American Institute of Mathematical Sciences

• Previous Article
A Liouville theorem for ancient solutions to a semilinear heat equation and its elliptic counterpart
• ERA Home
• This Issue
• Next Article
Global dynamics of the solution for a bistable reaction diffusion equation with nonlocal effect
doi: 10.3934/era.2021019
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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 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, doi: 10.3934/era.2021019
References:

show all references

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
 [1] Lihuai Du, Ting Zhang. Local and global strong solution to the stochastic 3-D incompressible anisotropic Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4745-4765. doi: 10.3934/dcds.2018209 [2] Yinnian He, Pengzhan Huang, Jian Li. H2-stability of some second order fully discrete schemes for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2745-2780. doi: 10.3934/dcdsb.2018273 [3] Jingrui Wang, Keyan Wang. Almost sure existence of global weak solutions to the 3D incompressible Navier-Stokes equation. Discrete & Continuous Dynamical Systems, 2017, 37 (9) : 5003-5019. doi: 10.3934/dcds.2017215 [4] Wendong Wang, Liqun Zhang, Zhifei Zhang. On the interior regularity criteria of the 3-D navier-stokes equations involving two velocity components. Discrete & Continuous Dynamical Systems, 2018, 38 (5) : 2609-2627. doi: 10.3934/dcds.2018110 [5] Yanlin Liu, Ping Zhang. Remark on 3-D Navier-Stokes system with strong dissipation in one direction. Communications on Pure & Applied Analysis, 2021, 20 (7&8) : 2765-2787. doi: 10.3934/cpaa.2020244 [6] Thomas Y. Hou, Ruo Li. Nonexistence of locally self-similar blow-up for the 3D incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 637-642. doi: 10.3934/dcds.2007.18.637 [7] Shijin Ding, Zhilin Lin, Dongjuan Niu. Boundary layer for 3D plane parallel channel flows of nonhomogeneous incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2020, 40 (8) : 4579-4596. doi: 10.3934/dcds.2020193 [8] Vladimir V. Chepyzhov, E. S. Titi, Mark I. Vishik. On the convergence of solutions of the Leray-$\alpha$ model to the trajectory attractor of the 3D Navier-Stokes system. Discrete & Continuous Dynamical Systems, 2007, 17 (3) : 481-500. doi: 10.3934/dcds.2007.17.481 [9] Roberta Bianchini, Roberto Natalini. Convergence of a vector-BGK approximation for the incompressible Navier-Stokes equations. Kinetic & Related Models, 2019, 12 (1) : 133-158. doi: 10.3934/krm.2019006 [10] Keyan Wang. On global regularity of incompressible Navier-Stokes equations in $\mathbf R^3$. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1067-1072. doi: 10.3934/cpaa.2009.8.1067 [11] Chongsheng Cao. Sufficient conditions for the regularity to the 3D Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1141-1151. doi: 10.3934/dcds.2010.26.1141 [12] Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241 [13] Xuanji Jia, Zaihong Jiang. An anisotropic regularity criterion for the 3D Navier-Stokes equations. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1299-1306. doi: 10.3934/cpaa.2013.12.1299 [14] Hui Chen, Daoyuan Fang, Ting Zhang. Regularity of 3D axisymmetric Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 1923-1939. doi: 10.3934/dcds.2017081 [15] Gaocheng Yue, Chengkui Zhong. On the global well-posedness to the 3-D Navier-Stokes-Maxwell system. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5817-5835. doi: 10.3934/dcds.2016056 [16] Francisco Guillén-González, Mouhamadou Samsidy Goudiaby. Stability and convergence at infinite time of several fully discrete schemes for a Ginzburg-Landau model for nematic liquid crystal flows. Discrete & Continuous Dynamical Systems, 2012, 32 (12) : 4229-4246. doi: 10.3934/dcds.2012.32.4229 [17] Fabio Ramos, Edriss S. Titi. Invariant measures for the $3$D Navier-Stokes-Voigt equations and their Navier-Stokes limit. Discrete & Continuous Dynamical Systems, 2010, 28 (1) : 375-403. doi: 10.3934/dcds.2010.28.375 [18] Stefano Scrobogna. Global existence and convergence of nondimensionalized incompressible Navier-Stokes equations in low Froude number regime. Discrete & Continuous Dynamical Systems, 2020, 40 (9) : 5471-5511. doi: 10.3934/dcds.2020235 [19] Jian Su, Yinnian He. The almost unconditional convergence of the Euler implicit/explicit scheme for the three dimensional nonstationary Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3421-3438. doi: 10.3934/dcdsb.2017173 [20] Xiaoli Lu, Pengzhan Huang, Yinnian He. Fully discrete finite element approximation of the 2D/3D unsteady incompressible magnetohydrodynamic-Voigt regularization flows. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 815-845. doi: 10.3934/dcdsb.2020143