Editorial Type:
Article
Article Type:
Research Article

A Conservative Semi-Lagrangian Discontinuous Galerkin Scheme on the Cubed Sphere

Wei Guo Department of Mathematics, University of Houston, Houston, Texas

Search for other papers by Wei Guo in
Current site
Google Scholar
PubMed Close
,
Ramachandran D. Nair National Center for Atmospheric Research, Boulder, Colorado

Search for other papers by Ramachandran D. Nair in
Current site
Google Scholar
PubMed Close
, and
Jing-Mei Qiu Department of Mathematics, University of Houston, Houston, Texas

Search for other papers by Jing-Mei Qiu in
Current site
Google Scholar
PubMed Close
Print Publication:
01 Jan 2014
DOI:
https://doi.org/10.1175/MWR-D-13-00048.1
Page(s):
457–475
Restricted access

Abstract

The discontinuous Galerkin (DG) methods designed for hyperbolic problems arising from a wide range of applications are known to enjoy many computational advantages. DG methods coupled with strong-stability-preserving explicit Runge–Kutta discontinuous Galerkin (RKDG) time discretizations provide a robust numerical approach suitable for geoscience applications including atmospheric modeling. However, a major drawback of the RKDG method is its stringent Courant–Friedrichs–Lewy (CFL) stability restriction associated with explicit time stepping. To address this issue, the authors adopt a dimension-splitting approach where a semi-Lagrangian (SL) time-stepping strategy is combined with the DG method. The resulting SLDG scheme employs a sequence of 1D operations for solving multidimensional transport equations. The SLDG scheme is inherently conservative and has the option to incorporate a local positivity-preserving filter for tracers. A novel feature of the SLDG algorithm is that it can be used for multitracer transport for global models employing spectral-element grids, without using an additional finite-volume grid system. The quality of the proposed method is demonstrated via benchmark tests on Cartesian and cubed-sphere geometry, which employs nonorthogonal, curvilinear coordinates.

Corresponding author address: J.-M. Qiu, Department of Mathematics, University of Houston, 4800 Calhoun Rd., Houston, TX 77204. E-mail: jingqiu@math.uh.edu

Abstract

The discontinuous Galerkin (DG) methods designed for hyperbolic problems arising from a wide range of applications are known to enjoy many computational advantages. DG methods coupled with strong-stability-preserving explicit Runge–Kutta discontinuous Galerkin (RKDG) time discretizations provide a robust numerical approach suitable for geoscience applications including atmospheric modeling. However, a major drawback of the RKDG method is its stringent Courant–Friedrichs–Lewy (CFL) stability restriction associated with explicit time stepping. To address this issue, the authors adopt a dimension-splitting approach where a semi-Lagrangian (SL) time-stepping strategy is combined with the DG method. The resulting SLDG scheme employs a sequence of 1D operations for solving multidimensional transport equations. The SLDG scheme is inherently conservative and has the option to incorporate a local positivity-preserving filter for tracers. A novel feature of the SLDG algorithm is that it can be used for multitracer transport for global models employing spectral-element grids, without using an additional finite-volume grid system. The quality of the proposed method is demonstrated via benchmark tests on Cartesian and cubed-sphere geometry, which employs nonorthogonal, curvilinear coordinates.

Corresponding author address: J.-M. Qiu, Department of Mathematics, University of Houston, 4800 Calhoun Rd., Houston, TX 77204. E-mail: jingqiu@math.uh.edu
Save
Cover Monthly Weather Review
Monthly Weather Review

  • Blossey, P., and D. Durran, 2008: Selective monotonicity preservation in scalar advection. J. Comput. Phys., 227 (10), 5160–5183.

  • Childs, P., and K. Morton, 1990: Characteristic Galerkin methods for scalar conservation laws in one dimension. SIAM J. Numer. Anal., 27 (3), 553–594.

    • Search Google Scholar
    • Export Citation

  • Cockburn, B., and C.-W. Shu, 2001: Runge–Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput., 16 (3), 173–261.

    • Search Google Scholar
    • Export Citation

  • Dennis, J., and Coauthors, 2011: CAM-SE: A scalable spectral-element dynamical core for the Community Atmosphere Model. Int. J. High Perform. Comput. Appl., 26, 74–89.

    • Search Google Scholar
    • Export Citation

  • Erath, C., and R. D. Nair, 2014: A conservative multi-tracer transport scheme for spectral-element spherical grids. J. Comput. Phys., 256, 118–134, doi:10.1016/j.jcp.2013.08.050.

