Editorial Type:
Article
Article Type:
Research Article

Accuracy Analysis of a Spectral Element Atmospheric Model Using a Fully Implicit Solution Framework

Katherine J. Evans Oak Ridge National Laboratory, Oak Ridge, Tennessee

Search for other papers by Katherine J. Evans in
Current site
Google Scholar
PubMed Close
,
Mark A. Taylor Sandia National Laboratories, Albuquerque, New Mexico

Search for other papers by Mark A. Taylor in
Current site
Google Scholar
PubMed Close
, and
John B. Drake Oak Ridge National Laboratory, Oak Ridge, Tennessee

Search for other papers by John B. Drake in
Current site
Google Scholar
PubMed Close
Print Publication:
01 Aug 2010
DOI:
https://doi.org/10.1175/2010MWR3288.1
Page(s):
3333–3341
Restricted access

Abstract

A fully implicit (FI) time integration method has been implemented into a spectral finite-element shallow-water equation model on a sphere, and it is compared to existing fully explicit leapfrog and semi-implicit methods for a suite of test cases. This experiment is designed to determine the time step sizes that minimize simulation time while maintaining sufficient accuracy for these problems. For test cases without an analytical solution from which to compare, it is demonstrated that time step sizes 30–60 times larger than the gravity wave stability limits and 6–20 times larger than the advective-scale stability limits are possible using the FI method without a loss in accuracy, depending on the problem being solved. For a steady-state test case, the FI method produces error within machine accuracy limits as with existing methods, but using an arbitrarily large time step size.

Corresponding author address: Katherine J. Evans, Oak Ridge National Laboratory, 1 Bethel Valley Rd., P.O. Box 2008, MS 6016, Oak Ridge, TN 37831. Email: evanskj@ornl.gov

Abstract

A fully implicit (FI) time integration method has been implemented into a spectral finite-element shallow-water equation model on a sphere, and it is compared to existing fully explicit leapfrog and semi-implicit methods for a suite of test cases. This experiment is designed to determine the time step sizes that minimize simulation time while maintaining sufficient accuracy for these problems. For test cases without an analytical solution from which to compare, it is demonstrated that time step sizes 30–60 times larger than the gravity wave stability limits and 6–20 times larger than the advective-scale stability limits are possible using the FI method without a loss in accuracy, depending on the problem being solved. For a steady-state test case, the FI method produces error within machine accuracy limits as with existing methods, but using an arbitrarily large time step size.

Corresponding author address: Katherine J. Evans, Oak Ridge National Laboratory, 1 Bethel Valley Rd., P.O. Box 2008, MS 6016, Oak Ridge, TN 37831. Email: evanskj@ornl.gov

Save
Cover Monthly Weather Review
Monthly Weather Review

  • Cai, X-C., D. D. E. Keyes, and L. Marcinkowski, 2002: Non-linear additive Schwarz preconditioners and application in computational fluid dynamics. Int. J. Numer. Methods Fluids, 40 , 1463–1470.

    • Search Google Scholar
    • Export Citation

  • Collins, W. D., and Coauthors, 2006: The Community Climate System Model version 3 (CCSM3). J. Climate, 19 , 2122–2143.

  • Dennis, J., A. Fournier, W. F. Spotz, A. St.-Cyr, M. A. Taylor, S. J. Thomas, and H. Tufo, 2005: High resolution mesh convergence properties and parallel efficiency of a spectral element dynamical core. Int. J. High Perform. Comput. Appl., 19 , 225–235.

    • Search Google Scholar
    • Export Citation

  • Elman, H. C., and D. Silvester, 1995: Fast nonsymmetric iterations and preconditioning for the Navier–Stokes equations. Numerical Analysis Rep. 263, 400 pp.

    • Search Google Scholar
    • Export Citation

  • Evans, K. J., and D. A. Knoll, 2007: Temporal accuracy analysis of phase change convection simulations using the JFNK-SIMPLE algorithm. Int. J. Numer. Methods Fluids, 55 , 637–653.

    • Search Google Scholar
    • Export Citation

  • Evans, K. J., D. A. Knoll, and M. A. Pernice, 2006: Development of a 2-D algorithm to simulate convection and phase transition efficiently. J. Comput. Phys., 219 , 404–417.

    • Search Google Scholar
    • Export Citation

  • Evans, K. J., D. Rouson, A. Salinger, M. Taylor, W. Weijer, and J. B. White III, 2009: A scalable and adaptable solution framework within components of the Community Climate System Model. Lect. Notes Comput. Sci., 5545 , 332–341. doi:10.1007/978-3-642-01973-9.

