The last decade has seen rapid progress in the theoretical understanding, algorithmic development, and use of lattice Boltzmann methods (LBM). As opposed to traditional computational fluid dynamics approaches, which compute macroscopic fluid dynamic properties by discretizing the continuum governing equations, the independent variables in a lattice Boltzmann approach consist of particle distribution functions in phase space, from which macroscopic continuum fluid properties can be recovered. Historically, LBM methods have been derived from lattice gas automata (LGA) methods. LGA methods model macroscopic fluid motion through the evolution of a set of discrete particles on a regular lattice (cartesian grid). Many of the deficiencies of LGA methods in reproducing accurate macroscopic behavior were resolved by the LBM approach, by neglecting individual particle motion, and adopting a particle distribution function approach. However, LBM methods have retained this conceptual link to particle methods such as LGA, which has inhibited the use of more sophisticated numerical techniques.

In this work, we first describe the lattice Boltzmann method as a particular finite-difference discretization in space and time of the Boltzmann equation. The LBM time-step is then shown to be equivalent to a first-order explicit time step (forward Euler), which is also equivalent to a Jacobi iteration for the steady-state form of the LBM equations. It is then shown how this iteration strategy may be used as a solver for the steady-state lattice Boltzmann equation, or as a solver for an implicit time-integration strategy. Finally, it is shown how these problems may be solved more efficiently using the Jacobi iteration as a smoother in a multigrid scheme for the steady-state lattice Boltzmann equation. This requires under-relaxation of the iteration scheme to achieve good high-frequency damping properties, and a careful matching of the LBM discretizations on the coarse grid levels, in order to ensure consistent coarse and fine grid problems. The final result is a multigrid solver which can make use of a pre-existing LBM routine in a modular fashion, by invoking the LBM routine on each grid level. Convergence rates which are independent of the mesh resolution are demonstrated for the driven cavity problem on a cartesian grid, although the convergence rates degrade with increasing Reynolds number.

[1] D. J. Mavriplis, "Multigrid Solution of the Steady-State Lattice Boltzmann Equation", Paper delivered at International Conference for Mesoscopic Methods in Engineering and Science (ICMMES), Braunschweig, Germany, July 2004. to appear, Computers and Fluids, 2005.