    • Search Google Scholar
    • Export Citation

  • Giraldo, F., and M. Restelli, 2008: A study of spectral element and discontinuous Galerkin method for the Navier–Stokes equations in nonhydrostatic mesoscale modeling: Equation sets and test cases. J. Comput. Phys., 227, 3847–3877.

    • Search Google Scholar
    • Export Citation

  • Gottlieb, D., 1972: Strang-type difference schemes for multidimensional problems. SIAM J. Numer. Anal., 9 (4), 650–661.

  • Hall, D. M., and R. D. Nair, 2013: Discontinuous Galerkin transport on the spherical Yin–Yang overset mesh. Mon. Wea. Rev., 141, 264–282.

    • Search Google Scholar
    • Export Citation

  • Lauritzen, P., R. Nair, and P. Ullrich, 2010: A conservative semi-Lagrangian multi-tracer transport scheme (CSLAM) on the cubed-sphere grid. J. Comput. Phys., 229 (5), 1401–1424.

    • Search Google Scholar
    • Export Citation

  • Nair, R., and C. Jablonowski, 2008: Moving vortices on the sphere: A test case for horizontal advection problems. Mon. Wea. Rev., 136, 699–711.

    • Search Google Scholar
    • Export Citation

  • Nair, R., and P. Lauritzen, 2010: A class of deformational flow test cases for linear transport problems on the sphere. J. Comput. Phys., 229 (23), 8868–8887.

    • Search Google Scholar
    • Export Citation

  • Nair, R., S. Thomas, and R. Loft, 2005: A discontinuous Galerkin transport scheme on the cubed sphere. Mon. Wea. Rev., 133, 814–828.

  • Nair, R., H.-W. Choi, and H. M. Tufo, 2009: Computational aspects of a scalable high-order discontinuous Galerkin atmospheric dynamical core. Comput. Fluids, 38, 309–319.

    • Search Google Scholar
    • Export Citation

  • Pudykiewicz, J., 2011: On numerical solution of the shallow water equations with chemical reactions on icosahedral geodesic grid. J. Comput. Phys., 230 (5), 1956–1991.

    • Search Google Scholar
    • Export Citation

  • Qiu, J., and C. Shu, 2011: Positivity preserving semi-Lagrangian discontinuous Galerkin formulation: Theoretical analysis and application to the Vlasov–Poisson system. J. Comput. Phys., 230 (23), 8386–8409.

    • Search Google Scholar
    • Export Citation

  • Restelli, M., L. Bonaventura, and R. Sacco, 2006: A semi-Lagrangian discontinuous Galerkin method for scalar advection by incompressible flows. J. Comput. Phys., 216, 195–215.

    • Search Google Scholar
    • Export Citation

  • Ronchi, C., R. Iacono, and P. Paolucci, 1996: The cubed sphere: A new method for the solution of partial differential equations in spherical geometry. J. Comput. Phys., 124, 93–114.

    • Search Google Scholar
    • Export Citation

  • Rossmanith, J., and D. Seal, 2011: A positivity-preserving high-order semi-Lagrangian discontinuous Galerkin scheme for the Vlasov–Poisson equations. J. Comput. Phys., 230, 6203–6232.

    • Search Google Scholar
    • Export Citation

  • Russell, T., and M. Celia, 2002: An overview of research on Eulerian–Lagrangian localized adjoint methods (ELLAM). Adv. Water Resour., 25 (8), 1215–1231.

    • 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, 136–144.

    • Search Google Scholar
    • Export Citation

  • Strang, G., 1968: On the construction and comparison of difference schemes. SIAM J. Numer. Anal., 5 (3), 506–517.

  • Tumolo, G., L. Bonaventura, and M. Restelli, 2013: A semi-implicit, semi-Lagrangian, p-adaptive discontinuous Galerkin method for the shallow water equations. J. Comput. Phys., 232, 46–67.

    • Search Google Scholar
    • Export Citation

  • Williamson, D., J. Drake, J. Hack, R. 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

  • Zhang, X., and C.-W. Shu, 2010a: On maximum-principle-satisfying high order schemes for scalar conservation laws. J. Comput. Phys., 229, 3091–3120.

    • Search Google Scholar
    • Export Citation

  • Zhang, X., and C.-W. Shu, 2010b: On positivity preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes. J. Comput. Phys., 229, 8918–8934.

    • Search Google Scholar
    • Export Citation

  • Zhang, Y., and R. D. Nair, 2012: A nonoscillatory discontinuous Galerkin transport scheme on the cubed sphere. Mon. Wea. Rev., 140, 3106–3126.

    • Search Google Scholar
    • Export Citation
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 274 114 11
PDF Downloads 231 111 14