# American Institute of Mathematical Sciences

November  2021, 29(5): 2915-2944. doi: 10.3934/era.2021019

## Convergence analysis of Fourier pseudo-spectral schemes for three-dimensional incompressible Navier-Stokes equations

 Department of Mathematics, The University of Massachusetts, North Dartmouth, MA 02747, USA

Received  October 2020 Revised  January 2021 Published  November 2021 Early access  March 2021

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 important role in the analysis. In addition, a few multi-step temporal discretization is applied to achieve higher order temporal accuracy, while the numerical stability 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 interpolation for the viscous diffusion term, up to the fourth order accuracy in time. Optimal rate convergence analysis and error estimates are established in details. 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.

Citation: Cheng Wang. Convergence analysis of Fourier pseudo-spectral schemes for three-dimensional incompressible Navier-Stokes equations. Electronic Research Archive, 2021, 29 (5) : 2915-2944. doi: 10.3934/era.2021019
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##### References:
The discrete $\ell^2$ and $\ell^\infty$ numerical errors vs. temporal resolution $N_T$ for $N_T = 100:100:1000$, with a spatial resolution $N = 256$. The numerical results are obtained by the computation using the proposed scheme (3.1), (5.2), (5.5), (5.6), in the first, second, third and fourth order temporal accuracy orders, respectively. The viscosity parameter is taken as $\nu = 0.5$. The numerical errors for the velocity variable $u$ are displayed. The data lie roughly on curves $CN_T^{-1}$, $C N_T^{-2}$, $C N_T^{-3}$ and $C N_T^{-4}$, respectively, for appropriate choices of $C$, confirming the full temporal accuracy orders of the proposed schemes
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