Upwind compact finite difference scheme for time-accurate solution of the incompressible Navier-Stokes equations

Abdullah Shah; Li Yuan; Aftab Khan

doi:10.1016/j.amc.2009.10.001

Upwind compact finite difference scheme for time-accurate solution of the incompressible Navier-Stokes equations

Abdullah Shah^*, Li Yuan, Aftab Khan

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

65 Scopus citations

Abstract

This article presents a time-accurate numerical method using high-order accurate compact finite difference scheme for the incompressible Navier-Stokes equations. The method relies on the artificial compressibility formulation, which endows the governing equations a hyperbolic-parabolic nature. The convective terms are discretized with a third-order upwind compact scheme based on flux-difference splitting, and the viscous terms are approximated with a fourth-order central compact scheme. Dual-time stepping is implemented for time-accurate calculation in conjunction with Beam-Warming approximate factorization scheme. The present compact scheme is compared with an established non-compact scheme via analysis in a model equation and numerical tests in four benchmark flow problems. Comparisons demonstrate that the present third-order upwind compact scheme is more accurate than the non-compact scheme while having the same computational cost as the latter.

Original language	English
Pages (from-to)	3201-3213
Number of pages	13
Journal	Applied Mathematics and Computation
Volume	215
Issue number	9
DOIs	https://doi.org/10.1016/j.amc.2009.10.001
State	Published - 1 Jan 2010
Externally published	Yes

Bibliographical note

Funding Information:
The work of A. Shah was financially supported by CIIT ( IDB-1306 ) and L. Yuan was supported by Natural Science Foundation of China ( G10531080 and G10729101 ) and State Key Program for Developing Basic Sciences ( 2005CB321703 ). We would like to thanks the referees for their comments and suggestions and Prof. Michele Napolitano, Department of Mechanical and Industrial Engineering, University of Bari, Italy for providing his FORTRAN subroutine [42] .

Keywords

Artificial compressibility method
Doubly periodic shear layer
Dual-time stepping
Flux-difference splitting
Incompressible Navier-Stokes equation
Kovasznay flow problem
Oscillating plate
Taylor's decaying vortices
Upwind compact finite difference

ASJC Scopus subject areas

Computational Mathematics
Applied Mathematics

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10.1016/j.amc.2009.10.001

Cite this

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title = "Upwind compact finite difference scheme for time-accurate solution of the incompressible Navier-Stokes equations",

abstract = "This article presents a time-accurate numerical method using high-order accurate compact finite difference scheme for the incompressible Navier-Stokes equations. The method relies on the artificial compressibility formulation, which endows the governing equations a hyperbolic-parabolic nature. The convective terms are discretized with a third-order upwind compact scheme based on flux-difference splitting, and the viscous terms are approximated with a fourth-order central compact scheme. Dual-time stepping is implemented for time-accurate calculation in conjunction with Beam-Warming approximate factorization scheme. The present compact scheme is compared with an established non-compact scheme via analysis in a model equation and numerical tests in four benchmark flow problems. Comparisons demonstrate that the present third-order upwind compact scheme is more accurate than the non-compact scheme while having the same computational cost as the latter.",

keywords = "Artificial compressibility method, Doubly periodic shear layer, Dual-time stepping, Flux-difference splitting, Incompressible Navier-Stokes equation, Kovasznay flow problem, Oscillating plate, Taylor's decaying vortices, Upwind compact finite difference",

author = "Abdullah Shah and Li Yuan and Aftab Khan",

note = "Funding Information: The work of A. Shah was financially supported by CIIT ( IDB-1306 ) and L. Yuan was supported by Natural Science Foundation of China ( G10531080 and G10729101 ) and State Key Program for Developing Basic Sciences ( 2005CB321703 ). We would like to thanks the referees for their comments and suggestions and Prof. Michele Napolitano, Department of Mechanical and Industrial Engineering, University of Bari, Italy for providing his FORTRAN subroutine [42] .",

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AU - Khan, Aftab

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N2 - This article presents a time-accurate numerical method using high-order accurate compact finite difference scheme for the incompressible Navier-Stokes equations. The method relies on the artificial compressibility formulation, which endows the governing equations a hyperbolic-parabolic nature. The convective terms are discretized with a third-order upwind compact scheme based on flux-difference splitting, and the viscous terms are approximated with a fourth-order central compact scheme. Dual-time stepping is implemented for time-accurate calculation in conjunction with Beam-Warming approximate factorization scheme. The present compact scheme is compared with an established non-compact scheme via analysis in a model equation and numerical tests in four benchmark flow problems. Comparisons demonstrate that the present third-order upwind compact scheme is more accurate than the non-compact scheme while having the same computational cost as the latter.

AB - This article presents a time-accurate numerical method using high-order accurate compact finite difference scheme for the incompressible Navier-Stokes equations. The method relies on the artificial compressibility formulation, which endows the governing equations a hyperbolic-parabolic nature. The convective terms are discretized with a third-order upwind compact scheme based on flux-difference splitting, and the viscous terms are approximated with a fourth-order central compact scheme. Dual-time stepping is implemented for time-accurate calculation in conjunction with Beam-Warming approximate factorization scheme. The present compact scheme is compared with an established non-compact scheme via analysis in a model equation and numerical tests in four benchmark flow problems. Comparisons demonstrate that the present third-order upwind compact scheme is more accurate than the non-compact scheme while having the same computational cost as the latter.

KW - Artificial compressibility method

KW - Doubly periodic shear layer

KW - Dual-time stepping

KW - Flux-difference splitting

KW - Incompressible Navier-Stokes equation

KW - Kovasznay flow problem

KW - Oscillating plate

KW - Taylor's decaying vortices

KW - Upwind compact finite difference

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JO - Applied Mathematics and Computation

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ER -

Upwind compact finite difference scheme for time-accurate solution of the incompressible Navier-Stokes equations

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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