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Spectral Matrix Conditioning in Cylindrical and Spherical Elliptic Equations

Published online by Cambridge University Press:  28 May 2015

F. Auteri  and
L. Quartapelle
F. Auteri*
Affiliation:
Politecnico di Milano, Dipartimento di Ingegneria Aerospaziale, Via La Masa 34, 20156 Milano, Italy
L. Quartapelle
Affiliation:
Politecnico di Milano, Dipartimento di Ingegneria Aerospaziale, Via La Masa 34, 20156 Milano, Italy
*
Corresponding author.Email address:[email protected]
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Abstract

In the spectral solution of 3-D Poisson equations in cylindrical and spherical coordinates including the axis or the center, it is convenient to employ radial basis functions that depend on the Fourier wavenumber or on the latitudinal mode. This idea has been adopted by Matsushima and Marcus and by Verkley for planar problems and pursued by the present authors for spherical ones. For the Dirichlet boundary value problem in both geometries, original bases have been introduced built upon Jacobi polynomials which lead to a purely diagonal representation of the radial second-order differential operator of all spectral modes. This note details the origin of such a diagonalization which extends to cylindrical and spherical regions the properties of the Legendre basis introduced by Jie Shen for Cartesian domains. Closed form expressions are derived for the diagonal elements of the stiffness matrices as well as for the elements of the tridiagonal mass matrices occurring in evolutionary problems. Furthermore, the bound on the condition number of the spectral matrices associated with the Helmholtz equation are determined, proving in a rigorous way one of the main advantages of the proposed radial bases.

Keywords

65M1078A48Poisson equationcylindrical and spherical coordinatesFourier expansionspherical harmonicsJacobi polynomialsspectral methodscondition number
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 4 , Issue 2 , May 2011 , pp. 113 - 141
Copyright
Copyright © Global Science Press Limited 2011

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References

[1]Abramowitz, M. and Stegun, I. A., Handbook of Mathematical Functions, Dover Publications, New York, 1965.Google Scholar
[2]Auteri, F. and Quartapelle, L., Spectral solvers for spherical elliptic problems, J. Comput. Phys., 227 (2007), pp. 36–54.CrossRefGoogle Scholar
[3]Auteri, F. and Quartapelle, L., Spectral elliptic solvers in a finite cylinder, Commun. Comput. Phys., 5 (2009), pp. 426–441.Google Scholar
[4]Bernardi, C., Dauge, M. and Maday, Y., Spectral Methods for Axisymmetric Domains, Elsevier, Paris, 1999.Google Scholar
[5]Boyd, J. B., Chebyshev and Fourier Spectral Methods, Dover Publications, Second Edition, Mineola, New York, 2001.Google Scholar
[6]Gustafson, K. E., Introduction to Partial Differential Equations and Hilbert Space Methods, Dover Publications, Third Edition, Calcutta and Charleston, Illinois, 1993.Google Scholar
[7]Haidvogel, D. B. and Zang, T. A., The accurate solution of Poisson’s equation by expansion in Chebuyshev polynomials, J. Comput. Phys., 30 (1979), pp. 167–180.CrossRefGoogle Scholar
[8]Heinrichs, W., Improved condition number for spectral methods, Math. Comp., 53 (1989), pp. 103–119.CrossRefGoogle Scholar
[9]Le Quéré, P. and Pécheux, J., A three-dimensional pseudo-spectral algorithm for the computation of convection in a rotating annulus, Spectral and High Order Methods for Partial Differential Equations, Proceedings of the ICOSAHOM ‘89 Conference, Villa Olmo, Como, Italy, 26-29 June 1989, pp. 261–271.CrossRefGoogle Scholar
[10]Lewis, H. R. and Bellan, P. M., Physical constraints on the coefficients of Fourier expansions in cylindrical coordinates, Journal of Mathematical Physics, 31 (1990), pp. 2592–2596.CrossRefGoogle Scholar
[11]Livermore, P. W., Jones, C. A. and Worland, S. J., Spectral radial basis functions for full sphere computations, J. Comput. Phys., 227 (2007), pp. 1209–1224.CrossRefGoogle Scholar
[12]Matsushima, T. and Marcus, P. S., A spectral method for polar coordinates, J. Comput. Phys., 120 (1995), pp. 365–374.CrossRefGoogle Scholar
[13]Shen, J., Efficient spectral–Galerkin method. I. Direct solvers of second- and fourth-order equations using Legendre polynomials, SIAM J. Sci. Comput., 15, (1994) pp. 1489–1505.CrossRefGoogle Scholar
[14]Shen, J., Efficient spectral–Galerkin methods. III. Polar and cylindrical geometries, SIAM J. Sci. Comput., 18 (1997), pp. 74–87.CrossRefGoogle Scholar
[15]Shen, J., Efficient spectral–Galerkin methods. IV. Spherical geometries, SIAM J. Sci. Comput., 20 (1999), pp. 1438–1455.CrossRefGoogle Scholar
[16]Verkley, W. T. M., A pseudo-spectral model for two-dimensional incompressible flow in a circular basin. I. Mathematical formulation, J. Comput. Phys., 136 (1997), pp. 100–114.CrossRefGoogle Scholar
[17]Verkley, W. T. M., A pseudo-spectral model for two-dimensional incompressible flow in a circular basin. II. Numerical examples, J. Comput. Phys., 136 (1997), pp. 115–131.CrossRefGoogle Scholar