On computing the long-time solution of the two-dimensional Navier-Stokes equations

The long-time behavior of the incompressible Navier-Stokes equations on a 2-torus is computed using several discretizations in space and in time. We compare several nonlinear Galerkin methods (NGMs) with the standard Galerkin method (SGM) in both their ability to capture the dynamic and geometric behavior of a turbulent flow accurately, and the computational efficiency with which they are solved using variable time-step integration schemes. To measure the convergence of the chaotic attractor we employ Poincare section plots as well as density functions of instantaneous Lyapunov exponents. As expected, for the smooth (single mode) force used, the NGMs fare no better than the SGM in reproducing the correct behavior. As the number of modes for each method is increased, however, a distinct advantage in computational cost emerges for the NGMs. This trend suggests that the correct solution may be computed at considerable savings for a nonsmooth force, at higher Reynolds numbers.