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

Xiaoming Wang

We investigate the long time behavior of the following efficient second
order in time scheme for the 2D Navier-Stokes equation in a periodic box:

$$
\frac{3\omega^{n+1}-4\omega^n+\omega^{n-1}}{2k} +
\nabla^\perp(2\psi^n-\psi^{n-1})\cdot\nabla(2\omega^n-\omega^{n-1}) -
\nu\Delta\omega^{n+1} = f^{n+1},
\quad -\Delta \psi^n = \omega^n.
$$

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 $\dot{L}^2$,
$\dot{H}^1$ and
$\dot{H}^2_{per}$ 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.