Abstract
The stability and convergence of the Fourier pseudo-spectral method are analyzed for the three dimensional incompressible Navier-Stokes equation, coupled with a variety of time-stepping methods, of up to fourth order temporal accuracy. An aliasing error control technique is applied in the error estimate for the nonlinear convection term, while an a-priori assumption for the numerical solution at the previous time steps will also play an impor-tant role in the analysis. In addition, a few multi-step temporal discretization is applied to achieve higher order temporal accuracy, while the numerical sta-bility is preserved. These semi-implicit numerical schemes use a combination of explicit Adams-Bashforth extrapolation for the nonlinear convection term, as well as the pressure gradient term, and implicit Adams-Moulton interpola-tion for the viscous diffusion term, up to the fourth order accuracy in time. Optimal rate convergence analysis and error estimates are established in de-tails. It is proved that, the Fourier pseudo-spectral method coupled with the carefully designed time-discretization is stable provided only that the time-step and spatial grid-size are bounded by two constants over a finite time. Some numerical results are also presented to verify the established convergence rates of the proposed schemes.