An Application of the Sinc-Collocation Method to a Three-Dimensional Oceanography Model

In this paper, we explore the applicability of the Sinc- Collocation method to a three-dimensional (3D) oceanography model. The model describes a wind-driven current with depth-dependent eddy viscosity in the complex-velocity system. In general, the Sinc-based methods excel over other traditional numerical methods due to their exponentially decaying errors, rapid convergence and handling problems in the presence of singularities in end-points. Together with these advantages, the Sinc-Collocation approach that we utilize exploits first derivative interpolation, whose integration is much less sensitive to numerical errors. We bring up several model problems to prove the accuracy, stability, and computational efficiency of the method. The approximate solutions determined by the Sinc-Collocation technique are compared to exact solutions and those obtained by the Sinc-Galerkin approach in earlier studies. Our findings indicate that the Sinc-Collocation method outperforms other Sinc-based methods in past studies.

Authors:

• Y. Mohseniahouei
• K. Abdella
• M. Pollanen

Keywords:

• Boundary Value Problems
• Differential Equations
• Sinc Numerical Methods
• Wind-Driven Currents

References:

[1] M. A. Noor and E. Al-Said, "Finite-difference method for a system of
third-order boundary-value problems," Journal of optimization theory
and applications, vol. 112, no. 3, pp. 627-637, 2002.
[2] O. Axelsson and V. A. Barker, Finite element solution of boundary value
problems: theory and computation. Society for Industrial and Applied
Mathematics, 1987, vol. 35.
[3] Z. Csendes, "A novel finite element method for two-point boundary
value problems," Mathematics and Computers in Simulation, vol. 20,
no. 3, pp. 197-203, 1978.
[4] K. Ruotsalainen and W. Wendland, "On the boundary element method
for some nonlinear boundary value problems," Numerische Mathematik,
vol. 53, no. 3, pp. 299-314, 1988.
[5] B. S. Attili and M. I. Syam, "Efficient shooting method for solving two
point boundary value problems," Chaos, Solitons & Fractals, vol. 35,
no. 5, pp. 895-903, 2008.
[6] J. Rashidinia and M. Ghasemi, "B-spline collocation for solution of twopoint
boundary value problems," Journal of computational and applied
mathematics, vol. 235, no. 8, pp. 2325-2342, 2011.
[7] F. Stenger, "Summary of sinc numerical methods," Journal of Computational
and Applied Mathematics, vol. 121, no. 1, pp. 379-420, 2000.
[8] F. Keinert, "Uniform approximation to x β by sinc functions," Journal
of approximation theory, vol. 66, no. 1, pp. 44-52, 1991.
[9] S. Narasimhan, J. Majdalani, and F. Stenger, "A first step in applying
the sinc collocation method to the nonlinear navier-stokes equations,"
Numerical Heat Transfer: Part B: Fundamentals, vol. 41, no. 5, pp.
447-462, 2002.
[10] A. Lippke, "Analytical solutions and sinc function approximations in
thermal conduction with nonlinear heat generation," Journal of Heat
Transfer (Transcations of the ASME (American Society of Mechanical
Engineers), Series C);(United States), vol. 113, no. 1, 1991.
[11] K. Al-Khaled, "Numerical approximations for population growth models,"
Applied mathematics and computation, vol. 160, no. 3, pp. 865-873,
2005.
[12] J. Lund and C. R. Vogel, "A fully-galerkin method for the numerical solution
of an inverse problem in a parabolic partial differential equation,"
Inverse Problems, vol. 6, no. 2, p. 205, 1999.
[13] R. C. Smith and K. L. Bowers, "Sinc-galerkin estimation of diffusivity
in parabolic problems," Inverse problems, vol. 9, no. 1, p. 113, 1999.
[14] K. Parand and A. Pirkhedri, "Sinc-collocation method for solving
astrophysics equations," New Astronomy, vol. 15, no. 6, pp. 533-537,
2010.
[15] M. El-Gamel and A. Zayed, "Sinc-galerkin method for solving nonlinear
boundary-value problems," Computers & Mathematics with Applications,
vol. 48, no. 9, pp. 1285-1298, 2004.
[16] F. Stenger and M. J. O-Reilly, "Computing solutions to medical problems
via sinc convolution," Automatic Control, IEEE Transactions on,
vol. 43, no. 6, pp. 843-848, 1998.
[17] K. Abdella, X. Yu, and I. Kucuk, "Application of the sinc method to a
dynamic elasto-plastic problem," Journal of Computational and Applied
Mathematics, vol. 223, no. 2, pp. 626-645, 2009.
[18] D. Winter, K. L. Bowers, and J. Lund, "Wind-driven currents in a sea
with a variable eddy viscosity calculated via a sinc-galerkin technique,"
International journal for numerical methods in fluids, vol. 33, no. 7, pp.
1041-1073, 2000.
[19] S. Koonprasert and K. L. Bowers, "Block matrix sinc-galerkin solution
of the wind-driven current problem," Applied mathematics and computation,
vol. 155, no. 3, pp. 607-635, 2004.
[20] E. Hesameddini and E. Asadolahifard, "The sinc-collocation method for
solving the telegraph equation," Journal of Computer Engineering and
Informatics.
[21] A. Saadatmandi, "Numerical study of second painlev'e equation," Communications
in Numerical Analysis, vol. 2012, 2012.
[22] B. Bialecki, "Sinc-collection methods for two-point boundary value
problems," IMA Journal of Numerical Analysis, vol. 11, no. 3, pp. 357-
375, 1991.
[23] V. W. Ekman, "On the influence of the earth\-s rotation on ocean
currents," Ark. Mat. Astron. Fys., vol. 2, pp. 1-53, 1905.
[24] N. Heaps et al., "On the numerical solution of the three-dimensional
hydrodynamical equations for tides and storm surges," M'emoires de la
Soci'et'e Royale des Sciences de Li`ege. Sixi`eme S'erie, 1972.
[25] N. Heaps, "Three-dimensional model for tides and surges with vertical
eddy viscosity prescribed in two layersi. mathematical formulation,"
Geophysical Journal of the Royal Astronomical Society, vol. 64, no. 1,
pp. 291-302, 1981.
[26] R. Lardner and Y. Song, "A hybrid spectral method for the threedimensional
numerical modelling of nonlinear flows in shallow seas,"
Journal of Computational Physics, vol. 100, no. 2, pp. 322-334, 1992.
[27] A. Davies, "The numerical solution of the three-dimensional hydrodynamic
equations, using a b-spline representation of the vertical current
profile," Elsevier Oceanography Series, vol. 19, pp. 1-25, 1977.
[28] A. Davies and A. Owen, "Three dimensional numerical sea model using
the galerkin method with a polynomial basis set," Applied mathematical
modelling, vol. 3, no. 6, pp. 421-428, 1979.
[29] A. M. Davies, "Solution of the 3d linear hydrodynamic equations
using an enhanced eigenfunction approach," International journal for
numerical methods in fluids, vol. 13, no. 2, pp. 235-250, 1991.
[30] K. Abdella, "Numerical solution of two-point boundary value problems
using sinc interpolation," in Proceedings of the American Conference
on Applied Mathematics (American-Math-12): Applied Mathematics in
Electrical and Computer Engineering, 2012, pp. 157-162.
[31] R. Burden and J. Faires, "Numerical analysis 7th ed., brooks/cole,
thomson learning," 2001.
[32] S. Koonprasert, "The sinc-galerkin method for problems," Ph.D. dissertation,
MONTANA STATE UNIVERSITY Bozeman, 2003.