Hostname: page-component-7dd5485656-pnlb5 Total loading time: 0.001 Render date: 2025-10-25T11:38:29.945Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

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