Editorial Type:
Article
Article Type:
Research Article

A Stability Analysis of Finite-Volume Advection Schemes Permitting Long Time Steps

Peter Hjort Lauritzen National Center for Atmospheric Research,* Boulder, Colorado

Search for other papers by Peter Hjort Lauritzen in
Current site
Google Scholar
PubMed Close
Print Publication:
01 Jul 2007
DOI:
https://doi.org/10.1175/MWR3425.1
Page(s):
2658–2673
Restricted access

Abstract

Finite-volume schemes developed in the meteorological community that permit long time steps are considered. These include Eulerian flux-form schemes as well as fully two-dimensional and cascade cell-integrated semi-Lagrangian (CISL) schemes. A one- and two-dimensional Von Neumann stability analysis of these finite-volume advection schemes is given. Contrary to previous analysis, no simplifications in terms of reducing the formal order of the schemes, which makes the analysis mathematically less complex, have been applied. An interscheme comparison of both dissipation and dispersion properties is given. The main finding is that the dissipation and dispersion properties of Eulerian flux-form schemes are sensitive to the choice of inner and outer operators applied in the scheme that can lead to increased numerical damping for large Courant numbers. This spurious dependence on the integer value of the Courant number disappears if the inner and outer operators are identical, in which case, under the assumptions used in the stability analysis, the Eulerian flux-form scheme becomes identical to the cascade scheme. To explain these properties a conceptual interpretation of the flux-based Eulerian schemes is provided. Of the two CISL schemes, the cascade scheme has superior stability properties.

* The National Center for Atmospheric Research is sponsored by the National Science Foundation

Corresponding author address: Peter Hjort Lauritzen, Climate Modeling Section, Climate and Global Dynamics Division, National Center for Atmospheric Research, 1850 Table Mesa Drive, Boulder, CO 80305. Email: pel@ucar.edu

Abstract

Finite-volume schemes developed in the meteorological community that permit long time steps are considered. These include Eulerian flux-form schemes as well as fully two-dimensional and cascade cell-integrated semi-Lagrangian (CISL) schemes. A one- and two-dimensional Von Neumann stability analysis of these finite-volume advection schemes is given. Contrary to previous analysis, no simplifications in terms of reducing the formal order of the schemes, which makes the analysis mathematically less complex, have been applied. An interscheme comparison of both dissipation and dispersion properties is given. The main finding is that the dissipation and dispersion properties of Eulerian flux-form schemes are sensitive to the choice of inner and outer operators applied in the scheme that can lead to increased numerical damping for large Courant numbers. This spurious dependence on the integer value of the Courant number disappears if the inner and outer operators are identical, in which case, under the assumptions used in the stability analysis, the Eulerian flux-form scheme becomes identical to the cascade scheme. To explain these properties a conceptual interpretation of the flux-based Eulerian schemes is provided. Of the two CISL schemes, the cascade scheme has superior stability properties.

* The National Center for Atmospheric Research is sponsored by the National Science Foundation

Corresponding author address: Peter Hjort Lauritzen, Climate Modeling Section, Climate and Global Dynamics Division, National Center for Atmospheric Research, 1850 Table Mesa Drive, Boulder, CO 80305. Email: pel@ucar.edu

Save
Cover Monthly Weather Review
Monthly Weather Review

  • Bates, J. R., and A. McDonald, 1982: Multiply-upstream, semi-Lagrangian advective schemes: Analysis and application to a multi-level primitive equation model. Mon. Wea. Rev., 110 , 1831–1842.

    • Search Google Scholar
    • Export Citation

  • Bott, A., 1989: A positive definite advection scheme obtained by nonlinear renomalization of advective fluxes. Mon. Wea. Rev., 117 , 1006–1015.

    • Search Google Scholar
    • Export Citation

  • Colella, P., and P. R. Woodward, 1984: The piecewise parabolic method (PPM) for gas-dynamical simulations. J. Comput. Phys., 54 , 174–201.

    • Search Google Scholar
    • Export Citation

  • Doswell III, C. A., 1984: A kinematic analysis of frontogenesis associated with a nondivergent vortex. J. Atmos. Sci., 41 , 1242–1248.

  • Durran, D. R., 1999: Numerical Methods for Wave Equations in Geophysical Fluid Dynamics. Springer-Verlag, 465 pp.

  • Godunov, S., 1959: A difference scheme for numerical computation of discontinuous solutions of equations in fluid dynamics. Mat. Sb., 47 , 271.

    • Search Google Scholar
    • Export Citation

  • Haltiner, G. J., and R. T. Williams, 1980: Numerical Prediction and Dynamic Meteorology. 2d ed. John Wiley & Sons, 477 pp.

  • Hólm, E. V., 1995: A fully two-dimensional, nonoscillatory advection scheme for momentum and scalar transport equations. Mon. Wea. Rev., 123 , 536–552.

    • Search Google Scholar
    • Export Citation

  • Hortal, M., 2002: The development and testing of a new two-time-level semi-Lagrangian scheme (SETTLS) in the ECMWF forecast model. Quart. J. Roy. Meteor. Soc., 128 , 1671–1687.

    • Search Google Scholar
    • Export Citation

  • Jablonowski, C., M. Herzog, J. E. Penner, R. C. Oehmke, Q. F. Stout, B. van Leer, and K. G. Powell, 2006: Block-structured adaptive grids on the sphere: Advection experiments. Mon. Wea. Rev., 134 , 3691–3713.

