A stabilized fractional-step finite element method for the time-dependent Navier–Stokes equations
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Yueqiang Shang
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Qing Liu
Abstract
We present a fractional-step finite element method based on a subgrid model for simulating the time-dependent incompressible Navier–Stokes equations. The method aims to the simulation of high Reynolds number flows and consists of two steps in which the nonlinearity and incompressibility are split into different steps. The first step of this method can be seen as a linearized Burger’s problem where a subgrid model based on an elliptic projection of the velocity into a lower-order finite element space is employed to stabilize the system, and the second step is a Stokes problem. Under mild regularity assumptions on the continuous solution, we obtain the stability of the numerical method, and derive error bound of the approximate velocity, which shows that first-order convergence rate in time and optimal convergence rate in space can be gotten by the method. Numerical experiments verify the theoretical predictions and demonstrate the promise of the proposed method, which show superiority of the proposed method to the compared method in the literature.
Funding source: Fundamental Research Funds for the Central Universities
Award Identifier / Grant number: XDJK2018B032
Funding source: The Natural Science Foundation of China
Award Identifier / Grant number: 11361016
Funding source: The Basic and Frontier Explore Program of Chongqing Municipality, China
Award Identifier / Grant number: cstc2018jcyjAX0305
Acknowledgments
The authors express their deep gratitude to the anonymous reviewers for the valuable comments and suggestions, which led to an improvement of the paper.
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Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
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Research funding: This work was supported by the Natural Science Foundation of China (No. 11361016), the Basic and Frontier Explore Program of Chongqing Municipality, China (No. cstc2018jcyjAX0305), and Fundamental Research Funds for the Central Universities (No. XDJK2018B032).
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Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
References
[1] R. Glowinski, “Finite element methods for incompressible viscous flow,” Numerical Methods for Fluids Part 3, vol. IX, P.G. Ciarlet and J.L. Lions, Eds., Amsterdam, Elsevier Science Publisher, 2003.Handbook of Numerical Analysis.10.1016/S1570-8659(03)09003-3Suche in Google Scholar
[2] V. John, Finite Element Methods for Incompressible Flow Problems, Berlin, Springer International Publishing AG, 2016.10.1007/978-3-319-45750-5Suche in Google Scholar
[3] A. J. Chorin, “Numerical solution of the Navier–Stokes equations,” Math. Comput., vol. 22, pp. 745–762, 1968. https://doi.org/10.1090/s0025-5718-1968-0242392-2.Suche in Google Scholar
[4] R. Temam, “Sur I’approximation de la soluiton des equations de Navier–Stokes par la methode des pas fractionaires, II (French),” Arch. Ration. Mech. Anal., vol. 33, pp. 377–385, 1969. https://doi.org/10.1007/bf00247696.Suche in Google Scholar
[5] K. Goda, “A multistep technique with implicit difference schemes for calculating two- or three dimensional cavity flows,” J. Comput. Phys., vol. 30, no. 1, pp. 76–95, 1979. https://doi.org/10.1016/0021-9991(79)90088-3.Suche in Google Scholar
[6] J. L. Guermond and J. Shen, “On the error estimates for the rotational pressure-correction projection methods,” Math. Comput., vol. 73, no. 248, pp. 1719–1738, 2003. https://doi.org/10.1090/s0025-5718-03-01621-1.Suche in Google Scholar
