Volume 7, Issue 4
A Fifth-Order Combined Compact Difference Scheme for Stokes Flow on Polar Geometries

Dongdong He & Kejia Pan

East Asian J. Appl. Math., 7 (2017), pp. 714-727.

Published online: 2018-02

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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 flow, combined compact difference (CCD) scheme, truncated Fourier series, shifted grid, coordinate singularity.

  • AMS Subject Headings

65N06

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{EAJAM-7-714, author = {}, title = {A Fifth-Order Combined Compact Difference Scheme for Stokes Flow on Polar Geometries}, journal = {East Asian Journal on Applied Mathematics}, year = {2018}, volume = {7}, number = {4}, pages = {714--727}, 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.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.200816.300517a}, url = {http://global-sci.org/intro/article_detail/eajam/10715.html} }
TY - JOUR T1 - A Fifth-Order Combined Compact Difference Scheme for Stokes Flow on Polar Geometries JO - East Asian Journal on Applied Mathematics VL - 4 SP - 714 EP - 727 PY - 2018 DA - 2018/02 SN - 7 DO - http://doi.org/10.4208/eajam.200816.300517a UR - https://global-sci.org/intro/article_detail/eajam/10715.html KW - Stokes flow, combined compact difference (CCD) scheme, truncated Fourier series, shifted grid, coordinate singularity. AB -

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.

Dongdong He & Kejia Pan. (2020). A Fifth-Order Combined Compact Difference Scheme for Stokes Flow on Polar Geometries. East Asian Journal on Applied Mathematics. 7 (4). 714-727. doi:10.4208/eajam.200816.300517a
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