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An Application of the Sinc-Collocation Method to a Three-Dimensional Oceanography Model
Abstract: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.
Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1331273Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1432
 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.
 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.
 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.
 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.
 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.
 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.
 F. Stenger, "Summary of sinc numerical methods," Journal of Computational and Applied Mathematics, vol. 121, no. 1, pp. 379-420, 2000.
 F. Keinert, "Uniform approximation to x β by sinc functions," Journal of approximation theory, vol. 66, no. 1, pp. 44-52, 1991.
 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.
 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.
 K. Al-Khaled, "Numerical approximations for population growth models," Applied mathematics and computation, vol. 160, no. 3, pp. 865-873, 2005.
 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.
 R. C. Smith and K. L. Bowers, "Sinc-galerkin estimation of diffusivity in parabolic problems," Inverse problems, vol. 9, no. 1, p. 113, 1999.
 K. Parand and A. Pirkhedri, "Sinc-collocation method for solving astrophysics equations," New Astronomy, vol. 15, no. 6, pp. 533-537, 2010.
 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.
 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.
 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.
 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.
 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.
 E. Hesameddini and E. Asadolahifard, "The sinc-collocation method for solving the telegraph equation," Journal of Computer Engineering and Informatics.
 A. Saadatmandi, "Numerical study of second painlev'e equation," Communications in Numerical Analysis, vol. 2012, 2012.
 B. Bialecki, "Sinc-collection methods for two-point boundary value problems," IMA Journal of Numerical Analysis, vol. 11, no. 3, pp. 357- 375, 1991.
 V. W. Ekman, "On the influence of the earth\-s rotation on ocean currents," Ark. Mat. Astron. Fys., vol. 2, pp. 1-53, 1905.
 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.
 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.
 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.
 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.
 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.
 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.
 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.
 R. Burden and J. Faires, "Numerical analysis 7th ed., brooks/cole, thomson learning," 2001.
 S. Koonprasert, "The sinc-galerkin method for problems," Ph.D. dissertation, MONTANA STATE UNIVERSITY Bozeman, 2003.