ACCURACY OF DISCRETE-VELOCITY BGK MODELS FOR THE SIMULATION OF THE INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

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 result s with those of the LB discretization and with finite-difference schem es applied to the incompressible Navier-Stokes equations in primitive- variable form. A Chapman-Enskog expansion of the PDE system predicts t hat 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 sch emes to the incompressible Navier-Stokes equations and quantify the er rors associated with compressibility and discretization effects. When compressibility error is smaller than discretization error, convergenc e in both grid spacing and time step is shown to be second-order for t he 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.