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

TL;DR

This paper proves the long-term stability of a classical implicit-explicit scheme for 2D Navier-Stokes equations under small time steps, ensuring convergence of attractors and measures to the true solutions.

Contribution

It establishes long-time stability and convergence of a popular numerical scheme for 2D Navier-Stokes equations with spectral methods.

Findings

Scheme is stable in $L^2$ and $H^1$ norms for small time steps.

Global attractors and invariant measures converge as time step vanishes.

Results apply to both semi-discrete and fully discrete spectral schemes.

Abstract

We prove that a popular classical implicit-explicit scheme for the 2D incompressible Navier--Stokes equations that treats the viscous term implicitly while the nonlinear advection term explicitly is long time stable provided that the time step is sufficiently small in the case with periodic boundary conditions. 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 NSE itself at vanishing time step. Both semi-discrete in time and fully discrete schemes with either Galerkin Fourier spectral or collocation Fourier spectral methods are considered.