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

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.

💡 Research Summary

This paper presents a third‑order accurate implicit‑explicit (IMEX) time‑stepping scheme for the two‑dimensional incompressible Navier–Stokes equations, combining a Fourier pseudo‑spectral spatial discretization with a backward differentiation formula of order three (BDF3) for the linear diffusion term and an Adams‑Bashforth extrapolation for the nonlinear convection term. The resulting fully discrete algorithm requires the solution of only a single Poisson‑like equation per time step, making it computationally attractive compared with lower‑order IMEX methods that often need multiple solves.

The authors first review existing first‑ and second‑order IMEX schemes, highlighting their unconditional stability but limited temporal accuracy. They then introduce the BDF3‑IMEX scheme: the vorticity equation is advanced using the BDF3 stencil, while the convection term is treated explicitly via a two‑step Adams‑Bashforth extrapolation. A zero‑average correction is added to preserve the mean‑free property of the vorticity under periodic boundary conditions. After updating the vorticity, the stream function is obtained by solving a discrete Poisson equation, and the velocity follows from the perpendicular gradient of the stream function.

A substantial part of the work is devoted to rigorous analysis. Lemma 2.1 establishes bounds for the Fourier collocation interpolation operator, ensuring that aliasing errors do not destroy stability. Lemma 2.2 provides a discrete Poincaré inequality and the skew‑symmetry of the discrete convection operator, which together guarantee that the nonlinear term does not contribute to the discrete energy growth. Lemma 2.3 gives a key estimate of the form
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