The LB method is a Lagrangian discretization of a discrete-velocity Boltzmann equation. We introduce an alternative, fourth-order discretization scheme and compare results with those of the LB discretization and with finite-difference schemes applied to the incompressible Navier-Stokes equations in primitive-variable form. A Chapman-Enskog expansion of the PDE system predicts that the macroscopic behavior corresponds to the incompressible Navier-Stokes equations with additional ‘compressibility error’ of order Mach number squared. We numerically demonstrate convergence of the BGK schemes to the incompressible Navier-Stokes equations and quantify the errors associated with compressibility and discretization effects. When compressibility error is smaller than discretization error, convergence in both grid spacing and time step is shown to be second-order for the LB method and is confirmed to be fourth-order for the fourth-order BGK solver. However, when the compressibility error is simultaneously reduced as the grid is refined, the LB method behaves as a first-order scheme in time.