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

TL;DR

This paper introduces a second order in time numerical scheme for the 2D Navier-Stokes equations that efficiently approximates long-term statistical properties, with proven stability and convergence results.

Contribution

It develops a new second order in time scheme combining BDF and Adams-Bashforth methods, ensuring stability and convergence for long-term statistical properties of 2D Navier-Stokes simulations.

Findings

Proves uniform in time bounds in multiple Sobolev norms.

Shows convergence of long-term statistics to the true NSE behavior.

Discusses fully discrete spectral implementations.

Abstract

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: $$ \frac{3\omega^{n+1}-4\omega^n+\omega^{n-1}}{2k} + \nabla^\perp(2\psi^n-\psi^{n-1})\cdot\nabla(2\omega^n-\omega^{n-1}) - \nu\Delta\omega^{n+1} = f^{n+1}, \quad -\Delta \psi^n = \om^n. $$ 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 $\dL2$, $\dH1$ 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.