A Dynamic hp-Adaptive Discontinuous Galerkin Method for Shallow-Water Flows on the Sphere with Application to a Global Tsunami Simulation

Sébastien Blaise Institute for Mathematics Applied to Geosciences, National Center for Atmospheric Research, Boulder, Colorado

Search for other papers by Sébastien Blaise in
Current site
Google Scholar
PubMed
Close
and
Amik St-Cyr Institute for Mathematics Applied to Geosciences, National Center for Atmospheric Research, Boulder, Colorado

Search for other papers by Amik St-Cyr in
Current site
Google Scholar
PubMed
Close
Restricted access

Abstract

A discontinuous Galerkin model solving the shallow-water equations on the sphere is presented. It captures the dynamically varying key aspects of the flows by having the advantageous ability to locally modify the mesh as well as the order of interpolation within each element. The computational load is efficiently distributed among processors in parallel using a weighted recursive coordinate bisection strategy. A simple error estimator, based on the discontinuity of the variables at the interfaces between elements, is used to select the elements to be refined or coarsened. The flows are expressed in three-dimensional Cartesian coordinates, but tangentially constrained to the sphere by adding a Lagrange multiplier to the system of equations. The model is validated on classic atmospheric test cases and on the simulation of the February 2010 Chilean tsunami propagation. The proposed multiscale strategy is able to reduce the computational time by an order of magnitude on the tsunami simulation, clearly demonstrating its potential toward multiresolution three-dimensional oceanic and atmospheric applications.

Current affiliation: Université Bordeaux 1, Talence, France.

Current affilation: Royal Dutch Shell, Houston, Texas.

The National Center for Atmospheric Research is sponsored by the National Science Foundation.

Corresponding author address: Sébastien Blaise, Université Bordeaux 1, Institut de Mathématiques de Bordeaux, 351, cours de la Libération, F-33405 Talence CEDEX, France. E-mail: sebastien.blaise@u-bordeaux1.fr

Abstract

A discontinuous Galerkin model solving the shallow-water equations on the sphere is presented. It captures the dynamically varying key aspects of the flows by having the advantageous ability to locally modify the mesh as well as the order of interpolation within each element. The computational load is efficiently distributed among processors in parallel using a weighted recursive coordinate bisection strategy. A simple error estimator, based on the discontinuity of the variables at the interfaces between elements, is used to select the elements to be refined or coarsened. The flows are expressed in three-dimensional Cartesian coordinates, but tangentially constrained to the sphere by adding a Lagrange multiplier to the system of equations. The model is validated on classic atmospheric test cases and on the simulation of the February 2010 Chilean tsunami propagation. The proposed multiscale strategy is able to reduce the computational time by an order of magnitude on the tsunami simulation, clearly demonstrating its potential toward multiresolution three-dimensional oceanic and atmospheric applications.

Current affiliation: Université Bordeaux 1, Talence, France.

Current affilation: Royal Dutch Shell, Houston, Texas.

The National Center for Atmospheric Research is sponsored by the National Science Foundation.

Corresponding author address: Sébastien Blaise, Université Bordeaux 1, Institut de Mathématiques de Bordeaux, 351, cours de la Libération, F-33405 Talence CEDEX, France. E-mail: sebastien.blaise@u-bordeaux1.fr
Save
  • Aizinger, V., and C. Dawson, 2002: A discontinuous Galerkin method for two-dimensional flow and transport in shallow water. Adv. Water Resour., 25, 6784, doi:10.1016/S0309-1708(01)00019-7.

    • Search Google Scholar
    • Export Citation
  • Amante, C., and B. W. Eakins, 2009: ETOPO1 1 arc-minute global relief model: Procedures, data sources and analysis. Tech. Rep., NOAA Tech. Memo. NESDIS NGDC-24, 19 pp.

    • Search Google Scholar
    • Export Citation
  • Bernard, P.-E., N. Chevaugeon, V. Legat, E. Deleersnijder, and J.-F. Remacle, 2007: High-order h-adaptive discontinuous Galerkin methods for ocean modelling. Ocean Dyn., 57, 109121.

