Open AccessConvergence analysis of Fourier pseudo-spectral schemes for three-dimensional incompressible Navier-Stokes equations
Author(s) -
Cheng Wang
Publication year - 2021
Publication title -
electronic research archive
Language(s) - English
Resource type - Journals
ISSN - 2688-1594
DOI - 10.3934/era.2021019
Subject(s) - discretization , temporal discretization , mathematics , nonlinear system , convergence (economics) , fourier transform , fourier analysis , aliasing , fourier series , extrapolation , spectral method , interpolation (computer graphics) , mathematical analysis , numerical analysis , bounded function , compressibility , rate of convergence , computer science , filter (signal processing) , physics , classical mechanics , mechanics , computer network , quantum mechanics , economics , computer vision , economic growth , motion (physics) , channel (broadcasting)
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.