A Discontinuous Galerkin Global Shallow Water Model

Ramachandran D. Nair Scientific Computing Division, National Center for Atmospheric Research,* Boulder, Colorado

Search for other papers by Ramachandran D. Nair in
Current site
Google Scholar
PubMed
Close
,
Stephen J. Thomas Scientific Computing Division, National Center for Atmospheric Research,* Boulder, Colorado

Search for other papers by Stephen J. Thomas in
Current site
Google Scholar
PubMed
Close
, and
Richard D. Loft Scientific Computing Division, National Center for Atmospheric Research,* Boulder, Colorado

Search for other papers by Richard D. Loft in
Current site
Google Scholar
PubMed
Close
Restricted access

We are aware of a technical issue preventing figures and tables from showing in some newly published articles in the full-text HTML view.
While we are resolving the problem, please use the online PDF version of these articles to view figures and tables.

Abstract

A discontinuous Galerkin shallow water model on the cubed sphere is developed, thereby extending the transport scheme developed by Nair et al. The continuous flux form nonlinear shallow water equations in curvilinear coordinates are employed. The spatial discretization employs a modal basis set consisting of Legendre polynomials. Fluxes along the element boundaries (internal interfaces) are approximated by a Lax–Friedrichs scheme. A third-order total variation diminishing Runge–Kutta scheme is applied for time integration, without any filter or limiter. Numerical results are reported for the standard shallow water test suite. The numerical solutions are very accurate, there are no spurious oscillations in test case 5, and the model conserves mass to machine precision. Although the scheme does not formally conserve global invariants such as total energy and potential enstrophy, conservation of these quantities is better preserved than in existing finite-volume models.

Corresponding author address: Ramachandran D. Nair, Scientific Computing Division, National Center for Atmospheric Research, 1850 Table Mesa Drive, Boulder, CO 80305. Email: rnair@ucar.edu

Abstract

A discontinuous Galerkin shallow water model on the cubed sphere is developed, thereby extending the transport scheme developed by Nair et al. The continuous flux form nonlinear shallow water equations in curvilinear coordinates are employed. The spatial discretization employs a modal basis set consisting of Legendre polynomials. Fluxes along the element boundaries (internal interfaces) are approximated by a Lax–Friedrichs scheme. A third-order total variation diminishing Runge–Kutta scheme is applied for time integration, without any filter or limiter. Numerical results are reported for the standard shallow water test suite. The numerical solutions are very accurate, there are no spurious oscillations in test case 5, and the model conserves mass to machine precision. Although the scheme does not formally conserve global invariants such as total energy and potential enstrophy, conservation of these quantities is better preserved than in existing finite-volume models.

Corresponding author address: Ramachandran D. Nair, Scientific Computing Division, National Center for Atmospheric Research, 1850 Table Mesa Drive, Boulder, CO 80305. Email: rnair@ucar.edu

Save
  • Arakawa, A., and V. R. Lamb, 1981: A potential enstrophy and energy conserving scheme for the shallow-water equations. Mon. Wea. Rev., 109 , 1836.

    • Search Google Scholar
    • Export Citation
  • Bacon, D. P., and Coauthors, 2000: A dynamically adapting weather and dispersion model: The Operational Multiscale Environment Model with Grid Adaptivity (OMEGA). Mon. Wea. Rev., 128 , 20442076.

    • Search Google Scholar
    • Export Citation
  • Bassi, F., and S. Rebay, 1997: A high-order accurate discontinuous finite element solution of the 2D Euler equations. J. Comput. Phys., 138 , 251285.

    • Search Google Scholar
    • Export Citation
  • Cockburn, B., and C. W. Shu, 1989: TVB Runge–Kutta local projection discontinuous Galerkin method for conservation laws II: General framework. Math. Comput., 52 , 411435.

    • Search Google Scholar
    • Export Citation
  • Cockburn, B., and C. W. Shu, 1998: The Runge–Kutta discontinuous Galerkin finite element method for conservation laws V: Multidimensional systems. J. Comput. Phys., 141 , 199224.

    • Search Google Scholar
    • Export Citation
  • Cockburn, B., and C. W. Shu, 2001: The Runge–Kutta discontinuous Galerkin method for convection-dominated problems. J. Sci. Comput., 16 , 173261.

    • Search Google Scholar
    • Export Citation
  • Cockburn, B., G. E. Karniadakis, and C. W. Shu, 2000: Discontinuous Galerkin Methods: Theory, Computation, and Applications. Lecture Notes in Computational Science and Engineering Vol. 11, Springer-Verlag, 470 pp.

    • Search Google Scholar
    • Export Citation
  • Fournier, A., M. A. Taylor, and J. J. Tribbia, 2004: A spectral element atmospheric model (SEAM): High-resolution parallel computation and localized resolution of regional dynamics. Mon. Wea. Rev., 132 , 726748.

