Multigrid solution of the steady-state lattice Boltzmann equation

Efficient solution strategies for the steady-state lattice Boltzmann equation are investigated. Stable iterative methods for the linearized lattice Boltzmann equation are formulated based on the linearization of the lattice Boltzmann time-stepping procedure. These are applied as relaxation methods within a linear multigrid scheme, which itself is used to drive a Newton solver for the full non-linear problem. Although the linear multigrid strategy provides rapid convergence, the cost of a linear residual evaluation is found to be substantially higher than the cost of evaluating the non-linear residual directly. Therefore, a non-linear multigrid approach is adopted, which makes use of the non-linear LBE time-stepping scheme on each grid level. Rapid convergence to steady-state is achieved by the non-linear algorithm, resulting in one or more orders of magnitude increase in solution efficiency over the LBE time-integration approach. Grid-independent convergence rates are demonstrated, although degradation with increasing Reynolds number is observed, as in the case of the original LBE time-stepping scheme. The multigrid solver is implemented in a modular fashion by calling an existing LBE time-stepping routine, and delivers the identical steady-state solution as the original LBE time-stepping approach.