Convergence analysis of high-order IMEX-BDF schemes for the incompressible Navier–Stokes equations.

In this paper, we consider developing high-order temporal integration schemes for the unsteady incompressible Navier–Stokes equations in bounded two-dimensional domain subjected to the periodic boundary conditions. Utilizing the k -step (k = 3 , 4 , 5) backward differentiation formula (BDF) coupled with the implicit–explicit (IMEX) treatment of the nonlinear convective term in an anti-symmetry form, a class of IMEX-BDF k schemes up to fifth-order in time are constructed and analyzed. By imposing a zero-mean constrain on the finite-dimensional space for the pressure, the proposed numerical schemes are proven to be uniquely solvable. Based on the recent theoretical framework consisting of a class of discrete orthogonal convolution kernels, rigorous L 2 norm error estimates for both the velocity and the pressure are established by using a novel divergence free projection system. The proposed schemes are then implemented in two benchmark experiments, including a Taylor–Green vortex problem and a double shear layer flow at various high Reynolds numbers. Numerical results demonstrate the expected solution accuracy and the computational effectiveness in simulating the realistic flow dynamics. • We consider developing high-order temporal integration schemes for the incompressible Navier–Stokes equations. • We introduce a class of discrete orthogonal convolution kernels to develop a unified framework for the L 2 norm convergence analysis. • Extensive benchmark experiments are performed to show the effectiveness of the high-order IMEX-BDF schemes. [ABSTRACT FROM AUTHOR]