    • Search Google Scholar
    • Export Citation
  • Bernard, P.-E., J.-F. Remacle, R. Comblen, V. Legat, and K. Hillewaert, 2009: High-order discontinuous Galerkin schemes on general 2D manifolds applied to the shallow water equations. J. Comput. Phys., 228 (17), 65146535.

    • Search Google Scholar
    • Export Citation
  • Bhanot, G., and Coauthors, 2008: Early experiences with the 360TF IBM Blue Gene/L platform. Int. J. Comput. Methods, 5, 237253.

  • Blaise, S., B. de Brye, A. de Brauwere, E. Deleersnijder, E. J. M. Delhez, and R. Comblen, 2010a: Capturing the residence time boundary layer–application to the Scheldt Estuary. Ocean Dyn., 60, 535554, doi:10.1007/s10236-010-0272-8.

    • Search Google Scholar
    • Export Citation
  • Blaise, S., R. Comblen, V. Legat, J.-F. Remacle, E. Deleersnijder, and J. Lambrechts, 2010b: A discontinuous finite element baroclinic marine model on unstructured prismatic meshes. Part I: Space discretization. Ocean Dyn., 60 (6), 13711393, doi:10.1007/s10236-010-0358-3.

    • Search Google Scholar
    • Export Citation
  • Boman, E., and Coauthors, cited 2011: Zoltan 3.0: Parallel partitioning, load-balancing, and data management services: User’s guide. Tech. Rep. SAND2007-4748W, Sandia National Laboratories, Albuquerque, NM. [Available online at http://www.cs.sandia.gov/Zoltan/ug_html/ug.html.]

    • Search Google Scholar
    • Export Citation
  • Burstedde, C., O. Ghattas, M. Gurnis, T. Isaac, G. Stadler, T. Warburton, and L. Wilcox, 2010: Extreme-scale AMR. Proc. 2010 ACM/IEEE Int. Conf. for High Performance Computing, Networking, Storage, and Analysis, Washington, DC, IEEE Computer Society, 1–12.

    • Search Google Scholar
    • Export Citation
  • Chen, C., F. Xiao, and X. Li, 2011: An adaptive multimoment global model on a cubed sphere. Mon. Wea. Rev., 139, 523548.

  • Cockburn, B., and C.-W. Shu, 2001: Runge–Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput., 16 (3), 173261.

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

    • Search Google Scholar
    • Export Citation
  • Comblen, R., S. Legrand, E. Deleersnijder, and V. Legat, 2009: A finite element method for solving the shallow water equations on the sphere. Ocean Modell., 28, 1223, doi:10.1016/j.ocemod.2008.05.004.

    • Search Google Scholar
    • Export Citation
  • Comblen, R., S. Blaise, V. Legat, J.-F. Remacle, E. Deleersnijder, and J. Lambrechts, 2010: A discontinuous finite element baroclinic marine model on unstructured prismatic meshes. Part II: Implicit/explicit time discretization. Ocean Dyn., 60 (6), 13951414, doi:10.1007/s10236-010-0357-4.

    • Search Google Scholar
    • Export Citation
  • Côté, J., 1988: A Lagrange multiplier approach for the metric terms of semi-Lagrangian models on the sphere. Quart. J. Roy. Meteor. Soc., 114, 13471352.

    • Search Google Scholar
    • Export Citation
  • Dennis, J., A. Fournier, W. F. Spotz, A. St-Cyr, M. A. Taylor, S. J. Thomas, and H. Tufo, 2005a: High-resolution mesh convergence properties and parallel efficiency of a spectral element atmospheric dynamical core. Int. J. High Perform. Comput. Appl., 19 (3), 225235.

    • Search Google Scholar
    • Export Citation
  • Dennis, J., M. Levy, R. D. Nair, H. M. Tufo, and T. Voran, 2005b: Towards an efficient and scalable discontinuous Galerkin atmospheric model. Proc. 19th IEEE Int. Parallel and Distributed Processing Symposium (IPDPS’05),Workshop 13, Vol. 14, Denver, CO, IEEE Computer Society, 257263.