[7] H. Johnston and J.-G. Liu, “Accurate, stable and efficient Navier–Stokes solvers based on explicit treatment of the pressure term,” J. Comput. Phys., vol. 199, no. 1, pp. 221–259, 2004. https://doi.org/10.1016/j.jcp.2004.02.009.Suche in Google Scholar
[8] E. Baensch and A. Brenner, “A posteriori error estimates for pressue-correction schemes,” SIAM J. Numer. Anal., vol. 54, no. 4, pp. 2323–2358, 2016. https://doi.org/10.1137/15m102753x.Suche in Google Scholar
[9] J. L. Guermond and J. Shen, “Velocity-correction projection methods for incompressible flows,” SIAM J. Numer. Anal., vol. 41, no. 1, pp. 112–134, 2003. https://doi.org/10.1137/s0036142901395400.Suche in Google Scholar
[10] J. L. Guermond, J. Shen, and X. F. Yang, “Error analysis of fully discrete velocity-correction methods for incompressible flows,” Math. Comput., vol. 77, no. 263, pp. 1387–1405, 2008. https://doi.org/10.1090/s0025-5718-08-02109-1.Suche in Google Scholar
[11] A. Poux, S. Glockner, E. Ahusborde, and M. Azaïez, “Open boundary conditions for the velocity-correction scheme of the Navier–Stokes equations,” Comput. Fluids, vol. 70, pp. 29–43, 2012. https://doi.org/10.1016/j.compfluid.2012.08.028.Suche in Google Scholar
[12] J. Shen, “On error estimates of the projection methods for the Navier–Stokes equations: second-order schemes,” Math. Comput., vol. 65, no. 215, pp. 1039–1065, 1996. https://doi.org/10.1090/s0025-5718-96-00750-8.Suche in Google Scholar
[13] J. L. Guermond and J. Shen, “A new class of truly consistent splitting schemes for incompressible flows,” J. Comput. Phys., vol. 192, no. 1, pp. 262–276, 2003. https://doi.org/10.1016/j.jcp.2003.07.009.Suche in Google Scholar
[14] Q. F. Liu, Y. R. Hou, Z. H. Wang, and J. K. Zhao, “A two-level consistent splitting scheme for the Navier–Stokes equations,” Comput. Fluids, vol. 140, pp. 167–174, 2016. https://doi.org/10.1016/j.compfluid.2016.09.010.Suche in Google Scholar
[15] J. L. Guermond, P. Minev, and J. Shen, “An overview of projection methods for incompressible flows,” Comput. Methods Appl. Mech. Eng., vol. 195, pp. 6011–6045, 2006. https://doi.org/10.1016/j.cma.2005.10.010.Suche in Google Scholar
[16] J. Shen, “On error estimates of some higher order projection and penalty-projection methods for Navier–Stokes equations,” Numer. Math., vol. 62, no. 1, pp. 49–73, 1992. https://doi.org/10.1007/bf01396220.Suche in Google Scholar
[17] M. Jobelin, C. Lapuerta, J.-C. Latché, P. Angot, and B. Piar, “A finite element penalty projection method for incompressible flows,” J. Comput. Phys., vol. 217, no. 2, pp. 502–518, 2006. https://doi.org/10.1016/j.jcp.2006.01.019.Suche in Google Scholar
[18] P. Angot and R. Cheayto, “On the error estimates of the vector penality-projection methods: second-order scheme,” Math. Comput., vol. 87, no. 313, pp. 2159–2187, 2017. https://doi.org/10.1090/mcom/3309.Suche in Google Scholar
[19] E. Hausenblas and T. A. Randrianasolo, “Time-discretization of stochastic 2-D Navier–Stokes equations with a penalty-projection method,” Numer. Math., vol. 143, no. 2, pp. 339–378, 2019. https://doi.org/10.1007/s00211-019-01057-3.Suche in Google Scholar
[20] J. Deteix and D. Yakoubi, “Improving the pressure accuracy in a projection scheme for incompressible fluids with variable viscosity,” Appl. Math. Lett., vol. 79, pp. 111–117, 2018. https://doi.org/10.1016/j.aml.2017.12.004.Suche in Google Scholar
[21] J. Deteix and D. Yakoubi, “Shear rate projection schemes for non-Newtonian fluids,” Comput. Methods Appl. Mech. Eng., vol. 354, pp. 620–636, 2019. https://doi.org/10.1016/j.cma.2019.06.006.Suche in Google Scholar
[22] X. L. Feng, Y. N. He, and D. M. Liu, “Convergence analysis of an implicit fractional-step method for the incompressible Navier–Stokes equations,” Appl. Math. Model., vol. 35, no. 12, pp. 5856–5871, 2011. https://doi.org/10.1016/j.apm.2011.05.042.Suche in Google Scholar
