Fast and accurate solution of time-dependent incompressible flow

The stream function and vorticity formulation of the governing equations for non-steady incompressible viscous flows is considered. The time accurate solution to the differential problem requires that the transients be resolved. It is shown that the time scales of the transients are related to the spatial scales of the different components of the transients. By taking advantage of this relation, a multi-grid method has been developed in which both the time steps and the mesh spacing vary simultaneously. The resulting method is not only accurate but very fast, and machine round-off accuracy (reduction of the residuals by 16 orders of magnitude) can be achieved by a computational effort which is equivalent to about 100 successive point relaxation sweeps on the difference equations. Some results for the non-steady driven cavity flow are also given.