A new linearly extrapolated Crank-Nicolson time-stepping scheme for the Navier-Stokes equations

Abstract

We investigate the stability of a fully-implicit, linearly extrapolated Crank-Nicolson (CNLE) time-stepping scheme for finite element spatial discretization of the Navier-Stokes equations. Although presented in 1976 by Baker and applied and analyzed in various contexts since then, all known convergence estimates of CNLE require a time-step restriction. We propose a new linear extrapolation of the convecting velocity for CNLE that ensures energetic stability without introducing an undesirable exponential Gronwall constant. Such a result is unknown for conventional CNLE for inhomogeneous boundary data (usual techniques fail!). Numerical illustrations are provided showing that our new extrapolation clearly improves upon stability and accuracy from conventional CNLE.