Editorial Type:
Article
Article Type:
Research Article

An Alternative Leapfrog Scheme for Surface Gravity Wave Equations

Weidong Zhou South China Sea Institute of Oceanology, Chinese Academy of Sciences, Guangzhou, China

Search for other papers by Weidong Zhou in
Current site
Google Scholar
PubMed Close
Print Publication:
01 Sep 2002
DOI:
https://doi.org/10.1175/1520-0426(2002)019<1415:AALSFS>2.0.CO;2
Page(s):
1415–1423
Restricted access

Abstract

An alternative leapfrog scheme using a staggered time grid system is proposed to solve surface gravity wave equations. In addition to the nondissipative second-order accuracy scheme that is inherent in the standard leapfrog scheme, the alternative leapfrog scheme has the advantages that 1) no separation of the solution occurs at the even and odd time levels, 2) it is twice as efficient as the standard leapfrog scheme, and 3) it requires only the one time level storage.

Numerical experiments show its numerical properties and computational efficiency. The wave amplitude is not damped and the total energy is conserved. The alternative leapfrog scheme is one of the most efficient schemes. It is applied to the surface gravity wave terms in ocean circulation models, and its usefulness is shown.

Corresponding author address: Weidong Zhou, South China Sea Institute of Oceanology, Chinese Academy of Sciences, 164W Xingang Rd., Guangzhou 510301, China. Email: wdzhou@scsio.ac.cn

Abstract

An alternative leapfrog scheme using a staggered time grid system is proposed to solve surface gravity wave equations. In addition to the nondissipative second-order accuracy scheme that is inherent in the standard leapfrog scheme, the alternative leapfrog scheme has the advantages that 1) no separation of the solution occurs at the even and odd time levels, 2) it is twice as efficient as the standard leapfrog scheme, and 3) it requires only the one time level storage.

Numerical experiments show its numerical properties and computational efficiency. The wave amplitude is not damped and the total energy is conserved. The alternative leapfrog scheme is one of the most efficient schemes. It is applied to the surface gravity wave terms in ocean circulation models, and its usefulness is shown.

Corresponding author address: Weidong Zhou, South China Sea Institute of Oceanology, Chinese Academy of Sciences, 164W Xingang Rd., Guangzhou 510301, China. Email: wdzhou@scsio.ac.cn

Save
Cover Journal of Atmospheric and Oceanic Technology
Journal of Atmospheric and Oceanic Technology

  • Anderson, D. A., Tannehill J. C. , and Pletcher R. H. , 1997: Computational Fluid Mechanics and Heat Transfer. 2d ed. Hemisphere Publishing, MacGraw-Hill, 792 pp.

    • Search Google Scholar
    • Export Citation

  • Asselin, R., 1972: Frequency filter for time integration. Mon. Wea. Rev., 100 , 187–490.

  • Blumberg, A. F., and Mellor G. L. , 1987: A description of a three-dimensional coastal ocean circulation model. Three-Dimensional Coastal Ocean Models, Vol. 4, N. Heaps, Ed., Amer. Geophys. Union, 208 pp.

    • Search Google Scholar
    • Export Citation

  • Bryan, K., 1969: A numerical method for the study of the circulation of the world ocean. J. Comput. Phys., 4 , 347–376.

  • Courant, R., Friedrichs K. O. , and Lewy H. , 1928: Über die partiellen differenzengleichungen der mathematischen physic. Math. Ann., 100 , 32–74.

  • Haidvogel, D. B., and Beckmann A. , 1999: Numerical Ocean Circulation Modeling. Series on Environmental and Management, Vol. 2,. Imperial College Press, 319 pp.

    • Search Google Scholar
    • Export Citation

  • Killworth, P. D., Stainforth D. , Webb D. , and Paterson S. , 1991: The development of a free-surface Bryan–Cox–Semtner ocean model. J. Phys. Oceanogr., 21 , 1333–1348.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Lilly, D. L., 1965: On the computational stability of numerical solutions of the time-dependent non-linear geophysical fluid dynamics problem. Mon. Wea. Rev., 93 , 11–26.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • O'Brien, J. J., 1986: Time integration schemes. Advanced Physical Oceanographic Numerical Modelling, J. J. O'Brien, Ed., NATO ASI Series, Reidel, 155–164.

    • Search Google Scholar
    • Export Citation

  • Sanz-Serna, J. M., 1985: Studies in numerical nonlinear instability. I. Why do leapfrog schemes go unstable? SIAM J. Sci. Stat. Comput., 6 , 923–938.

  • Sod, G. A., 1985: Numerical Methods in Fluid Dynamics. Cambridge University Press, 792 pp.

  • Song, Y., and Tang T. , 1993: Dispersion and group velocity in numerical schemes for three-dimensional hydrodynamic equations. J. Comput. Phys., 105 , 72–82.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Song, Y., . 1994: Staggered Turkel–Zwas schemes for two-dimensional shallow-water equations. Mon. Wea. Rev., 122 , 223–234.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Takano, K., 1974: A general circulation model for the World Ocean. Tech. Rep. No. 8, Numerical Prediction of Weather and Climate, Dept. of Meteorology, University of California, Los Angeles, 46 pp.

    • Search Google Scholar
    • Export Citation

  • Zhou, W., 1999: A free-surface ocean model. Advances in Numerical Mathematics: Proceedings of Fourth Japan–China Joint Seminar on Numerical Mathematics. H. Kawarada, et al., Eds., Gakuto International Series, Mathematical Sciences and Applications, Vol. 12, Gakuto, 271–286.

    • Search Google Scholar
    • Export Citation

  • Zhou, W., . 2002: A proper time integration with split stepping for the explicit free-surface modeling. Adv. Atmos. Sci., 19 , 255–265.

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 580 237 46
PDF Downloads 174 26 5