The RAW Filter: An Improvement to the Robert–Asselin Filter in Semi-Implicit Integrations

Paul D. Williams Department of Meteorology, University of Reading, Reading, United Kingdom

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Abstract

Errors caused by discrete time stepping may be an important component of total model error in contemporary atmospheric and oceanic simulations. To reduce time-stepping errors in leapfrog integrations, the Robert–Asselin–Williams (RAW) filter was proposed by the author as a simple improvement to the widely used Robert–Asselin (RA) filter. The present paper examines the behavior of the RAW filter in semi-implicit integrations. First, in a linear theoretical analysis, the stability and accuracy are interrogated by deriving analytic expressions for the amplitude errors and phase errors. Then, power-series expansions are used to interpret the leading-order errors for small time steps and hence to identify optimal values of the filter parameters. Finally, the RAW filter is tested in a realistic nonlinear setting, by applying it to semi-implicit integrations of the elastic pendulum equations. The results suggest that replacing the RA filter with the RAW filter could reduce time-stepping errors in semi-implicit integrations.

Corresponding author address: Paul D. Williams, Department of Meteorology, University of Reading, Earley Gate, Reading RG6 6BB, United Kingdom. E-mail: p.d.williams@reading.ac.uk

Abstract

Errors caused by discrete time stepping may be an important component of total model error in contemporary atmospheric and oceanic simulations. To reduce time-stepping errors in leapfrog integrations, the Robert–Asselin–Williams (RAW) filter was proposed by the author as a simple improvement to the widely used Robert–Asselin (RA) filter. The present paper examines the behavior of the RAW filter in semi-implicit integrations. First, in a linear theoretical analysis, the stability and accuracy are interrogated by deriving analytic expressions for the amplitude errors and phase errors. Then, power-series expansions are used to interpret the leading-order errors for small time steps and hence to identify optimal values of the filter parameters. Finally, the RAW filter is tested in a realistic nonlinear setting, by applying it to semi-implicit integrations of the elastic pendulum equations. The results suggest that replacing the RA filter with the RAW filter could reduce time-stepping errors in semi-implicit integrations.

Corresponding author address: Paul D. Williams, Department of Meteorology, University of Reading, Earley Gate, Reading RG6 6BB, United Kingdom. E-mail: p.d.williams@reading.ac.uk
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  • Amezcua, J., E. Kalnay, and P. D. Williams, 2011: The effects of the RAW filter on the climatology and forecast skill of the SPEEDY model. Mon. Wea. Rev., 139, 608619.

    • Search Google Scholar
    • Export Citation
  • 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.

  • Dietrich, D. E., and J. J. Wormeck, 1985: An optimized implicit scheme for compressible reactive gas flow. Numer. Heat Transfer, 8, 335348.

    • Search Google Scholar
    • Export Citation
  • 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. S. Medvedev, T. Kuroda, R. Saito, G. Villanueva, A. G. Feofilov, A. 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
  • Kalnay, E., 2003: Atmospheric Modeling, Data Assimilation and Predictability. Cambridge University Press, 364 pp.

  • 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
  • Kwizak, M., and A. J. Robert, 1971: A semi-implicit scheme for grid point atmospheric models of the primitive equations. Mon. Wea. Rev., 99, 3236.

    • Search Google Scholar
    • Export Citation
  • Leclair, M., and G. Madec, 2009: A conservative leapfrog time stepping method. Ocean Modell., 30, 8894.

  • Lilly, D. K., 1965: On the computational stability of numerical solutions of time-dependent non-linear geophysical fluid dynamics problems. Mon. Wea. Rev., 93, 1126.

    • Search Google Scholar
    • Export Citation
  • Lynch, P., 2002: The swinging spring: A simple model of atmospheric balance. Large-Scale Atmosphere–Ocean Dynamics II: Geometric Methods and Models, J. Norbury and I. Roulstone, Eds., Cambridge University Press, 64–108.

    • 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, Volume I. Global Atmospheric Research Programme Publications Series, No. 17, GARP, 64 pp.

    • Search Google Scholar
    • Export Citation
  • Molteni, F., 2003: Atmospheric simulations using a GCM with simplified physical parametrizations. I: Model climatology and variability in multi-decadal experiments. Climate Dyn., 20, 175191.

    • Search Google Scholar
    • Export Citation
  • Pfeffer, R. L., I. M. Navon, and X. L. 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
  • Roache, P. J., and D. E. Dietrich, 1988: Evaluation of the filtered Leapfrog–Trapezoidal time integration method. Numer. Heat Transfer, 14, 149164.

    • 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
  • Robert, A. J., 1969: The integration of a spectral model of the atmosphere by the implicit method. Proc. WMO–IUGG Symp. on NWP, Tokyo, Japan, Japan Meteorological Agency, 19–24.

    • Search Google Scholar
    • Export Citation
  • Robert, A. J., and M. Lépine, 1997: An anomaly in the behaviour of a time filter used with the leapfrog scheme in atmospheric models. 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, S3–S15.

    • Search Google Scholar
    • Export Citation
  • Schlesinger, R. E., L. W. Uccellini, and D. R. 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
  • Skamarock, W. C., and J. B. Klemp, 1992: The stability of time-split numerical methods for the hydrostatic and the nonhydrostatic elastic equations. Mon. Wea. Rev., 120, 21092127.

    • Search Google Scholar
    • Export Citation
  • Staniforth, A., 1997: André Robert (1929–1993): His pioneering contributions to numerical modelling. 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. K., 1987: Robert’s recursive frequency filter: A reexamination. Meteor. Atmos. Phys., 37, 4859.

  • Teixeira, J., C. A. Reynolds, and K. Judd, 2007: Time step sensitivity of nonlinear atmospheric models: Numerical convergence, truncation error growth, and ensemble design. J. Atmos. Sci., 64, 175189.

    • Search Google Scholar
    • Export Citation
  • Williams, P. D., 2009: A proposed modification to the Robert–Asselin time filter. Mon. Wea. Rev., 137, 25382546.

  • 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 Develop., 2, 1332.

    • Search Google Scholar
    • Export Citation
  • Williamson, D. L., and J. G. Olson, 2003: Dependence of aqua-planet simulations on time step. Quart. J. Roy. Meteor. Soc., 129, 20492064.

    • Search Google Scholar
    • Export Citation
  • Wolfram Research, 2008: Mathematica Version 7.0. Wolfram Research.

  • Young, J. A., 1968: Comparative properties of some time differencing schemes for linear and nonlinear oscillations. Mon. Wea. Rev., 96, 357364.

    • Search Google Scholar
    • Export Citation
  • Zhao, B., and Q. Zhong, 2009: The dynamical and climate tests of an atmospheric general circulation model using the second-order Adams–Bashforth method. Acta Meteor. Sin., 23, 738749.

    • Search Google Scholar
    • Export Citation
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