Spectral Preconditioners for Nonhydrostatic Atmospheric Models

Stephen J. Thomas National Center for Atmospheric Research, Boulder, Colorado

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Joshua P. Hacker National Center for Atmospheric Research, Boulder, Colorado

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Piotr K. Smolarkiewicz National Center for Atmospheric Research, Boulder, Colorado

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Roland B. Stull Department of Earth and Ocean Sciences, University of British Columbia, Vancouver, British Columbia, Canada

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Abstract

The elliptic problems in semi-implicit nonhydrostatic atmospheric models are difficult. Typically, they are poorly conditioned, nonseparable, contain cross-derivative terms, and are often nonsymmetric. Here, the resulting linear system is solved using a preconditioned Krylov subspace method—the generalized conjugate residual (GCR) algorithm. A horizontal spectral preconditioner is developed as an alternative to a more standard and much simpler line Jacobi relaxation scheme. However, the efficacy of the spectral preconditioner requires neglecting the cross-derivative terms and homogenization (e.g., averaging) metric coefficients over the computational domain. Because such a compromise causes a substantial departure of the preconditioner from the original elliptic operator, it is not obvious a priori whether it leads to a competitive solver. The robustness of the proposed approach over a broad range of representative meteorological applications is evaluated, in the context of a three-time-level semi-implicit semi-Lagrangian all-scale weather-prediction/research model.

Corresponding author address: Dr. Stephen J. Thomas, NCAR/SCD, P.O. Box 3000, Boulder, CO 80307-3000. Email: thomas@ucar.edu

Abstract

The elliptic problems in semi-implicit nonhydrostatic atmospheric models are difficult. Typically, they are poorly conditioned, nonseparable, contain cross-derivative terms, and are often nonsymmetric. Here, the resulting linear system is solved using a preconditioned Krylov subspace method—the generalized conjugate residual (GCR) algorithm. A horizontal spectral preconditioner is developed as an alternative to a more standard and much simpler line Jacobi relaxation scheme. However, the efficacy of the spectral preconditioner requires neglecting the cross-derivative terms and homogenization (e.g., averaging) metric coefficients over the computational domain. Because such a compromise causes a substantial departure of the preconditioner from the original elliptic operator, it is not obvious a priori whether it leads to a competitive solver. The robustness of the proposed approach over a broad range of representative meteorological applications is evaluated, in the context of a three-time-level semi-implicit semi-Lagrangian all-scale weather-prediction/research model.

Corresponding author address: Dr. Stephen J. Thomas, NCAR/SCD, P.O. Box 3000, Boulder, CO 80307-3000. Email: thomas@ucar.edu

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  • Arakawa, A., and V. Lamb, 1977: Computational design of the basic dynamical processes of the UCLA general circulation model. Methods in Computational Physics, Vol. 17, J. Chang, Ed., Academic Press, 174–265.

    • Search Google Scholar
    • Export Citation
  • Axelsson, O., 1994: Iterative Solution Methods. Cambridge University Press, 654 pp.

  • Barros, S., and T. Kauranne, 1994: On the parallelization of global spectral weather models. Parallel Comput., 20 , 13351356.

  • Benoit, R., M. Desgagné, P. Pellerin, S. Pellerin, Y. Chartier, and S. Desjardins, 1997: The Canadian MC2: A semi-Lagrangian, semi-implicit wide-band atmospheric model suited for fine-scale process studies and simulations. Mon. Wea. Rev., 125 , 23822415.

    • Search Google Scholar
    • Export Citation
  • Bernardet, P., 1995: The pressure term in the anelastic model: A symmetric elliptic solver for an Arakawa C grid in generalized coordinates. Mon. Wea. Rev., 123 , 24742490.

    • Search Google Scholar
    • Export Citation
  • Bourke, W., 1974: A multi-level spectral model. I. Formulation and hemispheric integrations. Mon. Wea. Rev., 102 , 687701.

  • Cullen, M., 1990: A test of a semi-implicit integration technique for a fully compressible nonhydrostatic model. Quart. J. Roy. Meteor. Soc., 116 , 12531258.

    • Search Google Scholar
    • Export Citation
  • Davies, H., 1976: A lateral boundary formulation for multi-level prediction models. Quart. J. Roy. Meteor. Soc., 102 , 405418.

  • Eisenstat, S., H. Elman, and M. Schultz, 1983: Variational iterative methods for nonsymmetric systems of linear equations. SIAM J. Numer. Anal., 2 , 345357.

    • Search Google Scholar
    • Export Citation
  • Elman, H., and D. O'Leary, 1998: Efficient iterative solution of the three-dimensional Helmholtz equation. J. Comput. Phys., 142 , 163181.

