A Proposed Modification to the Robert–Asselin Time Filter

Paul D. Williams Department of Meteorology, University of Reading, Reading, United Kingdom, and Kavli Institute for Theoretical Physics, University of California, Santa Barbara, Santa Barbara, California

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Abstract

The Robert–Asselin time filter is widely used in numerical models of weather and climate. It successfully suppresses the spurious computational mode associated with the leapfrog time-stepping scheme. Unfortunately, it also weakly suppresses the physical mode and severely degrades the numerical accuracy. These two concomitant problems are shown to occur because the filter does not conserve the mean state, averaged over the three time slices on which it operates. The author proposes a simple modification to the Robert–Asselin filter, which does conserve the three-time-level mean state. When used in conjunction with the leapfrog scheme, the modification vastly reduces the impacts on the physical mode and increases the numerical accuracy for amplitude errors by two orders, yielding third-order accuracy. The modified filter could easily be incorporated into existing general circulation models of the atmosphere and ocean. In principle, it should deliver more faithful simulations at almost no additional computational expense. Alternatively, it may permit the use of longer time steps with no loss of accuracy, reducing the computational expense of a given simulation.

Corresponding author address: Paul D. Williams, Department of Meteorology, University of Reading, P.O. Box 243, Earley Gate, Reading RG6 6BB, United Kingdom. Email: p.d.williams@reading.ac.uk

Abstract

The Robert–Asselin time filter is widely used in numerical models of weather and climate. It successfully suppresses the spurious computational mode associated with the leapfrog time-stepping scheme. Unfortunately, it also weakly suppresses the physical mode and severely degrades the numerical accuracy. These two concomitant problems are shown to occur because the filter does not conserve the mean state, averaged over the three time slices on which it operates. The author proposes a simple modification to the Robert–Asselin filter, which does conserve the three-time-level mean state. When used in conjunction with the leapfrog scheme, the modification vastly reduces the impacts on the physical mode and increases the numerical accuracy for amplitude errors by two orders, yielding third-order accuracy. The modified filter could easily be incorporated into existing general circulation models of the atmosphere and ocean. In principle, it should deliver more faithful simulations at almost no additional computational expense. Alternatively, it may permit the use of longer time steps with no loss of accuracy, reducing the computational expense of a given simulation.

Corresponding author address: Paul D. Williams, Department of Meteorology, University of Reading, P.O. Box 243, Earley Gate, Reading RG6 6BB, United Kingdom. Email: p.d.williams@reading.ac.uk

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  • Asselin, R., 1972: Frequency filter for time integrations. Mon. Wea. Rev., 100 , 487490.

  • Bartello, P., 2002: A comparison of time discretization schemes for two-timescale problems in geophysical fluid dynamics. J. Comput. Phys., 179 , 268285.

    • 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
  • Cordero, E., and A. Staniforth, 2004: A problem with the Robert–Asselin time filter for three-time-level semi-implicit semi-Lagrangian discretizations. Mon. Wea. Rev., 132 , 600610.

    • Search Google Scholar
    • Export Citation
  • Déqué, M., and D. Cariolle, 1986: Some destabilizing properties of the Asselin time filter. Mon. Wea. Rev., 114 , 880884.

  • Durran, D. R., 1991: The third-order Adams–Bashforth method: An attractive alternative to leapfrog time differencing. Mon. Wea. Rev., 119 , 702720.

    • Search Google Scholar
    • Export Citation
  • Durran, D. R., 1999: Numerical Methods for Wave Equations in Geophysical Fluid Dynamics. Springer-Verlag, 482 pp.

  • Ford, R., 1994: Gravity-wave radiation from vortex trains in rotating shallow-water. J. Fluid Mech., 281 , 81118.

  • Fraedrich, K., H. Jansen, E. Kirk, U. Luksch, and F. Lunkeit, 2005: The Planet Simulator: Towards a user friendly model. Meteor. Z., 14 , 299304.

    • Search Google Scholar
    • Export Citation
  • Griffies, S. M., R. C. Pacanowski, M. Schmidt, and V. Balaji, 2001: Tracer conservation with an explicit free surface method for z-coordinate ocean models. Mon. Wea. Rev., 129 , 10811098.

    • Search Google Scholar
    • Export Citation
  • Haltiner, G. J., and R. T. Williams, 1980: Numerical Prediction and Dynamic Meteorology. 2nd ed. Wiley, 496 pp.

  • Hartogh, P., A. Medvedev, T. Kuroda, R. Saito, G. Villanueva, A. Feofilov, A. Kutepov, and U. Berger, 2005: Description and climatology of a new general circulation model of the Martian atmosphere. J. Geophys. Res., 110 , E11008. doi:10.1029/2005JE002498.

    • Search Google Scholar
    • Export Citation
  • Kar, S. K., 2006: A semi-implicit Runge–Kutta time-difference scheme for the two-dimensional shallow-water equations. Mon. Wea. Rev., 134 , 29162926.

    • Search Google Scholar
    • Export Citation
  • Kurihara, Y., 1965: On the use of implicit and iterative methods for the time integration of the wave equation. Mon. Wea. Rev., 93 , 3346.

    • Search Google Scholar
    • Export Citation
  • Magazenkov, L. N., 1980: Time integration schemes for fluid dynamics equations, effectively damping the high frequency components (in Russian). Tr. Gl. Geofiz. Obs., 410 , 120129.

    • Search Google Scholar
    • Export Citation
  • Mesinger, F., and A. Arakawa, 1976: Numerical Methods Used in Atmospheric Models. Global Atmospheric Research Program (GARP) Publications Series 17, Vol. I, GARP, 64 pp.

    • Search Google Scholar
    • Export Citation
  • Pfeffer, R., I. Navon, and X. Zou, 1992: A comparison of the impact of two time-differencing schemes on the NASA GLAS climate model. Mon. Wea. Rev., 120 , 13811393.

    • Search Google Scholar
    • Export Citation
  • Robert, A. J., 1966: The integration of a low order spectral form of the primitive meteorological equations. J. Meteor. Soc. Japan, 44 , 237245.

    • Search Google Scholar
    • Export Citation
  • Schlesinger, R., L. Uccellini, and D. Johnson, 1983: The effects of the Asselin time filter on numerical solutions to the linearized shallow-water wave equations. Mon. Wea. Rev., 111 , 455467.

    • Search Google Scholar
    • Export Citation
  • Staniforth, A., 1997: André Robert (1929–1993): His pioneering contributions to numerical modeling. Numerical Methods in Atmospheric and Oceanic Modelling: The André J. Robert Memorial Volume, C. A. Lin, R. Laprise, and H. Ritchie, Eds., NRC Research Press, 25–54.

    • Search Google Scholar
    • Export Citation
  • Tandon, M., 1987: Robert’s recursive frequency filter: A reexamination. Meteor. Atmos. Phys., 37 , 4859.

  • Williams, P. D., T. W. N. Haine, P. L. Read, S. R. Lewis, and Y. H. Yamazaki, 2009: QUAGMIRE v1.3: A quasi-geostrophic model for investigating rotating fluids experiments. Geosci. Model Dev., 2 , 1332.

    • Search Google Scholar
    • Export Citation
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