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

## 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.

## Corresponding author

*Corresponding author. Email addresses:hedongdong@cuhk.edu.cn (D.D. He), pankejia@hotmail.com (K.J. Pan)

## References

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[1] Chu, P. and Fan, C., A three-point combined compact difference scheme, J. Comput. Phys. 140, 370399 (1998).
[2] Chen, B.Y., He, D.D. and Pan, K.J., A linearized high-order combined compact difference scheme for multi-dimensional coupled Burgers’ equations, Numer. Math.-Theory. Me. In press.
[3] Denni, S.C.R. and Ng, M., Nguyen, P., Numerical solution for the steady motion of a viscous fluid inside a circular boundary using integral conditions, J. Comput. Phys. 108, 142152 (1993).
[4] Ehrlich, L.W. and Gupta, M.M., Some difference schemes for the biharmonic equation, SIAM J. Numer. Anal. 12, 773789 (1975).
[5] Greengard, L. and Kropinski, M.C., An integral equation approach to the incompressible Navier-Stokes equations in two dimensions, SIAM J. Sci. Comput. 20, 318336 (1998).
[6] Gao, G.H. and Sun, H.W., Three-point combined compact alternating direction implicit difference schemes for two-dimensional time-fractional advection-diffusion equations, Commun. Comput. Phys. 17, 487509 (2015).
[7] Gao, G.H. and Sun, H.W., Three-point combined compact difference schemes for time-fractional advection-diffusion equations with smooth solutions, J. Comput. Phys. 298, 520538 (2015).
[8] He, D.D., An unconditionally stable spatial sixth-order CCD-ADI method for the two-dimensional linear telegraph equation, Numer. Algorithms 72, 11031117 (2016).
[9] He, D.D. and Pan, K.J., An unconditionally stable linearized CCD-ADI method for generalized nonlinear Schrödinger equations with variable coefficients in two and three dimensions, Comput. Math. Appl. 73, 23602374 (2017).
[10] Hanse, M.O.L. and Shen, W.Z., Vorticity-velocity formulation of the 3D Navier-Stokes equations in cylindrical co-ordinates, Int. J. Numer. Meth. Fl. 41, 2945 (2003).
[11] Huang, W. and Tang, T., Pseudospectral solutions for steady motion of a viscous fluid inside a circular boundary, Appl. Numer. Math. 33, 167173 (2000).
[12] Ito, K. and Qiao, Z.H., A high order compact MAC finite difference scheme for the Stokes equations: Augmented variable approach, J. Comput. Phys. 227, 81778190 (2008).
[13] Kohr, M. and Pop, I., Viscous incompressible flow for low Reynolds numbers, WIT, Southampton, Boston, 2004.
[14] Karageorghis, A. and Tang, T., A spectral domain decomposition approach for steady Navier-Stokes problems in circular geometries, Comput. Fluids 25, 541549 (1996).
[15] Lai, M.C., A simple compact fourth-order poisson solver on polar geometry, J. Comput. Phys. 182, 337345 (2002).
[16] Lai, M.C., Fourth-order finite difference scheme for the incompressible Navier-Stokes equations in a disk, Int. J. Numer. Meth. Fl. 42, 909922 (2003).
[17] Lai, M.C., Lin, W.W. and Wang, W., A fast spectral/difference method without pole conditions for Poisson-type equations in cylindrical and spherical geometries, IMA J. Numer. Anal. 22, 537548 (2002).
[18] Lai, M.C. and Liu, H.C., Fast direct solver for the biharmonic equation on a disk and its application to incompressible flows, Appl. Math. Comput. 164, 679695 (2005).
[19] Lopez, J.M., Marques, F. and Shen, J., An efficient spectral-projection method for the Navier-Stokes equations in cylindrical geometries: II. three-dimensional cases, J. Comput. Phys. 176, 384401 (2002).
[20] Lopez, J.M. and Shen, J., An efficient spectral-projection method for the Navier-Stokes equations in cylindrical geometries: I. axisymmetric cases, J. Comput. Phys. 139, 308326 (1998).
[21] Lai, M.C. and Wang, W.C., Fast direct solvers for Poisson equation on 2D polar and spherical geometries, Numer. Meth. Part. D. E. 18, 5668 (2002).
[22] Mahesh, K., A family of high order finite difference schemes with good spectral resolution, J. Comput. Phys. 145, 332358 (1998).
[23] Mohseni, K. and Colonius, T., Numerical treatment of polar coordinate singularities, J. Comput. Phys. 157, 787795 (2000).
[24] Nygård, F. and Andersson, H.I., On pragmatic parallelization of a serial Navier-Stokes solver in cylindrical coordinates, Int. J. Numer. Method. H. 22, 503511 (2012).
[25] Nihei, T. and Ishii, K., A fast solver of the shallow water equations on a sphere using a combined compact difference scheme, J. Comput. Phys. 187, 639659 (2003).
[26] Purcell, E.M., Life at low Reynolds number, Am. J. Phys. 45, 311 (1977).
[27] Pandit, S.K. and Karmakar, H., An efficient implicit compact streamfunction velocity formulation of two dimensional flows, J. Sci. Comput. 68, 653688 (2016).
[28] Pulicani, J.P. and Ouazzani, J., A Fourier-Chebyshev pseudospectral method for solving steady 3-D Navier-Stokes and heat equations in cylindrical cavities, Comput. Fluids 20, 93109 (1991).
[29] Sha, W., Nakabayashi, K. and Ueda, H., An accurate second-order approximation factorization method for time-dependent incompressible Navier-Stokes equations in spherical polar coordinates, J. Comput. Phys. 142, 4766 (1998).
[30] Sun, H.W. and Li, L.Z., A CCD-ADI method for unsteady convection-diffusion equations, Comput. Phys. Commun. 185, 790797 (2014).
[31] Lee, S.T., Liu, J. and Sun, H.W., Combined compact difference scheme for linear second-order partial differential equations with mixed derivative, J. Comput. Appl. Math. 264, 2337 (2014).
[32] Sengupta, T.K., Lakshmanan, V. and Vijay, V., A new combined stable and dispersion relation preserving compact scheme for non-periodic problems, J. Comput. Phys. 228, 30483071 (2009).
[33] Sengupta, T.K., Vijay, V. and Bhaumik, S., Further improvement and analysis of CCD scheme: dissipation discretization and de-aliasing properties, J. Comput. Phys. 228, 61506168 (2009).
[34] Torres, D.J. and Coutsias, E.A., Pseudospectral solution of the two-dimensional Navier-Stokes equations in a disk, SIAM J. Sci. Comput. 21, 378403 (1999).
[35] Tian, Z.F. and Yu, P.X., An efficient compact difference scheme for solving the streamfunction formulation of the incompressible Navier-Stokes equations, J. Comput. Phys. 230, 64046419 (2011).
[36] Verzicco, R. and Orlandi, P., A finite-difference scheme for three-dimensional incompressible flows in cylindrical coordinates, J. Comput. Phys. 123, 402414 (1996).
[37] Yu, P.X. and Tian, Z.F., A compact scheme for the streamfunction-velocity formulation of the 2D steady incompressible Navier-Stokes equations in polar coordinaes, J. Sci. Comput. 56, 165189 (2013).
[38] Zielinski, A.P., On trial functions applied in the generalized Trefftz method, Adv. Eng. Software 24, 147155 (1995).

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