Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-26T19:16:08.090Z Has data issue: false hasContentIssue false
  • English
  • Français

Pressure-Correction Projection FEM for Time-Dependent Natural Convection Problem

Part of: Equations of mathematical physics and other areas of application Partial differential equations, boundary value problems Incompressible viscous fluids

Published online by Cambridge University Press:  08 March 2017

Jilian Wu ,
Xinlong Feng  and
Fei Liu
Jilian Wu*
Affiliation:
College of Mathematics and Systems Science, Xinjiang University, Urumqi, 830046, P.R. China
Xinlong Feng*
Affiliation:
College of Mathematics and Systems Science, Xinjiang University, Urumqi, 830046, P.R. China
Fei Liu*
Affiliation:
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, P.R. China
*
*Corresponding author. Email addresses:happy_whw@163.com (J. Wu), fxlmath@gmail.com (X. Feng), liufei_2000@163.com (F. Liu)
*Corresponding author. Email addresses:happy_whw@163.com (J. Wu), fxlmath@gmail.com (X. Feng), liufei_2000@163.com (F. Liu)
*Corresponding author. Email addresses:happy_whw@163.com (J. Wu), fxlmath@gmail.com (X. Feng), liufei_2000@163.com (F. Liu)
Get access

Abstract

Pressure-correction projection finite element methods (FEMs) are proposed to solve nonstationary natural convection problems in this paper. The first-order and second-order backward difference formulas are applied for time derivative, the stability analysis and error estimates of the semi-discrete schemes are presented using energy method. Compared with characteristic variational multiscale FEM, pressure-correction projection FEMs are more efficient and unconditionally energy stable. Ample numerical results are presented to demonstrate the effectiveness of the pressure-correction projection FEMs for solving these problems.

Keywords

Time-dependent natural convection problemspressure-correction projection FEMsbackward difference formulastability analysiserror estimate

MSC classification

Secondary: 65N15: Error bounds 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 76D07: Stokes and related (Oseen, etc.) flows 65N12: Stability and convergence of numerical methods 35Q79: PDEs in connection with classical thermodynamics and heat transfer
Type
Research Article
Information
Communications in Computational Physics , Volume 21 , Issue 4 , April 2017 , pp. 1090 - 1117
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Communicated by Jie Shen

References

[1] Boland, J. and Layton, W.. An analysis of the finite element method for natural convection problems. Numer. Meth. Part. Differ. Equ., 6(2): 115–126, 1990.Google Scholar
[2] Chorin, A.. Numerical solution of the Navier-Stokes equations. Math. Comp., 22: 745–762, 1968.CrossRefGoogle Scholar
[3] De Vahl Davis, G. Natural convection of air in a square cavity: a bench mark numerical solution. Int. J. Numer. Meth. Fluids, 3(3): 249–264, 1983.Google Scholar
[4] W. E, and Liu, J.. Projection method. I. Convergence and numerical boundary layers. SIAM J. Numer. Anal., 32(4): 1017–1057, 1995.Google Scholar
[5] W. E, and Liu, J.. Gauge method for viscous incompressible flows. Commun. Math. Sci., 1(2): 317–332, 2003.Google Scholar
[6] Goda, K.. A multistep technique with implicit difference schemes for calculating two-or three-dimensional cavity flows. J. Comput. Phys., 30(1): 76–95, 1979.CrossRefGoogle Scholar
[7] Guermond, J., Minev, P., and Shen, J.. An overview of projection methods for incompressible flows. Comput. Meth. Appl. Mech. Engrg., 195(44-47): 6011–6045, 2006.CrossRefGoogle Scholar
[8] Guermond, J. and Shen, J.. A new class of truly consistent splitting schemes for incompressible flows. J. Comput. Phys., 192(1): 262–276, 2003.Google Scholar
[9] Guermond, J. and Shen, J.. Velocity-correction projection methods for incompressible flows. SIAM J. Numer. Anal., 41(1): 112–134, 2003.Google Scholar
[10] Guermond, J. and Shen, J.. On the error estimates for the rotational pressure-correction projection methods. Math. Comp., 73(248): 1719–1737, 2004.Google Scholar
[11] He, Y.. Stability and convergence of iterative methods related to viscosities for the 2D/3D steady Navier-Stokes equations. J. Math. Anal. Appl., 423(2): 1129–1149, 2015.Google Scholar
[12] He, Y. and Li, J.. A stabilized finite element method based on local polynomial pressure projection for the stationary Navier-Stokes equations. Appl. Numer.Math., 58(10): 1503–1514, 2008.Google Scholar
[13] Luo, Z., Zhu, J., Xie, Z., and Zhang, G.. Difference scheme and numerical simulation based on mixed finite elementmethod for natural convection problem. Appl. Math. Mech., 24(9): 1100–1110, 2003.Google Scholar
[14] Nochetto, R. and Pyo, J.. Error estimates for semi-discrete gauge methods for the Navier-Stokes equations. Math. Comp., 74(250): 521–542, 2005.Google Scholar
[15] Pyo, J.. Error estimates for the second order semi-discrete stabilized gauge-Uzawa method for the Navier-Stokes equations. Int. J. Numer. Anal. Model., 10(1): 24–41, 2013.Google Scholar
[16] Shen, J.. On error estimates of projection methods for Navier-Stokes equations: first-order schemes. SIAM J. Numer. Anal., 29(1): 57–77, 1992.Google Scholar
[17] Shen, J.. Modeling and numerical approximation of two-phase incompressible flows by a phase-field approach. In Multiscale modeling and analysis for materials simulation, volume 22 of Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., pages 147–195. World Sci. Publ., Hackensack, NJ, 2012.CrossRefGoogle Scholar
[18] Si, Z., Song, X., and Huang, P.. Modified characteristics gauge-Uzawa finite element method for time dependent conduction-convection problems. J. Sci. Comput., 58(1): 1–24, 2014.CrossRefGoogle Scholar
[19] Su, H., Qian, L., Gui, D., and Feng, X.. Second order fully discrete and divergence free conserving scheme for time-dependent conduction–convection equations. Int. Comm. Heat Mass Trans., 59: 120–129, 2014.CrossRefGoogle Scholar
[20] Temam, R.. Sur l’approximation de la solution des équations de navier-stokes par la méthode des pas fractionnaires II. Archive for Rational Mechanics and Analysis, 33(5): 377–385, 1969.Google Scholar
[21] Timmermans, L., Minev, P., and Van De Vosse, F.. An approximate projection scheme for incompressible flow using spectral elements. Inter. J. Numer. Meth. Fluids, 22(7): 673–688, 1996.Google Scholar
[22] Wan, D., Patnaik, B., and Wei, G.. A new benchmark quality solution for the buoyancy-driven cavity by discrete singular convolution. Numer. Heat Trans., Part B, 40(3): 199–228, 2001.Google Scholar
[23] Wu, J., Gui, D., Liu, D., and Feng, X.. The characteristic variational multiscale method for time dependent conduction–convection problems. Int. Comm. Heat Mass Trans., 68: 58–68, 2015.Google Scholar
[24] Zhang, T. and Tao, Z.. Decoupled scheme for time-dependent natural convection problem II: Time semidiscreteness. Math. Probl. Eng., 2014.Google Scholar
[25] Zhang, Y., Hou, Y., and Zheng, H.. A finite element variational multiscale method for steady-state natural convection problem based on two local gauss integrations. Numer. Meth. Part. Differ. Equ., 30(2): 361–375, 2014.Google Scholar