Editorial Type:
Article
Article Type:
Research Article

Theoretical and Numerical Analysis of a Class of Semi-Implicit Semi-Lagrangian Schemes Potentially Applicable to Atmospheric Models

Abdelaziz Beljadid Department of Civil Engineering, University of Ottawa, Ottawa, Ontario, Canada

Search for other papers by Abdelaziz Beljadid in
Current site
Google Scholar
PubMed Close
,
Abdolmajid Mohammadian Department of Civil Engineering, University of Ottawa, Ottawa, Ontario, Canada

Search for other papers by Abdolmajid Mohammadian in
Current site
Google Scholar
PubMed Close
,
Martin Charron Recherche en Prévision Numérique Atmosphérique, Environnement Canada, Dorval, Quebec, Canada

Search for other papers by Martin Charron in
Current site
Google Scholar
PubMed Close
, and
Claude Girard Recherche en Prévision Numérique Atmosphérique, Environnement Canada, Dorval, Quebec, Canada

Search for other papers by Claude Girard in
Current site
Google Scholar
PubMed Close
Print Publication:
01 Dec 2014
DOI:
https://doi.org/10.1175/MWR-D-13-00302.1
Page(s):
4458–4476
Restricted access

Abstract

In this paper, theoretical and numerical analyses of the properties of some complex semi-Lagrangian methods are performed to deal with the issues of the instability associated with the treatment of the nonlinear part of the forcing term. A class of semi-Lagrangian semi-implicit schemes is proposed using a modified TR-BDF2 method, which is the combination of the trapezoidal rule (TR) and the second-order backward differentiation formula (BDF2). The process used for the nonlinear term includes two stages as predictor and corrector in the trapezoidal method and one stage for the BDF2 method. For the treatment of the linear term, the implicit trapezoidal method is employed in the first step, the explicit trapezoidal method in the second step, and the implicit BDF2 method in the third step. The combination of these techniques leads to a family of schemes that has a large region of absolute stability, performs well for the purely oscillatory cases, and has good qualities in terms of accuracy and convergence. The use of the explicit method for the linear term in the second step makes the proposed class of schemes competitive in terms of efficiency compared to some well-known schemes that use two steps. Numerical experiments presented herein confirm that the proposed class of schemes performs well in terms of stability, accuracy, convergence, and efficiency in comparison with other, previously known, semi-Lagrangian semi-implicit schemes and semi-implicit predictor–corrector methods. The potential practical application of the proposed class of schemes to a weather prediction model or any other atmospheric model is not discussed and could be the subject of other forthcoming studies.

Corresponding author address: Abdelaziz Beljadid, Dept. of Civil Engineering, University of Ottawa, 550 Rue Cumberland, Ottawa, ON K1N 6N5, Canada. E-mail: abelj016@uottawa.ca

Abstract

In this paper, theoretical and numerical analyses of the properties of some complex semi-Lagrangian methods are performed to deal with the issues of the instability associated with the treatment of the nonlinear part of the forcing term. A class of semi-Lagrangian semi-implicit schemes is proposed using a modified TR-BDF2 method, which is the combination of the trapezoidal rule (TR) and the second-order backward differentiation formula (BDF2). The process used for the nonlinear term includes two stages as predictor and corrector in the trapezoidal method and one stage for the BDF2 method. For the treatment of the linear term, the implicit trapezoidal method is employed in the first step, the explicit trapezoidal method in the second step, and the implicit BDF2 method in the third step. The combination of these techniques leads to a family of schemes that has a large region of absolute stability, performs well for the purely oscillatory cases, and has good qualities in terms of accuracy and convergence. The use of the explicit method for the linear term in the second step makes the proposed class of schemes competitive in terms of efficiency compared to some well-known schemes that use two steps. Numerical experiments presented herein confirm that the proposed class of schemes performs well in terms of stability, accuracy, convergence, and efficiency in comparison with other, previously known, semi-Lagrangian semi-implicit schemes and semi-implicit predictor–corrector methods. The potential practical application of the proposed class of schemes to a weather prediction model or any other atmospheric model is not discussed and could be the subject of other forthcoming studies.

Corresponding author address: Abdelaziz Beljadid, Dept. of Civil Engineering, University of Ottawa, 550 Rue Cumberland, Ottawa, ON K1N 6N5, Canada. E-mail: abelj016@uottawa.ca
Save
Cover Monthly Weather Review
Monthly Weather Review

  • Bonaventura, L., 2000: A semi-implicit, semi-Lagrangian scheme using the height coordinate for a nonhydrostatic and fully elastic model of atmospheric flows. J. Comput. Phys., 158, 186–213, doi:10.1006/jcph.1999.6414.

