Iterative projection method for unsteady Navier-Stokes equations with high Reynolds numbers

A new iterative projection method is proposed to solve the unsteady Navier-Stokes equations with high Reynolds numbers. The convectional projection method attempts to project the intermediate velocity to the divergence free space only once per time step. However, such a velocity is not genuinely divergence free in general practice, which can yield large errors when the Reynolds number is high. The new method has several important features: the BDF2 time discretization, the skew-symmetric convection in a semi-implicit form, two modulating parameters, and the iterative projections in each time step. A major difficulty in the proof of iteration convergence is the nonlinear convection. We solve this problem by first analyzing the non-convective scheme with a focus on the spectral properties of the iterative matrix, and then employing a delicate perturbation analysis for the convective scheme. The work achieves the weakly divergence free velocity (strongly divergence free for divergence free finite element spaces), and the rigorous stability and error analysis when the iterations converge. The three dimensional numerical tests confirm that this new method can effectively treat high Reynolds numbers with only a few iterations per time step, where the convectional projection method and the iterative projection method with the explicit convection would fail.