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 in-compressible 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 L-2 and H-1 norms. We provide numerical verification of these results. We also demonstrate that divergence-free finite elements are critical for long-time H-1 stability. (c) 2018 Elsevier Inc. All rights reserved.