    • Search Google Scholar
    • Export Citation

  • Heroux, M. A., and Coauthors, 2005: An overview of the Trilinos project. ACM Trans. Math. Softw., 31 (3) 397–423.

  • Holdaway, D., J. Thruburn, and N. Wood, 2007: On the relation between spectral order of accuracy, convergence rate and spectral slope for linear numerical methods applied to multiscale problems. Int. J. Numer. Methods Fluids, 56 , 1297–1303.

    • Search Google Scholar
    • Export Citation

  • Hughes, T., G. Engel, L. Mazzei, and M. Larson, 2000: The continuous Galerkin method is locally conservative. J. Comput. Phys., 163 , 467–488.

    • Search Google Scholar
    • Export Citation

  • Jakob, R., J. J. Hack, and D. L. Williamson, 1995: Spectral transform solution of the shallow water test set. J. Comput. Phys., 119 , 164–187.

    • Search Google Scholar
    • Export Citation

  • Keyes, D., D. Reynolds, and C. Woodward, 2006: Implicit solvers for large-scale nonlinear problems. J. Phys. Conf. Series, 46 , 433–442.

    • Search Google Scholar
    • Export Citation

  • Knoll, D., and D. Keyes, 2004: Jacobian-free Newton–Krylov methods: A survey of approaches and applications. J. Comput. Phys., 193 , 357–397.

    • Search Google Scholar
    • Export Citation

  • Maltrud, M. E., and J. L. McClean, 2005: An eddy-resolving global 1/10° ocean simulation. Ocean Modell., 8 , 31–54.

  • Pernice, M., and M. Tocci, 2001: A multigrid-preconditioned Newton–Krylov method for the incompressible Navier–Stokes equations. SIAM J. Sci. Comput., 23 , 398–418.

    • Search Google Scholar
    • Export Citation

  • Robert, A., 1966: The integration of a low order spectral form of the primitive meteorological equations. J. Meteor. Soc. Japan, 44 , 237–245.

    • Search Google Scholar
    • Export Citation

  • Saad, Y., and M. H. Schultz, 1986: GMRES: Generalized minimum residual algorithm for solving nonsymmetric systems. SIAM J. Sci. Stat. Comput., 7 , 856–869.

    • Search Google Scholar
    • Export Citation

  • Taylor, M. A., and A. Fournier, 2010: A compatible and conservative spectral element method on unstructured grids. J. Comput. Phys., 229 , 5879–5895. doi:10.1016/j.jcp.2010.04.008.

    • Search Google Scholar
    • Export Citation

  • Taylor, M. A., J. Tribbia, and M. Iskandarani, 1997: The spectral element method for the shallow water equations on the sphere. J. Comput. Phys., 130 , 92–108.

    • Search Google Scholar
    • Export Citation

  • Taylor, M. A., J. Edwards, S. Thomas, and R. Nair, 2007: A mass and energy conserving spectral element atmospheric dynamical core on the cubed-sphere grid. J. Phys. Conf. Series, 78 , 12–24.

    • Search Google Scholar
    • Export Citation

  • Taylor, M. A., J. Edwards, and A. St.-Cyr, 2008: Petascale atmospheric models for the Community Climate System Model: New developments and evaluation of scalable dynamic cores. J. Phys. Conf. Series, 125 , 12–23.

    • Search Google Scholar
    • Export Citation

  • Thomas, S. J., and R. D. Loft, 2002: Semi-implicit spectral element model. SIAM J. Sci. Comput., 17 , 339–350.

  • Thomas, S. J., and R. D. Loft, 2005: The NCAR spectral element climate dynamical core: Semi-implicit Eulerian formulation. SIAM J. Sci. Comput., 25 , 307–322.

    • Search Google Scholar
    • Export Citation

  • Williamson, D. L., J. B. Drake, J. J. Hack, R. J. Jakob, and P. Swarztrauber, 1992: A standard test set for numerical approximations to the shallow water equations in spherical geometry. J. Comput. Phys., 102 , 211–224.

    • Search Google Scholar
    • Export Citation

  • Yang, C., J. Cao, and X-C. Cai, 2010: A fully implicit domain decomposition algorithm for shallow water equations on the cubed sphere. SIAM J. Sci. Comput., 32 , 418–438.

    • Search Google Scholar
    • Export Citation
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 205 80 11
PDF Downloads 85 29 9