    • Search Google Scholar
    • Export Citation
  • Giraldo, F. X., and T. E. Rosmond, 2004: A scalable spectral element Eulerian atmospheric model (SEE-AM) for NWP: Dynamical core test. Mon. Wea. Rev., 132 , 133153.

    • Search Google Scholar
    • Export Citation
  • Giraldo, F. X., J. S. Hesthaven, and T. Warburton, 2002: Nodal high-order discontinuous Galerkin methods for spherical shallow water equations. J. Comput. Phys., 181 , 499525.

    • Search Google Scholar
    • Export Citation
  • Gottlieb, S., C. W. Shu, and E. Tadmor, 2001: Strong stability-preserving high-order time discretization methods. SIAM Rev., 43 , 89112.

    • Search Google Scholar
    • Export Citation
  • Heikes, R., and D. A. Randall, 1995: Numerical integration of the shallow water equations on a twisted icosahedral grid. Part I: Basic design and results of tests. Mon. Wea. Rev., 123 , 18621880.

    • Search Google Scholar
    • Export Citation
  • Iskandrani, M., D. B. Haidvogal, J. C. Levin, E. Curchister, and C. A. Edwards, 2002: Multiscale geophysical modeling using the spectral element method. Comput. Sci. Eng., 4 , 4248.

    • Search Google Scholar
    • Export Citation
  • Jablonowski, C., 2004: Adaptive grids in weather and climate modeling. Ph.D. thesis, University of Michigan, 272 pp.

  • Jakob-Chien, R., J. J. Hack, and D. L. Williamson, 1995: Spectral transform solutions to the shallow water test set. J. Comput. Phys., 119 , 164187.

    • Search Google Scholar
    • Export Citation
  • Karniadakis, G. E., and S. J. Sherwin, 1999: Spectral/hp Element Methods for CFD. Oxford University Press, 390 pp.

  • Leveque, R. J., 2002: Finite Volume Methods for Hyperbolic Problems. Cambridge Texts in Applied Mathematics, Cambridge University Press, 558 pp.

    • Search Google Scholar
    • Export Citation
  • Lin, S-J., and B. Rood, 1997: An explicit flux-form semi-Lagrangian shallow water model on the sphere. Quart. J. Roy. Meteor. Soc., 123 , 25312533.

    • Search Google Scholar
    • Export Citation
  • Loft, R. D., S. J. Thomas, and J. M. Dennis, 2001: Terascale spectral element dynamical core for atmospheric general circulation models. Proc. ACM/IEEE Supercomputing 2001 Conf., Denver, CO, ACM/IEEE, CD-ROM.

  • McDonald, A., and J. R. Bates, 1989: Semi-Lagrangian integration of a shallow water model on the sphere. Mon. Wea. Rev., 117 , 130137.

    • Search Google Scholar
    • Export Citation
  • Nair, R. D., S. J. Thomas, and R. D. Loft, 2005: A discontinuous Galerkin transport scheme on the cubed sphere. Mon. Wea. Rev, 133 , 827841.

    • Search Google Scholar
    • Export Citation
  • Rančić, M. R., J. Purser, and F. Mesinger, 1996: A global-shallow water model using an expanded spherical cube. Quart. J. Roy. Meteor. Soc., 122 , 959982.

    • Search Google Scholar
    • Export Citation
  • Remacle, J-F., J. E. Flaherty, and M. S. Shephard, 2003: An adaptive discontinuous Galerkin technique with an orthogonal basis applied to compressible flow problems. SIAM Rev., 45 , 5372.

    • Search Google Scholar
    • Export Citation
  • Ronchi, C., R. Iacono, and P. S. Paolucci, 1996: The “cubed sphere”: A new method for the solution of partial differential equations in spherical geometry. J. Comput. Phys., 124 , 93114.

    • Search Google Scholar
    • Export Citation
  • Sadourny, R., 1972: Conservative finite-difference approximations of the primitive equations on quasi-uniform spherical grids. Mon. Wea. Rev., 100 , 136144.

    • Search Google Scholar
    • Export Citation
  • Taylor, M., J. Tribbia, and M. Iskandrani, 1997: The spectral element method for the shallow water equations on the sphere. J. Comput. Phys., 130 , 92108.

    • Search Google Scholar
    • Export Citation
  • Thomas, S. J., and R. D. Loft, 2002: Semi-implicit spectral element model. J. Sci. Comput., 17 , 339350.

  • Thuburn, J., 1997: A PV-based shallow-water model on a hexagonal-icosahedral grid. Mon. Wea. Rev., 125 , 23282336.

  • Williamson, D. L., J. B. Drake, J. Hack, R. Jakob, and P. N. Swartztrauber, 1992: A standard test set for numerical approximations to the shallow water equations in spherical geometry. J. Comput. Phys., 102 , 211224.

    • Search Google Scholar
    • Export Citation
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 994 395 66
PDF Downloads 718 209 16