Solution of the unsteady Navier-Stokes equations by spectral methods

The two-dimensional unsteady Navier-Stokes equations for incompressible fluid flows are solved numerically using a spectral method with an alternating-direction time discretization. The resulting subproblems (a linear diffusion problem involving the solution of the Stokes equations and a nonlinear convection problem) are reduced to a cascade of linear Helmholtz problems by an optimal-control procedure, and the results obtained by this approach are compared with those obtained using a finite-element approximation. The size of the ill-conditioned linear systems to be solved is reduced by coordinating the subdomains of the global domain on the basis of the two-dimensional Helmholtz problem. Sample numerical results are presented graphically.