[23] X. L. Feng, Y. N. He, and P. Huang, “A stabilized implicit fractional-step method for the time-dependent Navier–Stokes equations using equal-order pairs,” J. Math. Anal. Appl., vol. 392, no. 2, pp. 209–224, 2012. https://doi.org/10.1016/j.jmaa.2012.03.026.Suche in Google Scholar
[24] J. L. Wu, L. L. Wei, and X. L. Feng, “Novel fractional time-stepping algorithms for natural convection problems with variable density,” Appl. Numer. Math., vol. 151, pp. 64–84, 2020. https://doi.org/10.1016/j.apnum.2019.12.012.Suche in Google Scholar
[25] O. Nikan, Z. Avazzadeh, and J. A. Tenreiro Machado, “Numerical investigation of fractional nonlinear sine-Gordon and Klein–Gordon models arising in relativistic quantum mechanics,” Eng. Anal. Bound. Elem., vol. 120, pp. 223–237, 2020. https://doi.org/10.1016/j.enganabound.2020.08.017.Suche in Google Scholar
[26] O. Nikan, J. A. Tenreiro Machado, Z. Avazzadeh, and H. Jafari, “Numerical evaluation of fractional Tricomi-type model arising from physical problems of gas dynamics,” J. Adv. Res., vol. 25, pp. 205–216, 2020. https://doi.org/10.1016/j.jare.2020.06.018.Suche in Google Scholar PubMed PubMed Central
[27] O. Nikan, H. Jafari, and A. Golbabai, “Numerical analysis of the fractional evolution model for heat flow in materials with memory,” Alexandria Eng. J., vol. 59, no. 4, pp. 2627–2637, 2020. https://doi.org/10.1016/j.aej.2020.04.026.Suche in Google Scholar
[28] O. Nikan, A. Golbabai, J. A. Tenreiro Machado, and T. Nikazad, “Numerical approximation of the time fractional cable model arising in neuronal dynamics,” Eng. Comput., 2020, https://doi.org/10.1007/s00366-020-01033-8.Suche in Google Scholar
[29] O. Nikan, J. A. Tenreiro Machado, and A. Golbabai, “Numerical solution of time-fractional fourth-order reaction-diffusion model arising in composite environments,” Appl. Math. Model., vol. 89, pp. 819–836, 2021. https://doi.org/10.1016/j.apm.2020.07.021.Suche in Google Scholar
[30] D. Meidner and T. Richter, “A posteriori error estimation for the fractional step theta discretization of the incompressible Navier–Stokes equations,” Comput. Methods Appl. Mech. Eng., vol. 288, pp. 45–59, 2015. https://doi.org/10.1016/j.cma.2014.11.031.Suche in Google Scholar
[31] R. L. T. Bevan, E. Boileau, R. van Loon, R. W. Lewis, and P. Nithiarasu, “A comparative study of fractional step method in its quasi-implicit, semi-implicit and fully-explicit forms for incompressible flows,” Int. J. Numer. Methods Heat Fluid Flow, vol. 26, nos 3–4, pp. 595–623, 2016. https://doi.org/10.1108/hff-06-2015-0233.Suche in Google Scholar
[32] A. Viguerie and A. Veneziani, “Algebraic splitting methods for the steady incompressible Navier–Stokes equations at moderate Reynolds numbers,” Comput. Methods Appl. Mech. Eng., vol. 330, pp. 271–291, 2018. https://doi.org/10.1016/j.cma.2017.10.028.Suche in Google Scholar
[33] T. Zhang, Y. X. Qian, and J. Y. Yuan, “The fully discrete fractional-step method for the Oldroyd model,” Appl. Numer. Math., vol. 129, pp. 83–103, 2018. https://doi.org/10.1016/j.apnum.2018.03.003.Suche in Google Scholar
[34] J. Li, P. Z. Huang, C. Zhang, and G. Guo, “A linear, decoupled fractional time-stepping method for the nonlinear fluid-fluid interaction,” Numer. Meth. PDEs, vol. 35, no. 5, pp. 1873–1889, 2019. https://doi.org/10.1002/num.22382.Suche in Google Scholar
[35] Y. Zhang and Y. N. He, “Assessment of subgrid-scale models for the incompressible Navier–Stokes equations,” J. Comput. Appl. Math., vol. 234, pp. 593–604, 2010. https://doi.org/10.1016/j.cam.2009.12.051.Suche in Google Scholar
[36] Y. Q. Shang, “A two-level subgrid stabilized Oseen iterative method for the steady Navier–Stokes equations,” J. Comput. Phys., vol. 233, no. 1, pp. 210–226, 2013. https://doi.org/10.1016/j.jcp.2012.08.024.Suche in Google Scholar