    • Search Google Scholar
    • Export Citation
  • Freund, R., 1993: A transpose-free quasi-mimimum residual algorithm for non-Hermitian linear systems. SIAM J. Sci. Stat. Comput., 14 , 470482.

    • Search Google Scholar
    • Export Citation
  • Gal-Chen, T., and R. Sommerville, 1975: On the use of a coordinate transformation for the solution of the Navier–Stokes equations. J. Comput. Phys., 17 , 209228.

    • Search Google Scholar
    • Export Citation
  • Golding, B., 1992: An efficient nonhydrostatic forecast model. Meteor. Atmos. Phys., 50 , 89103.

  • Grabowski, W., and P. Smolarkiewicz, 2002: A multiscale model for meteorological research. Mon. Wea. Rev., 130 , 939956.

  • Greenbaum, A., 1997: Iterative Methods for Solving Linear Systems. Society of Industrial and Applied Mathematics, 220 pp.

  • Hestenes, M., and E. Stiefel, 1952: Method of conjugate gradients for solving linear systems. J. Res. Natl. Bur. Stand. (U.S.), 49 , 409436.

    • Search Google Scholar
    • Export Citation
  • Kadogliu, H., and S. Mudrick, 1992: On the implementation of the GMRES(m) method to elliptic equations in meteorology. J. Comput. Phys., 102 , 348359.

    • Search Google Scholar
    • Export Citation
  • Kapitza, H., and D. Eppel, 1992: The nonhydrostatic mesoscale model GESIMA. Part I: Dynamical equations and tests. Beitr. Phys. Atmos., 65 , 129146.

    • Search Google Scholar
    • Export Citation
  • Lafore, J., and Coauthors. 1998: The Meso–NH atmospheric modeling system. Part I: Adiabatic formulation and control simulations. Ann. Geophys., 16 , 90109.

    • Search Google Scholar
    • Export Citation
  • Laprise, R., D. Caya, G. Bergeron, and M. Giguère, 1997: The formulation of the André Robert MC2 (Mesoscale Compressible Community) model. Atmos.–Ocean, 35 , 127152.

    • Search Google Scholar
    • Export Citation
  • Lynch, R., J. Rice, and D. Thomas, 1964: Direct solution of partial difference equations by tensor product methods. Numer. Math., 6 , 185199.

    • Search Google Scholar
    • Export Citation
  • Machenhauer, B., and R. Daley, 1972: A baroclinic primitive equation model with a spectral representation in three dimensions. Tech. Rep. 4, Institute of Theoretical Meteorology, University of Copenhagen, 63 pp.

    • Search Google Scholar
    • Export Citation
  • Marshall, J., C. Hill, L. Perelman, and A. Adcroft, 1997: Hydrostatic, quasi-hydrostatic and non-hydrostatic ocean modeling. J. Geophys. Res., 102 , 57335752.

    • Search Google Scholar
    • Export Citation
  • Navon, I., and Y. Cai, 1993: Domain decomposition and parallel processing of a finite element model of the shallow water equations. Comput. Methods Appl. Mech. Eng., 106 , 179212.

    • Search Google Scholar
    • Export Citation
  • Prusa, J., P. Smolarkiewicz, and R. Garcia, 1996: On the propagation and breaking at high altitudes of gravity waves excited by tropospheric forcing. J. Atmos. Sci., 53 , 21862216.

    • Search Google Scholar
    • Export Citation
  • Roache, P., 1972: Computational Fluid Dynamics. Hermosa, 446 pp.

  • Robert, A., 1969: Integration of a spectral model of the atmosphere by the implicit method. Proc. WMO/IUGG Symp. on NWP, Vol. 7, Tokyo, Japan, Japan Meteorological Agency, 19–24.

    • Search Google Scholar
    • Export Citation
  • Robert, A., 1981: A stable numerical integration scheme for the primitive meteorological equations. Atmos.–Ocean, 19 , 3546.

  • Robert, A., 1993: Bubble convection experiments with a semi-implicit formulation of the Euler equations. J. Atmos. Sci., 50 , 18651873.

    • Search Google Scholar
    • Export Citation
  • Robert, A., T. Yee, and H. Ritchie, 1985: A semi-Lagrangian semi-implicit numerical integration scheme for multilevel atmospheric models. Mon. Wea. Rev., 113 , 388394.

    • Search Google Scholar
    • Export Citation
  • Saad, Y., 1993: A flexible inner-outer preconditioned GMRES algorithm. SIAM J. Sci. Stat. Comput., 14 , 461469.

  • Saad, Y., 1996: Iterative Methods for Sparse Linear Systems. PWS, 447 pp.

  • Saad, Y., and M. Schultz, 1986: GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput., 7 , 856869.

