Commun. Comput. Phys., 5 (2009), pp. 712-729.


A Third-Order Upwind Compact Scheme on Curvilinear Meshes for the Incompressible Navier-Stokes Equations

Abdullah Shah 1, Hong Guo 2, Li Yuan 2*

1 LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China; and Department of Mathematics, COMSATS Institute of Information Technology, Islamabad, Pakistan.
2 LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China.

Received 30 September 2007; Accepted (in revised version) 12 April 2008
Available online 1 August 2008

Abstract

This paper presents a new version of the upwind compact finite difference scheme for solving the incompressible Navier-Stokes equations in generalized curvilinear coordinates. The artificial compressibility approach is used, which transforms the elliptic-parabolic equations into the hyperbolic-parabolic ones so that flux difference splitting can be applied. The convective terms are approximated by a third-order upwind compact scheme implemented with flux difference splitting, and the viscous terms are approximated by a fourth-order central compact scheme. The solution algorithm used is the Beam-Warming approximate factorization scheme. Numerical solutions to benchmark problems of the steady plane Couette-Poiseuille flow, the lid-driven cavity flow, and the constricting channel flow with varying geometry are presented. The computed results are found in good agreement with established analytical and numerical results. The third-order accuracy of the scheme is verified on uniform rectangular meshes.

AMS subject classifications: 76D05, 65N06

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Key words: Upwind compact difference, flux difference splitting, incompressible Navier-Stokes equations, artificial compressibility, lid-driven cavity flow.

*Corresponding author.
Email: abdullah@lsec.cc.ac.cn (A. Shah), guoh@lsec.cc.ac.cn (H. Guo), lyuan@lsec.cc.ac.cn (L. Yuan)
 

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