    • 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., doi:10.1177/1094342011428142, in press.

    • Search Google Scholar
    • Export Citation
  • Deville, M. O., P. F. Fischer, and E. H. Mund, 2002: High-Order Methods for Incompressible Fluid Flow. Appl. Comput. Math. Monogr., No. 9, Cambridge University Press, 528 pp.

    • Search Google Scholar
    • Export Citation
  • Devine, K., E. Boman, R. Heaphy, B. Hendrickson, and C. Vaughan, 2002: Zoltan data management services for parallel dynamic applications. Comput. Sci. Eng., 4 (2), 9097.

    • Search Google Scholar
    • Export Citation
  • Edwards, H. C., 2002: SIERRA framework version 3: Core services theory and design. Tech. Rep. SAND2002-3616, Sandia National Laboratories, Albuquerque, NM, 97 pp.

    • Search Google Scholar
    • Export Citation
  • Emanuel, K., 2005: Increasing destructiveness of tropical cyclones over the past 30 years. Nature, 436, 686688.

  • Eskilsson, C., 2011: An hp-adaptive discontinuous Galerkin method for shallow water flows. Int. J. Numer. Methods Fluids, 67 (11), 16051623, doi:10.1002/fld.2434.

    • Search Google Scholar
    • Export Citation
  • Geuzaine, C., and J.-F. Remacle, 2009: Gmsh: A three-dimensional finite element mesh generator with built-in pre- and post-processing facilities. Int. J. Numer. Methods Eng., 11, 13091331.

    • Search Google Scholar
    • Export Citation
  • Giraldo, F. X., 2001: A spectral element shallow water model on spherical geodesic grids. Int. J. Numer. Methods Fluids, 35 (8), 869901.

    • Search Google Scholar
    • Export Citation
  • Giraldo, F. X., 2006: High-order triangle-based discontinuous Galerkin methods for hyperbolic equations on a rotating sphere. J. Comput. Phys., 214 (2), 447465.

    • Search Google Scholar
    • Export Citation
  • Giraldo, F. X., and M. Restelli, 2008: A study of spectral element and discontinuous Galerkin methods for the Navier–Stokes equations in nonhydrostatic mesoscale atmospheric modeling: Equation sets and test cases. J. Comput. Phys., 227, 38493877, doi:10.1016/j.jcp.2007.12.009.

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

    • Search Google Scholar
    • Export Citation
  • Gottlieb, S., and C.-W. Shu, 1998: Total variation diminishing Runge–Kutta schemes. Math. Comput., 67, 7385, doi:10.1090/S0025-5718-98-00913-2.

    • Search Google Scholar
    • Export Citation
  • Gottlieb, S., D. Ketcheson, and C.-W. Shu, 2011: Strong Stability Preserving Runge–Kutta and Multistep Time Discretizations. World Scientific Publishing Company, 188 pp.

    • Search Google Scholar
    • Export Citation
  • Grabowski, W. W., 2001: Coupling cloud processes with the large-scale dynamics using the cloud resolving convection parameterization (CRCP). J. Atmos. Sci., 58, 978997.

    • Search Google Scholar
    • Export Citation
  • Hesthaven, J. S., and T. Wartburton, 2008: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Texts in Applied Mathematics, Vol. 54, Springer, 515 pp.

    • Search Google Scholar
    • Export Citation
  • 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, 2005: Spectral/hp Element Methods for Computational Fluid Dynamics (Numerical Mathematics and Scientific Computation). Oxford University Press, 686 pp.

    • Search Google Scholar
    • Export Citation
  • Khairoutdinov, M. F., and D. A. Randall, 2001: A cloud resolving model as a cloud parameterization in the NCAR community climate system model: Preliminary results. Geophys. Res. Lett., 28, 36173620.

