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An FFT Based Fast Poisson Solver on Spherical Shells

Published online by Cambridge University Press:  20 August 2015

Yin-Liang Huang ,
Jian-Guo Liu  and
Wei-Cheng Wang
Yin-Liang Huang*
Affiliation:
Department of Applied Mathematics, National University of Tainan, Tainan 70005, Taiwan
Jian-Guo Liu*
Affiliation:
Department of Physics and Department of Mathematics, Duke University, Durham, NC 27708, USA
Wei-Cheng Wang*
Affiliation:
Department of Mathematics, National Tsing Hua University, Hsinchu 30013, Taiwan
*
Corresponding author.Email:liang@mail.nutn.edu.tw
Email:jian-guo.liu@duke.edu
Email:wangwc@math.nthu.edu.tw
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Abstract

We present a fast Poisson solver on spherical shells. With a special change of variable, the radial part of the Laplacian transforms to a constant coefficient differential operator. As a result, the Fast Fourier Transform can be applied to solve the Poisson equation with operations. Numerical examples have confirmed the accuracy and robustness of the new scheme.

Keywords

35Q8665N0665N1565N2265T50Poisson equationspherical coordinateFFTspectral-finite difference methodfast diagonalizationhigh order accuracyerror estimatetrapezoidal ruleEuler-Maclaurin formulaBernoulli numbers

Information

Type
Research Article
Information
Communications in Computational Physics , Volume 9 , Issue 3 , March 2011 , pp. 649 - 667
Copyright
Copyright © Global Science Press Limited 2011

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References

[1]Auteri, F. and Quartapelle, L., Spectral solvers for spherical elliptic problems, J. Comput. Phys., 227 (2007), 36–54.Google Scholar
[2]Carpenter, M. H., Gottlieb, D. and Abarbanel, S., Stable and accurate boundary treatments for compact, high-order finite difference schemes, Appl. Numer. Math., 12(1-3) (1993), 55–87.CrossRefGoogle Scholar
[3]E, W. and Liu, J.-G., Essentially compact schemes for unsteady viscous incompressible flows, J. Comput. Phys., 126 (1996), 122–138.CrossRefGoogle Scholar
[4]Fornberg, B., A Practical Guide to Pseudo-Spectral Methods, Cambridge University Press, New York, 1998.Google Scholar
[5]Glatzmaier, G. A. and Roberts, P. H., A three-dimensional convective dynamo solution with rotating and finitely conducting inner core and mantle, Int. Phys. Earth. Planet., 91 (1995), 63–75.Google Scholar
[6]Gottlieb, D. and Orszag, S. A., Numerical Analysis of Spectral Methods Theory and Applications, SIAM, Philadelphia, 1977.Google Scholar
[7]Hesthaven, J. S., Gottlieb, S. and Gottlieb, D., Spectral Methods for Time-Dependent Problems, Cambridge University Press, New York, 2007.Google Scholar
[8]Li, M., Tang, T. and Fornberg, B., A compact fourth-order finite difference scheme for the steady incompressible Navier-Stokes equations, Int. J. Numer. Method. Fluids., 20 (1995), 1137–1151.Google Scholar
[9]Orszag, S., Fourier series on spheres, Weather. Rev., 102 (1974), 56–75.Google Scholar
[10]Kress, R., Numerical Analysis, Springer, New York, 1998.Google Scholar
[11]Ronchi, C., Iacono, R. and Paolucci, P. S., The “cubed sphere”: A new method for the solution of partial differential equations in spherical geometry, J. Comput. Phys., 124 (1996), 93–114.Google Scholar
[12]Shen, J., Efficient spectral-Galerkin methods IV, spherical geometries, SIAM J. Sci. Comput., 20 (1999), 1438–1455.Google Scholar
[13]Shen, J. and Tang, T., Spectral and High-Order Methods with Applications, Science Press, Beijing, 2006.Google Scholar
[14]Tilgner, A., Spectral methods for the simulation of incompressible flows in spherical shells, Int. J. Numer. Method. Fluids., 30 (1999), 713–724.3.0.CO;2-Y>CrossRefGoogle Scholar
[15]Wiegelmann, T., Computing nonlinear force-free coronal magnetic fields in spherical geometry, Solar. Phys., 240 (2007), 227–239.CrossRefGoogle Scholar
[16]Yee, S. Y. K., Solution of Poisson’s equation on a sphere by truncated double Fourier series, Month. Weather. Rev., 109 (1981), 501–505.Google Scholar
[17]Yoshida, M. and Kageyama, A., Application of the Yin-Yang grid to a thermal convection of a Boussinesq fluid with infinite Prandtl number in a three-dimensional spherical shell, Geophys. Res. Lett., 31 (2004), L12609.CrossRefGoogle Scholar