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An application of one-sided Jacobi polynomials for spectral modeling of vector fields in polar coordinates. (English) Zbl 1175.65120
Summary: A spectral tau-method is proposed for solving vector field equations defined in polar coordinates. The method employs one-sided Jacobi polynomials as radial expansion functions and Fourier exponentials as azimuthal expansion functions. All the regularity requirements of the vector field at the origin and the physical boundary conditions at a circumferential boundary are exactly satisfied by adjusting the additional tau-coefficients of the radial expansion polynomials of the highest order. The proposed method is applied to linear and nonlinear-dispersive time evolution equations of hyperbolic-type describing internal Kelvin and Poincaré waves in a shallow, stratified lake on a rotating plane.

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35L02 First-order hyperbolic equations
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76M22 Spectral methods applied to problems in fluid mechanics
Full Text: DOI
[1] Abramowitz, M.; Stegun, I.A., Handbook of mathematical functions, (1972), Dover · Zbl 0515.33001
[2] Auteri, F.; Quartapelle, L., Spectral solvers for spherical elliptic problems, J. comput. phys., 227, 36-54, (2007) · Zbl 1127.65088
[3] Auteri, F.; Quartapelle, L., Spectral elliptic solvers in a finite cylinder, Commun. comput. phys., 5, 426-441, (2009) · Zbl 1364.65231
[4] Born, M.; Wolf, E., Principles of optics, (1999), Cambridge University
[5] Boyd, J.P., Chebyshev and Fourier spectral methods, (2001), Dover · Zbl 0987.65122
[6] Csanady, G.T., Large-scale motion in the great lakes, J. geophys. res., 72, 16, 4151-4162, (1967)
[7] Eisen, H.; Heinrichs, W.; Witsch, K., Spectral collocation methods and polar coordinate singularities, J. comput. phys., 96, 241-257, (1991) · Zbl 0731.65095
[8] D. Gottlieb, S. Orszag, Numerical analysis of spectral methods; theory and applications, CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia, PA, 1977. · Zbl 0412.65058
[9] Ishioka, K., Spectral model for shallow-water equation on a disk. I: basic formulation, J. jpn. soc. fluid mech., 22, 345-358, (2003)
[10] Ishioka, K., Spectral model for shallow-water equation on a disk. II: numerical examples, J. jpn. soc. fluid mech., 22, 429-441, (2003)
[11] Krylov, V.I., Approximated calculation of integrals, (2006), Dover
[12] Lamb, Sir H., Hydrodynamics, (1932), Dover
[13] A. Leonard, A. Wray, A new numerical method for the simulation of three-dimensional flow in a pipe, in: E. Krause (Ed.), Proceedings of the Eighth International Conference on Numerical Methods in Fluid Dynamics, Aachen, Germany, Springer-Verlag, Berlin, June 28-July 2, 1982.
[14] Lewis, H.R.; Bellan, P.M., Physical constraints on the coefficients of Fourier expansions in cylindrical coordinates, J. math. phys., 31, 11, 2592-2596, (1990) · Zbl 0781.42005
[15] Livermore, P.W.; Jones, C.A.; Worland, S.J., Spectral radial basis functions for full sphere computations, J. comput. phys., 227, 1209-1224, (2007) · Zbl 1128.65016
[16] Lopez, J.M.; Marques, F.; Shen, Jie, An efficient spectral-projection method for the navier – stokes equations in cylindrical geometries. II: three-dimensional cases, J. comput. phys., 176, 384-401, (2002) · Zbl 1130.76392
[17] Matsushima, T.; Marcus, P.S., A spectral method for polar coordinates, J. comput. phys., 120, 365-374, (1995) · Zbl 0842.65051
[18] Merilees, P.E., An alternative scheme for a summation of a series of spherical harmonics, J. appl. meteor., 12, 224-227, (1973)
[19] Noll, R.J., Zernike polynomials and atmospheric turbulence, J. opt. soc. am., 66, 207-211, (1976)
[20] Orszag, S.A., Fourier series on spheres, Mon. weather rev., 102, 56-75, (1974)
[21] Orszag, S.A.; Patera, A.T., Secondary instability of wall-bounded shear flows, J. fluid mech., 128, 347-385, (1983) · Zbl 0556.76039
[22] Priymak, V.G.; Miyazaki, T., Accurate navier – stokes investigation of transitional and turbulent flows in circular pipe, J. comput. phys., 142, 370-411, (1998) · Zbl 0933.76064
[23] Verkley, W.T.M., A spectral model for two-dimensional incompressible fluid flow in a circular basin. I: mathematical foundation, J. comput. phys., 136, 100-114, (1997) · Zbl 0889.76071
[24] Verkley, W.T.M., A spectral model for two-dimensional incompressible fluid flow in a circular basin. II: numerical examples, J. comput. phys., 136, 115-131, (1997) · Zbl 0889.76072
[25] Zernike, von F., Beugungstheorie des schneidenverfahrens und seiner verbesserten form, der phasenkontrastmethode, Physica, 1, 689-704, (1934) · Zbl 0009.28101
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