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A Fifth-Order Combined Compact Difference Scheme for Stokes Flow on Polar Geometries

Part of: Partial differential equations, boundary value problems

Published online by Cambridge University Press:  31 January 2018

Dongdong He  and
Kejia Pan
Dongdong He*
Affiliation:
School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen, 518172, China
Kejia Pan*
Affiliation:
School of Mathematics and Statistics, Central South University, Changsha 410083, China
*
*Corresponding author. Email addresses:hedongdong@cuhk.edu.cn (D.D. He), pankejia@hotmail.com (K.J. Pan)
*Corresponding author. Email addresses:hedongdong@cuhk.edu.cn (D.D. He), pankejia@hotmail.com (K.J. Pan)
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Abstract

Incompressible flows with zero Reynolds number can be modeled by the Stokes equations. When numerically solving the Stokes flow in stream-vorticity formulation with high-order accuracy, it will be important to solve both the stream function and velocity components with the high-order accuracy simultaneously. In this work, we will develop a fifth-order spectral/combined compact difference (CCD) method for the Stokes equation in stream-vorticity formulation on the polar geometries, including a unit disk and an annular domain. We first use the truncated Fourier series to derive a coupled system of singular ordinary differential equations for the Fourier coefficients, then use a shifted grid to handle the coordinate singularity without pole condition. More importantly, a three-point CCD scheme is developed to solve the obtained system of differential equations. Numerical results are presented to show that the proposed spectral/CCD method can obtain all physical quantities in the Stokes flow, including the stream function and vorticity function as well as all velocity components, with fifth-order accuracy, which is much more accurate and efficient than low-order methods in the literature.

Keywords

Stokes flowcombined compact difference (CCD) schemetruncated Fourier seriesshifted gridcoordinate singularity

MSC classification

Secondary: 65N06: Finite difference methods
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 7 , Issue 4 , November 2017 , pp. 714 - 727
Copyright
Copyright © Global-Science Press 2017 

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