Unconditional long-time stability of a velocity–vorticity method for the 2D Navier–Stokes equations

Abstract

We prove unconditional long-time stability for a particular velocity---vorticity discretization of the 2D Navier---Stokes equations. The scheme begins with a formulation that uses the Lamb vector to couple the usual velocity---pressure system to the vorticity dynamics equation, and then discretizes with the finite element method in space and implicit---explicit BDF2 in time, with the vorticity equation decoupling at each time step. We prove the method's vorticity and velocity are both long-time stable in the $$L^2$$L2 and $$H^1$$H1 norms, without any timestep restriction. Moreover, our analysis avoids the use of Gronwall-type estimates, which leads us to stability bounds with only polynomial (instead of exponential) dependence on the Reynolds number. Numerical experiments are given that demonstrate the effectiveness of the method.