Long time stability of a classical efficient scheme for two dimensional Navier--Stokes equations

Sigal Gottlieb, Florentina Tone, Cheng Wang, Xiaoming Wang, Djoko Wirosoetisno

This paper considers the long-time stability property of a popular semi-implicit scheme for the 2D incompressible Navier--Stokes equations in a periodic box that treats the viscous term implicitly and the nonlinear advection term explicitly. We consider both the semi-discrete (discrete in time but continuous in space) and fully discrete schemes with either Fourier Galerkin spectral or Fourier pseudospectral (collocation) methods. We prove that, in all cases, the scheme is long time stable provided that the timestep is sufficiently small. The long time stability in the $L^2$ and $H^1$ norms further leads to the convergence of the global attractors and invariant measures of the scheme to those of the Navier--Stokes equations at vanishing timestep.