An efficient second order in time scheme for approximating long time statistical properties of the two dimensional Navier-Stokes equations

We investigate the long tim behavior of the following efficient second order in time scheme for the 2D Navier-Stokes equation in a periodic box: $$\frac{3{\omega }^{n+1}-4{\omega }^n+{\omega }^{n-1}}{2k}+{\mathrm{\nabla }}^{\perp }(2{\psi }^n-{\psi }^{n-1})\cdot \mathrm{\nabla }(2{\omega }^n-{\omega }^{n-1})-\nu \mathrm{\Delta }{\omega }^{n+1}=f^{n+1},-\mathrm{\Delta }{\psi }^n={\text{om}}^n\mathrm{.}$$ The scheme is a combination of a 2nd order in time backward-differentiation (BDF) and a special explicit Adams-Bashforth treatment of the advection term. Therefore only a linear constant coefficient Poisson type problem needs to be solved at each time step. We prove uniform in time bounds on this scheme in $\text{dL}2$, $\text{dH}1$ and ${\dot{H}}_{per}^2$ provided that the time-step is sufficiently small. These time uniform estimates further lead to the convergence of long time statistics (stationary statistical properties) of the scheme to that of the NSE itself at vanishing time-step. Fully discrete schemes with either Galerkin Fourier or collocation Fourier spectral method are also discussed.