The long-time L2 and H1 stability of linearly extrapolated second-order time-stepping schemes for the 2D incompressible Navier-Stokes equations

Herein we present a study on the long-time stability of finite element discretizations of a generalized class of semi-implicit second-order time-stepping schemes for the 2D incompressible Navier-Stokes equations. These remarkably efficient schemes require only a single linear solve per time-step through the use of a linearly-extrapolated advective term. Our result develops a class of sufficient conditions such that if external forcing is uniformly bounded in time, velocity solutions are uniformly bounded in time in both the L2 and H1 norms. We provide numerical verification of these results. We also demonstrate that divergence-free finite elements are critical for long-time H1 stability.