Numerical study of three multilevel preconditioners for solving 2D unsteady Navier-Stokes equations

Numerical experiments for solving 2D Navier-Stokes equations in the stream function (PS) formulation described by a fourth-order equation for PS (which is a natural formulation from a physical point of view) are presented. We use standard finite difference discretization for fourth-order elliptic equations where the non-linear convective terms are taken from the preceding time step. The resulting symmetric linear systems of equations are very ill conditioned. We overcome this difficulty by a proper preconditioning technique. We investigate the performance of three preconditioners constructed on the basis of optimal-order multilevel preconditioners for second-order elliptic operators. The numerical tests presented illustrate the limitations and advantages of the proposed preconditioners when we vary the Reynolds number, the mesh size (in space) and the time step.