Spectral Direction Splitting Schemes for the Incompressible Navier-Stokes Equations

We propose and analyze spectral direction splitting schemes for the incompressible Navier-Stokes equations. The schemes combine a Legendre-spectral method for the spatial discretization and a pressure-stabilization/direction splitting scheme for. the temporal discretization, leading to a sequence of one-dimensional elliptic equations at each time step while preserving the same order of accuracy as the usual pressurestabilization schemes. We prove that these schemes are unconditionally stable, and present numerical results which demonstrate the stability accuracy, and efficiency of the proposed methods.