Hostname: page-component-76fb5796d-skm99 Total loading time: 0 Render date: 2024-04-27T01:41:42.763Z Has data issue: false hasContentIssue false
  • English
  • Français

A Unstructured Nodal Spectral-Element Method for the Navier-Stokes Equations

Published online by Cambridge University Press:  20 August 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
*
Email:xunyicao2009@yeah.net
Email:shen7@purdue.edu
Corresponding author.Email:cjxu@xmu.edu.en
Get access

Abstract

An unstructured nodal spectral-element method for the Navier-Stokes equations is developed in this paper. The method is based on a triangular and tetrahedral rational approximation and an easy-to-implement nodal basis which fully enjoys the tensorial product property. It allows arbitrary triangular and tetrahedral mesh, affording greater flexibility in handling complex domains while maintaining all essential features of the usual spectral-element method. The details of the implementation and some numerical examples are provided to validate the efficiency and flexibility of the proposed method.

Keywords

65N3565N2265F0535J05Navier-Stokes equationsunstructured meshtriangular spectral element method
Type
Research Article
Information
Communications in Computational Physics , Volume 12 , Issue 1 , July 2012 , pp. 315 - 336
Copyright
Copyright © Global Science Press Limited 2012

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]Boyd, J. P.. Chebyshev and Fourier spectral methods. Dover Publications Inc., Mineola, NY, second edition, 2001.Google Scholar
[2]Braess, D. and Schwab, C.. Approximation on simplices with respect to weighted Sobolev norms. J. Approx. Theory, 103(2):329–337, 2000.Google Scholar
[3]Chen, L., Shen, J., and Xu, C.. A triangular spectral method for the Stokes equations. Numer. Math.: Theory, Methods and Applications, to appear.Google Scholar
[4]Deville, M. O. and Mund, E. H.. Chebyshev pseudospectral solution of second-order elliptic equations with finite element preconditioning. J. Comput. Phys., 60:517–533, 1985.CrossRefGoogle Scholar
[5]Dubiner, M.. Spectral methods on triangles and other domains. J. Sci. Comput., 6(4):345–390, 1991.Google Scholar
[6]Gordon, W. J. and Hall, C. A.. Transfinite element methods: blending-function interpolation over arbitrary curved element domains. Numer. Math., 21:109–129, 1973/74.Google Scholar
[7]Heinrichs, W. and Loch, B. I.. Spectral schemes on triangular elements. J. Comput. Phys., 173(1):279–301, 2001.Google Scholar
[8]Hesthaven, J. S.. From electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex. SIAM J. Numer. Anal., 35(2):655–676 (electronic), 1998.Google Scholar
[9]Karniadakis, G. E. and Sherwin, S. J.. Spectral/hp element methods for CFD. Numerical Mathematics and Scientific Computation. Oxford University Press, New York, 1999.Google Scholar
[10]Koornwinder, T.. Two-variable analogues of the classical orthogonal polynomials. In Theory and application of special functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), pages 435–495. Math. Res. Center, Univ. Wisconsin, Publ. No. 35. Academic Press, New York, 1975.Google Scholar
[11]Maday, Y. and Patera, A. T.. Spectral element methods for the incompressible Navier-Stokes equations. In IN: State-of-the-art surveys on computational mechanics (A90-47176 21-64). New York, American Society of Mechanical Engineers, 1989, p. 71-143. Research supported by DARPA., volume 1, pages 71–143, 1989.Google Scholar
[12]Maday, Y., Meiron, D., Patera, A. T., and Rønquist, E. M.. Analysis of iterative methods for the steady and unsteady Stokes problem: application to spectral element discretizations. SIAM J. Sci. Comput., 14(2):310–337, 1993.Google Scholar
[13]Orszag, S. A.. Spectral methods for complex geometries. J. Comput. Phys., 37:70–92, 1980.Google Scholar
[14]Owens, R. G.. Spectral approximations on the triangle. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 454(1971):857–872, 1998.Google Scholar
[15]Pasquetti, R. and Rapetti, F.. Spectral element methods on triangles and quadrilaterals: comparisons and applications. Journal of Computational Physics, 198(1):349–362, 2004.Google Scholar
[16]Pasquetti, R. and Rapetti, F.. Spectral element methods on unstructured meshes: comparisons and recent advances. J. Sci. Comput., 27(1-3):377–387, 2006.Google Scholar
[17]Patera, A. T.. A spectral element method for fluid dynamics: laminar flow in a channel expansion. Journal of Computational Physics, 54(3):468–488, 1984.Google Scholar
[18]Shen, J., Wang, L.-L., and Li, H.. A triangular spectral element method using fully tensorial rational basis functions. SIAM J. Numer. Anal., 47(3):1619–1650, 2009.CrossRefGoogle Scholar
[19]Sherwin, S. J. and Karniadakis, G. E.. A triangular spectral element method; applications to the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg., 123(1-4):189–229, 1995.Google Scholar
[20]Taylor, M. A., Wingate, B. A., and Vincent, R. E.. An algorithm for computing Fekete points in the triangle. SIAM J. Numer. Anal., 38(5):1707–1720 (electronic), 2000.CrossRefGoogle Scholar
[21]Xu, C. and Pasquetti, R.. On the efficiency of semi-implicit and semi-Lagrangian spectral methods for the calculation of incompressible flows. International Journal for Numerical Methods in Fluids, 35(3):319–340,2001.Google Scholar
[22]Xu, C. and Maday, Y.. A spectral element method for the time-dependent two-dimensional euler equations: applications to flow simulations. J. Comput. Appl. Math., 91(1):63–85,1998.Google Scholar