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Two discretizations of a 9-velocity Boltzmann equation with a BGK collision operator are studied. A Chapman-Enskog expansion of the PDE system predicts that the macroscopic behavior corresponds to the incompressible Navier-Stokes equations with additional terms of order Mach number squared. We introduce a fourth-order scheme and compare results with those of the commonly used lattice Boltzmann discretization and with finite-difference schemes applied to the incompressible Navier-StokesarXiv:comp-gas/9307003v1 fatcat:qjbldykma5avvnqnffgdbxd2mi