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

TL;DR

The paper introduces an iterative projection framework for the unsteady Navier–Stokes equations at high Reynolds numbers, integrating a fully implicit BDF2 time discretization with a semi-implicit, skew-symmetric convection and two tunable parameters $\alpha$ and $\rho$ to drive repeated projections toward a divergence-free velocity. It provides a two-pronged theoretical analysis: a normal-mode study of the non-convective baseline and a perturbation-based convergence argument when convection is treated implicitly, complemented by a stability and error analysis of the limit scheme in a mixed FEM setting. The results show the method yields weakly divergence-free velocities upon convergence and robust error behavior, particularly for divergence-free FEMs, while enabling stable, accurate simulations at high Reynolds numbers with only a few iterations per time step. Numerical experiments in 3D with Taylor–Hood elements demonstrate fast convergence of the projection iterations and accurate steady or quasi-steady flows, outperforming standard projection and IMEX variants at high Re and illustrating the method’s practical impact for robust high-Re incompressible flow simulations.

Abstract

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.

Figures (5)

• Figure 1: Weak and strong measures of velocity divergence over iterations of Problem 1. First row: weak measure. Second row: strong measure.
• Figure 6: Problem 2: Weak and strong measures of velocity divergence over iterations of problem 2. First row: weak measure. Second row: strong measure.
• Figure 7: Velocity[a] and pressure[b] on the x-y plane of $z=0$ at time $t=20$. $Re=1000$, max speed=22.79. Produced by the conventional projection method.
• Figure 8: Velocity[a] and pressure[b] on the x-y plane of $z=0$ at time $t=20$. $Re=1000$, max speed=1. Produced by the iterative scheme.
• Figure 9: Comparison of normalized velocities, $v/2$ (red) and $u/2$ (blue) on the center lines $(x, 0, 0)$ and $(0, y, 0)$ on the x-y plane $z=0$. The whole velocity vector is $(u,v,w)$. The square symbols are extracted from figures of ALBENSOEDER2005536. The solid lines are from the iterative projection method of this work. [a]: $N=20$. [b]: $N=40$. $Re = 1000$, $t=20$.

Theorems & Definitions (15)

• Definition 1
• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Theorem 1: Iteration convergence without convection
• Lemma 5
• Lemma 6
• Theorem 2: Iteration Convergence of Iterative Projection Method at a single time step
• Definition 2