Time-grid independent error analysis of adaptive predictor-corrector BDF2 scheme for the unsteady Navier–Stokes equations with high Reynolds number

Abstract This article presents a novel adaptive predictor-corrector BDF2 (PC-BDF2) method for solving the unsteady incompressible Navier–Stokes equations with high Reynolds number. The PC-BDF scheme is shown to be unconditional $L^{2}$-norm stable on general nonuniform time grids. A grad-div stabilization and the skew-symmetric technique are introduced to develop a new error estimate of the convection term within the discrete convolution kernel framework. This new estimate effectively removes the dependence of convergence constants on the Reynolds number brought by the nonlinear convection and pressure terms. Consequently, a tailored discrete Gronwall inequality leads to the rigorous convergence analysis of the PC-BDF2 scheme, which is also independent of both the time-step ratios and the Reynolds numbers. The proposed PC-BDF2 scheme achieves robust and accurate long-term simulations of incompressible flows at high Reynolds numbers with adaptive time-stepping strategies. Numerical experiments, including accuracy tests up to $\text{Re}=10^{8}$ and lid-driven cavity flow simulations up to $\text{Re}=10^{4}$ with a final time $\text{T}=500$, confirm the robustness of the PC-BDF2 scheme against variations in time steps and high Reynolds numbers.