Stability of Semi-Implicit and Iterative Centered-Implicit Time Discretizations for Various Equation Systems Used in NWP

P. Bénard Centre National de Recherches Météorologiques, Météo-France, Toulouse, France

Search for other papers by P. Bénard in
Current site
Google Scholar
PubMed
Close
Restricted access

Abstract

The stability of the classical semi-implicit scheme and some more advanced iterative schemes recently proposed for NWP purposes is examined. In all of these schemes, the solution of the centered-implicit nonlinear equation is approached by an iterative fixed-point algorithm, preconditioned by a simple, constant in time, linear operator. A general methodology for assessing analytically the stability of these schemes on canonical problems for a vertically unbounded atmosphere is presented. The proposed method is valid for all the equation systems usually employed in NWP. However, as in earlier studies, the method can be applied only in simplified meteorological contexts, thus overestimating the actual stability that would occur in more realistic meteorological contexts. The analysis is performed in the spatially continuous framework, hence allowing the elimination of the spatial discretization or the boundary conditions as possible causes of the fundamental instabilities linked to the time scheme itself. The general method is then shown concretely to apply to various time-discretization schemes and equation systems (namely, shallow-water and fully compressible Euler equations). Analytical results found in the literature are obtained from the proposed method, and some original results are presented.

Corresponding author address: Pierre Bénard, CNRM/GMAP, 42, Avenue G. Coriolis, F-31057 Toulouse Cedex, France. Email: pierre.benard@meteo.fr

Abstract

The stability of the classical semi-implicit scheme and some more advanced iterative schemes recently proposed for NWP purposes is examined. In all of these schemes, the solution of the centered-implicit nonlinear equation is approached by an iterative fixed-point algorithm, preconditioned by a simple, constant in time, linear operator. A general methodology for assessing analytically the stability of these schemes on canonical problems for a vertically unbounded atmosphere is presented. The proposed method is valid for all the equation systems usually employed in NWP. However, as in earlier studies, the method can be applied only in simplified meteorological contexts, thus overestimating the actual stability that would occur in more realistic meteorological contexts. The analysis is performed in the spatially continuous framework, hence allowing the elimination of the spatial discretization or the boundary conditions as possible causes of the fundamental instabilities linked to the time scheme itself. The general method is then shown concretely to apply to various time-discretization schemes and equation systems (namely, shallow-water and fully compressible Euler equations). Analytical results found in the literature are obtained from the proposed method, and some original results are presented.

Corresponding author address: Pierre Bénard, CNRM/GMAP, 42, Avenue G. Coriolis, F-31057 Toulouse Cedex, France. Email: pierre.benard@meteo.fr

Save
  • Bubnová, R., G. Hello, P. Bénard, and J-F. Geleyn, 1995: Integration of the fully elastic equations cast in the hydrostatic pressure terrain-following coordinate in the framework of the ARPEGE/Aladin NWP system. Mon. Wea. Rev., 123 , 515535.

    • Search Google Scholar
    • Export Citation
  • Caya, D., and R. Laprise, 1999: A semi-implicit semi-Lagrangian regional climate model: The Canadian RCM. Mon. Wea. Rev., 127 , 341362.

    • Search Google Scholar
    • Export Citation
  • Côté, J., M. Béland, and A. Staniforth, 1983: Stability of vertical discretization schemes for semi-implicit primitive equation models: Theory and application. Mon. Wea. Rev., 111 , 11891207.

    • Search Google Scholar
    • Export Citation
  • Côté, J., S. Gravel, A. Méthot, A. Patoine, M. Roch, and A. Staniforth, 1998: The Operational CMC-MRB Global Environmental Multiscale (GEM) Model. Part I: Design considerations and formulation. Mon. Wea. Rev., 126 , 13731395.

    • Search Google Scholar
    • Export Citation
  • Cullen, M. J. P., 2001: Alternative implementations of the semi-Lagrangian semi-implicit schemes in the ECMWF model. Quart. J. Roy. Meteor. Soc., 127 , 27872802.

    • Search Google Scholar
    • Export Citation
  • Hereil, P., and R. Laprise, 1996: Sensitivity of internal gravity waves solutions to the time step of a semi-implicit semi-Lagrangian nonhydrostatic model. Mon. Wea. Rev., 124 , 972999.

    • Search Google Scholar
    • Export Citation
  • Laprise, R., 1992: The Euler equations of motion with hydrostatic pressure as an independent variable. Mon. Wea. Rev., 120 , 197207.

  • Qian, J-H., F. H. M. Semazzi, and J. S. Scroggs, 1998: A global nonhydrostatic semi-Lagrangian atmospheric model with orography. Mon. Wea. Rev., 126 , 747771.

    • Search Google Scholar
    • Export Citation
  • Robert, A. J., J. Henderson, and C. Turnbull, 1972: An implicit time integration scheme for baroclinic models of the atmosphere. Mon. Wea. Rev., 100 , 329335.

    • Search Google Scholar
    • Export Citation
  • Semazzi, F. H. M., J. H. Qian, and J. S. Scroggs, 1995: A global nonhydrostatic semi-Lagrangian atmospheric model without orography. Mon. Wea. Rev., 123 , 25342550.

    • Search Google Scholar
    • Export Citation
  • Simmons, A. J., and C. Temperton, 1997: Stability of a two-time-level semi-implicit integration scheme for gravity wave motion. Mon. Wea. Rev., 125 , 600615.

    • Search Google Scholar
    • Export Citation
  • Simmons, A. J., B. Hoskins, and D. Burridge, 1978: Stability of the semi-implicit method of time integration. Mon. Wea. Rev., 106 , 405412.

    • Search Google Scholar
    • Export Citation
  • Tanguay, M., A. Robert, and R. Laprise, 1990: A semi-implicit semi-Lagrangian fully compressible regional forecast model. Mon. Wea. Rev., 118 , 19701980.

    • Search Google Scholar
    • Export Citation
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 334 95 11
PDF Downloads 251 70 5