    • Search Google Scholar
    • Export Citation
  • Saito, K., U. Schättler, and U. Steppler, 1998: 3D mountain waves by the Lokal-Modell of DWD and the MRI mesoscale nonhydrostatic model. Pap. Meteor. Geophys., 49 , 719.

    • Search Google Scholar
    • Export Citation
  • Schumann, U., and H. Volkert, 1987: Three-dimensional mass- and momentum-consistent Helmholtz-equation in terrain-following coordinates. Notes on Numerical Fluid Mechanics, W. Hackbusch, Ed., Vol. 10, Springer-Verlag, 109–131.

    • Search Google Scholar
    • Export Citation
  • Schumann, U., and R. Sweet, 1988: Fast Fourier transforms for direct solution of Poisson's equation with staggered boundary conditions. J. Comput. Phys., 75 , 123137.

    • Search Google Scholar
    • Export Citation
  • Skålin, R., 1997: Scalability of parallel gridpoint limited-area atmospheric models. Part II: Semi-implicit time-integration schemes. J. Atmos. Oceanic Technol., 14 , 442455.

    • Search Google Scholar
    • Export Citation
  • Skamarock, W., P. Smolarkiewicz, and J. Klemp, 1997: Preconditioned conjugate-residual solvers for Helmholtz equations in nonhydrostatic models. Mon. Wea. Rev., 125 , 587599.

    • Search Google Scholar
    • Export Citation
  • Smith, R., 1980: Linear theory of stratified hydrostatic flow past an isolated mountain. Tellus, 32 , 348364.

  • Smolarkiewicz, P., and J. Pudykiewicz, 1992: A class of semi-Lagrangian approximations for fluids. J. Atmos. Sci., 49 , 20822096.

  • Smolarkiewicz, P., and L. Margolin, 1994: Variational solver for elliptic problems in atmospheric flows. Appl. Math. Comput. Sci., 4 , 527551.

    • Search Google Scholar
    • Export Citation
  • Smolarkiewicz, P., and L. Margolin, 1997: On forward-in-time differencing for fluids: An Eulerian/semi-Lagrangian nonhydrostatic model for stratified flows. Atmos.–Ocean, 35 , 127152.

    • Search Google Scholar
    • Export Citation
  • Smolarkiewicz, P., and L. Margolin, 2000: Variational methods for elliptic problems in fluid models. Proc. ECMWF Workshop on Developments in Numerical Methods for Very High Resolution Global Models, Reading, United Kingdom, ECMWF, 137–159.

    • Search Google Scholar
    • Export Citation
  • Smolarkiewicz, P., and J. Prusa, 2002: VLES modelling of geophysical fluids with nonoscillatory forward-in-time schemes. Int. J. Numer. Methods Fluids, 39 , 779819.

    • Search Google Scholar
    • Export Citation
  • Smolarkiewicz, P., V. Grubisǐć, and L. Margolin, 1997: On forward-in-time differencing for fluids: Stopping criteria for iterative solutions of anelastic pressure equations. Mon. Wea. Rev., 125 , 647654.

    • Search Google Scholar
    • Export Citation
  • Sweet, R., 1973: Direct methods for the solution of Poisson's equation on a staggered grid. J. Comput. Phys., 12 , 422428.

  • 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
  • Tapp, M., and P. White, 1976: A nonhydrostatic mesoscale model. Quart. J. Roy. Meteor. Soc., 102 , 277296.

  • Thomas, S., C. Girard, R. Benoit, M. Desgagné, and P. Pellerin, 1998: A new adiabatic kernel for the MC2 model. Atmos.–Ocean, 36 , 241270.

    • Search Google Scholar
    • Export Citation
  • Thomas, S., C. Girard, G. Doms, and U. Schättler, 2000: Semi-implicit scheme for the DWD Lokal-Modell. Meteor. Atmos. Phys., 73 , 105125.

    • Search Google Scholar
    • Export Citation
  • Thomas, S., J. Hacker, M. Desgagné, and R. Stull, 2002: An ensemble analysis of forecast errors related to floating point performance. Wea. Forecasting, 17 , 898906.

    • Search Google Scholar
    • Export Citation
  • Tokioka, T., 1978: Some considerations on vertical differencing. J. Meteor. Soc. Japan, 56 , 98111.

  • van der Vorst, H., 1992: Bi-CGStab: A fast and smoothly converging variant of Bi-CG for the solution of non-symmetric linear systems. SIAM J. Sci. Stat. Comput., 12 , 631644.

    • Search Google Scholar
    • Export Citation
  • Yeh, K-S., J. Cote, S. Gravel, A. Methot, A. Patoine, M. Roch, and A. Staniforth, 2002: The CMC–MRB Global Environmental Multiscale (GEM) model. Part III: Nonhydrostatic formulation. Mon. Wea. Rev., 130 , 339356.

    • Search Google Scholar
    • Export Citation
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