• Alpert, B. K., , and V. Rokhlin, 1991: A fast algorithm for the evaluation of Legendre expansions. SIAM J. Sci. Stat. Comput., 12 , 158179.

    • Search Google Scholar
    • Export Citation
  • Driscoll, J. R., , and D. M. Healy, 1994: Computing Fourier transforms and convolutions on the 2-Sphere. Adv. Appl. Math., 15 , 202250.

  • Greengard, L., , and V. Rokhlin, 1987: A fast algorithm for particle simulations. J. Comput. Phys., 73 , 325348.

  • Hack, J. J., , and R. Jakob, 1992: Description of a global shallow water model based on the spectral transform method. NCAR Tech. Note TN-343+STR, 39 pp.

  • Healy, D. M., , D. Rockmore, , P. J. Kostelec, , and S. S. B. Moore, Jr. 1996: FFTs for 2-Sphere—Improvements and variations. Tech. Rep. PCS-TR96-292, Dartmouth College, 43 pp.

  • Jakob, R., , J. J. Hack, , and D. L. Williamson, 1993: Solutions to the shallow water test set using the spectral transform method. NCAR Tech. Note TN-388+STR, 82 pp.

  • Mohlenkamp, M. J., 1999: A fast transform for spherical harmonics. J. Fourier Anal. Appl., 2 , 159184.

  • Mori, A., , R. Suda, , and M. Sugihara, 1999: An improvement on Orszag's fast algorithm for Legendre polynomial transform (in Japanese). Trans. Inf. Proc. Soc. Japan, 40 , 36123615.

    • Search Google Scholar
    • Export Citation
  • Orszag, S. A., 1986: Fast eigenfunction transforms. Science and Computers, G.-C. Rota, Ed., Vol. 10, Advances in Mathematics: Supplementary Studies, Academic Press, 23–30.

  • Potts, D., , G. Steidl, , and M. Tasche, 1998: Fast and stable algorithms for discrete spherical Fourier transforms. Linear Algebra Appl., 275–276 , 433450.

    • Search Google Scholar
    • Export Citation
  • Rivier, L., , R. Loft, , and L. M. Polvani, 2002: An efficient spectral dynamical core for distributed memory computers. Mon. Wea. Rev., 130 , 13841396.

    • Search Google Scholar
    • Export Citation
  • Suda, R., 2004: Stability analysis of the fast Legendre transform algorithm based on the fast multipole method. Proc. Estonian Acad. Sci. Phys. Math., 53 , 107115.

    • Search Google Scholar
    • Export Citation
  • Suda, R., , and M. Takami, 2001: Error analysis and control of fast spherical harmonics transform (in Japanese). Trans. Inf. Proc. Soc. Japan, 42 , (HPS4). 4959.

    • Search Google Scholar
    • Export Citation
  • Suda, R., , and M. Takami, 2002: A fast spherical harmonics transform algorithm. Math. Comput., 71 , 703715.

  • Suda, R., , and S. Kuriyama, 2004: Another preprocessing algorithm for generalized one-dimensional fast multipole method. J. Comput. Phys., 195 , 790803.

    • Search Google Scholar
    • Export Citation
  • Temperton, C., 1991: On scalar and vector transform methods for global spectral models. Mon. Wea. Rev., 119 , 13031307.

  • Williamson, D. L., , J. B. Drake, , J. J. Hack, , R. Jacob, , and P. N. Swarztrauber, 1992: A standard test set for numerical approximations to the shallow water equations in spherical geometry. J. Comput. Phys., 102 , 211224.

    • Search Google Scholar
    • Export Citation
  • Yarvin, N., , and V. Rokhlin, 1998: A generalized one-dimensional fast multipole method with application to filtering of spherical harmonics. J. Comput. Phys., 147 , 594609.

    • Search Google Scholar
    • Export Citation
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Fast Spherical Harmonic Transform Routine FLTSS Applied to the Shallow Water Test Set

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  • 1 Department of Computer Science, University of Tokyo, Tokyo, and CREST, Japan Science and Technology Agency, Saitama, Japan
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Abstract

The fast spherical harmonic transform algorithm proposed by Suda and Takami is evaluated in the solutions of the shallow water equation test set defined by Williamson et al. through replacing the Legendre transforms of the NCAR spectral transform shallow water model (STSWM) with routines of the fast Legendre transform with stable sampling (FLTSS), which is the first implementation of the Suda–Takami algorithm. The Suda–Takami algorithm is an approximate algorithm with the computational complexity O(T2 log T log ε−1), with T being the maximum wavenumber and ε the accuracy parameter of the FLTSS. The influence of the approximation errors of the FLTSS upon the numerical solutions is investigated. For all test cases of the Williamson et al. test set, the FLTSS stably solved the equations with the results that can be explained well with the accuracy ε. The stability in longer time integrations is also assessed, where test case 7 of an analyzed atmospheric initial condition was stably integrated for 1 yr. The FLTSS was faster than the STSWM at T170 and had higher resolutions on an Intel Mobile Pentium 4, where the lower space complexity (memory requirements) of the FLTSS was advantageous in addition to the lower computational complexity.

Corresponding author address: Reiji Suda, Department of Computer Science, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan. Email: reiji@is.s.u-tokyo.ac.jp

Abstract

The fast spherical harmonic transform algorithm proposed by Suda and Takami is evaluated in the solutions of the shallow water equation test set defined by Williamson et al. through replacing the Legendre transforms of the NCAR spectral transform shallow water model (STSWM) with routines of the fast Legendre transform with stable sampling (FLTSS), which is the first implementation of the Suda–Takami algorithm. The Suda–Takami algorithm is an approximate algorithm with the computational complexity O(T2 log T log ε−1), with T being the maximum wavenumber and ε the accuracy parameter of the FLTSS. The influence of the approximation errors of the FLTSS upon the numerical solutions is investigated. For all test cases of the Williamson et al. test set, the FLTSS stably solved the equations with the results that can be explained well with the accuracy ε. The stability in longer time integrations is also assessed, where test case 7 of an analyzed atmospheric initial condition was stably integrated for 1 yr. The FLTSS was faster than the STSWM at T170 and had higher resolutions on an Intel Mobile Pentium 4, where the lower space complexity (memory requirements) of the FLTSS was advantageous in addition to the lower computational complexity.

Corresponding author address: Reiji Suda, Department of Computer Science, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan. Email: reiji@is.s.u-tokyo.ac.jp

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