    • Search Google Scholar
    • Export Citation

  • Clancy, C., and J. A. Pudykiewicz, 2013: A class of semi-implicit predictor-corrector schemes for the time integration of atmospheric models. J. Comput. Phys., 250, 665–684, doi:10.1016/j.jcp.2012.08.032.

    • 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, 2787–2802, doi:10.1002/qj.49712757814.

    • Search Google Scholar
    • Export Citation

  • Dharmaraja, S., 2007: An analysis of the TR-BDF2 integration scheme. M.S. thesis, School of Engineering, Massachusetts Institute of Technology, 76 pp.

    • Search Google Scholar
    • Export Citation

  • Durran, D. R., 2010: Numerical Methods for Fluid Dynamics: With Applications to Geophysics. 2nd ed. Texts in Applied Mathematics, Vol. 32, Springer, 516 pp.

  • Durran, D. R., and P. A. Reinecke, 2004: Instability in a class of explicit two-time-level semi-Lagrangian schemes. Quart. J. Roy. Meteor. Soc., 130, 365–369, doi:10.1256/qj.03.14.

    • Search Google Scholar
    • Export Citation

  • Gospodinov, I. G., V. G. Spiridonov, and J.-F. Geleyn, 2001: Second-order accuracy of two-time-level semi-Lagrangian schemes. Quart. J. Roy. Meteor. Soc., 127, 1017–1033, doi:10.1002/qj.49712757317.

    • 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, doi:10.1002/qj.200212858314.

    • Search Google Scholar
    • Export Citation

  • McDonald, A., and J. R. Bates, 1987: Improving the estimate of the departure point in a two-time-level semi-Lagrangian and semi-implicit model. Mon. Wea. Rev., 115, 737–739, doi:10.1175/1520-0493(1987)115<0737:ITEOTD>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • McDonald, A., and J. Haugen, 1992: A two-time-level, three-dimensional semi-Lagrangian, semi-implicit, limited-area gridpoint model of the primitive equations. Mon. Wea. Rev., 120, 2603–2621, doi:10.1175/1520-0493(1992)120<2603:ATTLTD>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Ritchie, H., 1991: Application of the semi-Lagrangian method to a multilevel spectral primitive equations model. Quart. J. Roy. Meteor. Soc., 117, 91–106, doi:10.1002/qj.49711749705.

    • Search Google Scholar
    • Export Citation

  • Ritchie, H., and C. Beaudoin, 1994: Approximations and sensitivity experiments with a baroclinic semi-Lagrangian spectral model. Mon. Wea. Rev., 122, 2391–2399, doi:10.1175/1520-0493(1994)122<2391:AASEWA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Ritchie, H., C. Temperton, A. Simmons, M. Hortal, T. Davies, D. Dent, and M. Hamrud, 1995: Implementation of the semi-Lagrangian method in a high-resolution version of the ECMWF forecast model. Mon. Wea. Rev., 123, 489–514, doi:10.1175/1520-0493(1995)123<0489:IOTSLM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Robert, A., 1981: A stable numerical integration scheme for the primitive meteorological equations. Atmos.–Ocean, 19, 35–46, doi:10.1080/07055900.1981.9649098.

    • Search Google Scholar
    • Export Citation

  • Smolarkiewicz, P., and J. Pudykiewicz, 1992: A class of semi-Lagrangian approximations for fluids. J. Atmos. Sci., 49, 2082–2096, doi:10.1175/1520-0469(1992)049<2082:ACOSLA>2.0.CO;2.

    • 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, doi:10.1175/1520-0493(1991)119<2206:SLISFA>2.0.CO;2.

    • 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, 1970–1980, doi:10.1175/1520-0493(1990)118<1970:ASISLF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Temperton, C., and A. Staniforth, 1987: An efficient two-time-level semi-Lagrangian semi-implicit integration scheme. Quart. J. Roy. Meteor. Soc., 113, 1025–1039, doi:10.1002/qj.49711347714.

    • Search Google Scholar
    • Export Citation

  • Temperton, C., M. Hortal, and A. Simmons, 2001: A two-time-level semi-Lagrangian global spectral model. Quart. J. Roy. Meteor. Soc., 127, 111–126, doi:10.1002/qj.49712757107.

    • Search Google Scholar
    • Export Citation

  • White, J. B., III, and J. J. Dongarra, 2011: High-performance high-resolution semi-Lagrangian tracer transport on a sphere. J. Comput. Phys., 230, 6778–6799, doi:10.1016/j.jcp.2011.05.008.

    • Search Google Scholar
    • Export Citation
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 5746 5461 2543
PDF Downloads 313 77 6