Long-time stability and convergence analysis of an IMEX BDF3 scheme for 2-D incompressible Navier-Stokes equation

Abstract

High-order time-stepping schemes are crucial for simulating incompressible fluid flows due to their ability to capture complex turbulent behavior and unsteady motion. In this work, we propose a third-order accurate numerical scheme for the two-dimensional incompressible Navier-Stokes equation. Spatial and temporal discretization is achieved using Fourier pseudo-spectral approximation and the BDF3 stencil, combined with the Adams-Bashforth extrapolation for the nonlinear convection term, resulting in a semi-implicit, fully discrete formulation. This approach requires solving only a single Poisson-like equation per time step while maintaining the desired temporal accuracy. Classical numerical experiments demonstrate the advantage of our scheme in terms of permissible time step sizes. Moreover, we establish uniform-in-time bounds for the vorticity in both $L^2$ and higher-order $H^m$ norms ($m \geq 1$), provided the time step is sufficiently small. These bounds, in turn, facilitate the derivation of optimal convergence rates.