Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-28T05:09:22.085Z Has data issue: false hasContentIssue false
  • English
  • Français

Spectral Direction Splitting Schemes for the Incompressible Navier-Stokes Equations

Published online by Cambridge University Press:  28 May 2015

Lizhen Chen ,
Jie Shen  and
Chuanju Xu
Lizhen Chen
Affiliation:
School of Mathematical Sciences, Xiamen University, 361005 Xiamen, China
Jie Shen
Affiliation:
School of Mathematical Sciences, Xiamen University, 361005 Xiamen, China Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA
Chuanju Xu*
Affiliation:
School of Mathematical Sciences, Xiamen University, 361005 Xiamen, China
*
Corresponding author. Email: cjxu@xmu.edu.cn
Get access

Abstract

We propose and analyze spectral direction splitting schemes for the incompressible Navier-Stokes equations. The schemes combine a Legendre-spectral method for the spatial discretization and a pressure-stabilization/direction splitting scheme for the temporal discretization, leading to a sequence of one-dimensional elliptic equations at each time step while preserving the same order of accuracy as the usual pressure-stabilization schemes. We prove that these schemes are unconditionally stable, and present numerical results which demonstrate the stability, accuracy, and efficiency of the proposed methods.

Keywords

Navier-Stokes equationsprojection methoddirection splittingspectral methods
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 1 , Issue 3 , August 2011 , pp. 215 - 234
Copyright
Copyright © Global-Science Press 2011

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.)

References

[1]Bernardi, C. and Maday, Y.. Approximations Spectrales de Problèmes aux Limites Elliptiques. Springer-Verlag, Paris, 1992.Google Scholar
[2]Chorin, Alexandre Joel. Numerical solution of the Navier-Stokes equations. Math. Comp., 22:745–762. 1968.Google Scholar
[3]Guermond, J. L. and Shen, Jie. On the error estimates for the rotational pressure-correction projection methods. Math. Comp., 73(248):1719–1737 (electronic), 2004.Google Scholar
[4]Guermond, Jean-Luc and Minev, Peter D.. A new class of fractional step techniques for the incompressible Navier-Stokes equations using direction splitting. C. R. Math. Acad. Sci. Paris, 348(9-10):581–585, 2010.CrossRefGoogle Scholar
[5]Guermond, J.L., Minev, P., and Shen, Jie. An overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Engrg., 195:6011–6045, 2006.Google Scholar
[6]Guermond, J.L., Minev, P.D., and Salgado, A.J.Convergence analysis of a class of massively parallel direction splitting algorithms for the Navier-Stokes equations. Preprint, 2011.Google Scholar
[7]Peaceman, D. W. and Rachford, H. H. Jr.The numerical solution of parabolic and elliptic differential equations. J. Soc. Indust. Appl. Math., 3:28–41, 1955.Google Scholar
[8]Shen, Jie. On error estimates of projection methods for the Navier-Stokes equations: second- order schemes. Math. Comp, 65:1039–1065, July 1996.Google Scholar
[9]Témam, R.. Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires. I. Arch. Rational Mech. Anal., 32:135–153, 1969.Google Scholar
[10]Timmermans, L. J. P., Minev, P. D., and Vosse, F. N. Van De. An approximate projection scheme for incompressible flow using spectral elements. Int. J. Numer. Methods Fluids, 22:673–688, 1996.Google Scholar