[37] H. E. Jia, K. M. Teng, and K. T. Li, “On the error estimates of a new operate splitting method for the Navier–Stokes equations,” J. Comput. Math., vol. 32, no. 1, pp. 75–92, 2014. https://doi.org/10.4208/jcm.1310-m4211.Suche in Google Scholar
[38] V. Girault and P. A. Raviart, Finite Element Methods for Navier–Stokes Equations: Theory and Algorithms, Berlin Heidelberg, Springer-Verlag, 1986.10.1007/978-3-642-61623-5Suche in Google Scholar
[39] J. G. Heywood and R. Rannacher, “Finite element approximations of the nonstationary Navier–Stokes problem: IV. Error estimates for second-order time discretization,” SIAM J. Numer. Anal., vol. 27, pp. 353–384, 1990. https://doi.org/10.1137/0727022.Suche in Google Scholar
[40] F. Hecht, “New development in freefem++,” J. Numer. Math., vol. 20, nos 3-4, pp. 251–266, 2012. https://doi.org/10.1515/jnum-2012-0013.Suche in Google Scholar
[41] U. Ghia, K. Ghia, and C. Shin, “High-Re solutions for incompressible flow using the Navier–Stokes equations and multigrid method,” J. Comput. Phys., vol. 48, no. 3, pp. 387–411, 1982. https://doi.org/10.1016/0021-9991(82)90058-4.Suche in Google Scholar
© 2021 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Original Research Articles
- Research on bifurcation and chaos characteristics of planet gear transmission system with mixed elastohydrodynamic lubrication (EHL) friction
- Nonlinear modeling and simulation of flywheel energy storage rotor system with looseness and rub-impact coupling hitch
- Application of the piecewise constant method in nonlinear dynamics of drill string
- Fractional Kirchhoff-type problems with exponential growth without the Ambrosetti–Rabinowitz condition
- A stabilized fractional-step finite element method for the time-dependent Navier–Stokes equations
- Self-adaptive inertial subgradient extragradient scheme for pseudomonotone variational inequality problem
- An accurate and novel numerical simulation with convergence analysis for nonlinear partial differential equations of Burgers–Fisher type arising in applied sciences
- Numerical study of multi-dimensional hyperbolic telegraph equations arising in nuclear material science via an efficient local meshless method
- N-soliton solutions and the Hirota conditions in (1 + 1)-dimensions
- Quintic B-spline collocation method for the numerical solution of the Bona–Smith family of Boussinesq equation type
- Existence of positive periodic solutions for a class of second-order neutral functional differential equations
Artikel in diesem Heft
- Frontmatter
- Original Research Articles
- Research on bifurcation and chaos characteristics of planet gear transmission system with mixed elastohydrodynamic lubrication (EHL) friction
- Nonlinear modeling and simulation of flywheel energy storage rotor system with looseness and rub-impact coupling hitch
- Application of the piecewise constant method in nonlinear dynamics of drill string
- Fractional Kirchhoff-type problems with exponential growth without the Ambrosetti–Rabinowitz condition
- A stabilized fractional-step finite element method for the time-dependent Navier–Stokes equations
- Self-adaptive inertial subgradient extragradient scheme for pseudomonotone variational inequality problem
- An accurate and novel numerical simulation with convergence analysis for nonlinear partial differential equations of Burgers–Fisher type arising in applied sciences
- Numerical study of multi-dimensional hyperbolic telegraph equations arising in nuclear material science via an efficient local meshless method
- N-soliton solutions and the Hirota conditions in (1 + 1)-dimensions
- Quintic B-spline collocation method for the numerical solution of the Bona–Smith family of Boussinesq equation type
- Existence of positive periodic solutions for a class of second-order neutral functional differential equations
