Asymptotic analysis of the lattice Boltzmann equation

In this article we analyze the lattice Boltzmann equation (LBE) by using the asymptotic expansion technique. We first relate the LBE to the finite discrete-velocity model (FDVM) of the Boltzmann equation with the diffusive scaling. The analysis of this model directly leads to the incompressible Navier-Stokes equations, as opposed to the compressible Navier-Stokes equations obtained by the Chapman-Enskog analysis with convective scaling. We also apply the asymptotic analysis directly to the fully discrete LBE, as opposed to the usual practice of analyzing a continuous equation obtained through the Taylor-expansion of the LBE. This leads to a consistency analysis which provides order-by-order information about the numerical solution of the LBE. The asymptotic technique enables us to analyze the structure of the leading order errors and the accuracy of numerically derived quantities, such as vorticity. It also justifies the use of Richardson's extrapolation method. As an example, a two-dimensional Taylor-vortex flow is used to validate our analysis. The numerical results agree very well with our analytic predictions.