    • Search Google Scholar
    • Export Citation

  • Jöckel, P., R. von Kuhlmann, M. G. Lawrence, B. Steil, C. Brenninkmeijer, P. J. Crutzen, P. J. Rasch, and B. Eaton, 2001: On a fundamental problem in implementing flux-form advection schemes for tracer transport in 3-dimensional general circulation and chemistry transport models. Quart. J. Roy. Meteor. Soc., 127 , 1035–1052.

    • Search Google Scholar
    • Export Citation

  • Laprise, J. P. R., and A. Plante, 1995: A class of semi-Lagrangian integrated-mass (SLIM) numerical transport algorithms. Mon. Wea. Rev., 123 , 553–565.

    • Search Google Scholar
    • Export Citation

  • Lauritzen, P. H., E. Kaas, and B. Machenhauer, 2006a: A mass-conservative semi-implicit semi-Lagrangian limited-area shallow-water model on the sphere. Mon. Wea. Rev., 134 , 1205–1221.

    • Search Google Scholar
    • Export Citation

  • Leonard, B. P., A. P. Lock, and M. K. MacVean, 1996: Conservative explicit unrestricted-time-step multidimensional constancy-preserving advection schemes. Mon. Wea. Rev., 124 , 2588–2606.

    • Search Google Scholar
    • Export Citation

  • Lin, S-J., 2004: A “vertically Lagrangian” finite-volume dynamical core for global models. Mon. Wea. Rev., 132 , 2293–2307.

  • Lin, S-J., and R. B. Rood, 1996: Multidimensional flux-form semi-Lagrangian transport schemes. Mon. Wea. Rev., 124 , 2046–2070.

  • Machenhauer, B., and M. Olk, 1998: Design of a semi-implicit cell-integrated semi-Lagrangian model. Max Planck Institute for Meteorology Tech. Rep. 265, Hamburg, Germany, 76–85.

  • Machenhauer, B., E. Kaas, and P. H. Lauritzen, 2007: Finite volume methods in meteorology. Special Volume on Computational Methods for the Ocean and Atmosphere, Elsevier, in press.

  • Nair, R. D., and B. Machenhauer, 2002: The mass-conservative cell-integrated semi-Lagrangian advection scheme on the sphere. Mon. Wea. Rev., 130 , 649–667.

    • Search Google Scholar
    • Export Citation

  • Nair, R. D., J. S. Scroggs, and F. H. M. Semazzi, 2002: Efficient conservative global transport schemes for climate and atmospheric chemistry models. Mon. Wea. Rev., 130 , 2059–2073.

    • Search Google Scholar
    • Export Citation

  • Nair, R. D., J. S. Scroggs, and F. H. M. Semazzi, 2003: A forward-trajectory global semi-Lagrangian transport scheme. J. Comput. Phys., 190 , 275–294.

    • Search Google Scholar
    • Export Citation

  • Plante, A., 1993: Transport de substabces par schémas numériques semi-Lagrangiens intégrés par celluls. M.S. thesis, Physics Department, University of Québec at Montréal, 127 pp. [Available from J. P. Laprise, Physics Department, UQAM, P.O. Box 8888, Stn. Downtown, Montréal, QC H3C 3P8, Canada.].

  • Purser, R. J., and L. M. Leslie, 1991: An efficient interpolation procedure for high-order three-dimensional semi-Lagrangian models. Mon. Wea. Rev., 119 , 2492–2498.

    • Search Google Scholar
    • Export Citation

  • Rančić, M., 1992: Semi-Lagrangian piecewise biparabolic scheme for two-dimensional horizontal advection of a passive scalar. Mon. Wea. Rev., 120 , 1394–1406.

    • Search Google Scholar
    • Export Citation

  • Rančić, M., 1995: An efficient, conservative, monotonic remapping for semi-Lagrangian transport algorithms. Mon. Wea. Rev., 123 , 1213–1217.

    • Search Google Scholar
    • Export Citation

  • Skamarock, W. C., 2006: Positive-definite and monotonic limiters for unrestricted-time-step transport schemes. Mon. Wea. Rev., 134 , 2241–2250.

    • Search Google Scholar
    • Export Citation

  • Staniforth, A., and J. Côté, 1991: Semi-Lagrangian integration schemes for atmospheric models—A review. Mon. Wea. Rev., 119 , 2206–2223.

    • Search Google Scholar
    • Export Citation

  • van Leer, B., 1977: Towards the ultimate conservative difference scheme. IV: A new approach to numerical convection. J. Comput. Phys., 23 , 276–299.

    • Search Google Scholar
    • Export Citation

  • Zerroukat, M., N. Wood, and A. Staniforth, 2002: SLICE: A semi-Lagrangian inherently conserving and efficient scheme for transport problems. Quart. J. Roy. Meteor. Soc., 128 , 2801–2820.

    • Search Google Scholar
    • Export Citation

  • Zerroukat, M., N. Wood, and A. Staniforth, 2005: A monotonic and positive-definite filter for a semi-Lagrangian inherently conserving and efficient (SLICE) scheme. Quart. J. Roy. Meteor. Soc., 131 , 2923–2936.

    • Search Google Scholar
    • Export Citation

  • Zerroukat, M., N. Wood, and A. Staniforth, 2006: The parabolic spline method (PSM) for conservative transport problems. Int. J. Numer. Methods Fluids, 51 , 1297–1318.

    • Search Google Scholar
    • Export Citation
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 216 51 5
PDF Downloads 184 50 7