Long Time Stability of High Order MultiStep Numerical Schemes for Two-Dimensional Incompressible Navier--Stokes Equations

Abstract

The long-time stability properties of a few multistep numerical schemes for the two-dimensional incompressible Navier--Stokes equations (formulated in vorticity-stream function) are investigated in this article. These semi-implicit numerical schemes use a combination of explicit Adams--Bashforth extrapolation for the nonlinear convection term and implicit Adams--Moulton interpolation for the viscous diffusion term, up to fourth order accuracy in time. As a result, only two Poisson solvers are needed at each time step to achieve the desired temporal accuracy. The fully discrete schemes, with Fourier pseudospectral approximation in space, are analyzed in detail. With the help of a priori analysis and aliasing error control techniques, we prove uniform in time bounds for these high order schemes in both $L^2$ and $H^m$ norms, for $m \ge 1$, provided that the time step is bounded by a given constant. Such a long time stability is also demonstrated by the numerical experiments.