Stability and convergence of the Crank-Nicolson/Adams-Bashforth scheme for the time-dependent Navier-Stokes equations

In this paper, we study the stability and convergence of the Crank-Nicolson/Adams-Bashforth scheme for the two-dimensional nonstationary Navier-Stokes equations. A finite element method is applied for the spatial approximation of the velocity and pressure. The time discretization is based on the Crank-Nicolson scheme for the linear term and the explicit Adams-Bashforth scheme for the nonlinear term. Moreover, we present optimal error estimates and prove that the scheme is almost unconditionally stable and convergent, i.e., stable and convergent when the time step is less than or equal to a constant. © 2007 Society for Industrial and Applied Mathematics.