    • Search Google Scholar
    • Export Citation
  • Khouider, B., A. St-Cyr, A. J. Majda, and J. Tribbia, 2011: MJO and convectively coupled waves in a coarse resolution GCM with a simple multicloud parameterization. J. Atmos. Sci., 68, 240264.

    • Search Google Scholar
    • Export Citation
  • Kopriva, D. A., and G. Gassner, 2010: On the quadrature and weak form choices in collocation type discontinuous Galerkin spectral element methods. J. Sci. Comput., 44, 136155.

    • Search Google Scholar
    • Export Citation
  • Kopriva, D. A., S. L. Woodruff, and M. Y. Hussaini, 2002: Computation of electromagnetic scattering with a non-conforming discontinuous spectral element method. Int. J. Numer. Methods Eng., 53, 105122.

    • Search Google Scholar
    • Export Citation
  • Kubatko, E. J., J. J. Westerink, and C. Dawson, 2006: hp discontinuous Galerkin methods for advection dominated problems in shallow water flows. Comput. Methods Appl. Mech. Eng., 196, 437451, doi:10.1016/j.cma.2006.05.002.

    • Search Google Scholar
    • Export Citation
  • Kubatko, E. J., S. Bunya, C. Dawson, and J. J. Westerink, 2009: Dynamic p-adaptive Runge–Kutta discontinuous Galerkin methods for the shallow water equations. Comput. Methods Appl. Mech. Eng., 198 (21–26), 17661774.

    • Search Google Scholar
    • Export Citation
  • Läuter, M., D. Handorf, N. Rakowsky, J. Behrens, S. Frickenhaus, M. Best, K. Dethloff, and W. Hiller, 2007: A parallel adaptive barotropic model of the atmosphere. J. Comput. Phys., 223 (2), 609628.

    • Search Google Scholar
    • Export Citation
  • Löhner, R., and J. Baum, 1991: Numerical simulation of time-dependent 3-d flows using adaptive unstructured grids. Advances in the Free-Lagrange Method Including Contributions on Adaptive Gridding and the Smooth Particle Hydrodynamics Method, H. Trease, M. Fritts, and W. Crowley, Eds., Vol. 395, Springer, 47–56.

    • Search Google Scholar
    • Export Citation
  • Lörcher, F., G. Gassner, and C. Munz, 2008: An explicit discontinuous Galerkin scheme with local time-stepping for general unsteady diffusion equations. J. Comput. Phys., 227, 56495670.

    • Search Google Scholar
    • Export Citation
  • Mann, M. E., and K. A. Emanuel, 2006: Atlantic hurricane trends linked to climate change. Eos, Trans. Amer. Geophys. Union, 87 (24), 223244.

    • Search Google Scholar
    • Export Citation
  • McClean, J. L., and Coauthors, 2011: A prototype two-decade fully-coupled fine-resolution CCSM simulation. Ocean Modell., 39, 1030.

  • Mohseni, K., and T. Colonius, 2000: Numerical treatment of polar coordinate singularities. J. Comput. Phys., 157, 787795.

  • Nair, R. D., S. J. Thomas, and R. D. Loft, 2005a: A discontinuous Galerkin global shallow water model. Mon. Wea. Rev., 133, 876888.

  • Nair, R. D., S. J. Thomas, and R. D. Loft, 2005b: A discontinuous Galerkin transport scheme on the cubed sphere. Mon. Wea. Rev., 133, 814828.

    • Search Google Scholar
    • Export Citation
  • Nair, R. D., H.-W. Choi, and H. Tufo, 2009: Computational aspects of a scalable high-order discontinuous Galerkin atmospheric dynamical core. Comput. Fluids, 38 (2), 309319.

    • Search Google Scholar
    • Export Citation
  • Neale, R. B., and Coauthors, 2010: Description of the NCAR community atmosphere model (CAM 4.0). Tech. Rep., NCAR, 212 pp.

  • Okada, Y., 1985: Surface deformation due to shear and tensile faults in a half-space. Bull. Seismol. Soc. Amer., 75, 11351154.

  • Pedlosky, J., 1986: Geophysical Fluid Dynamics. 2nd ed. Springer, 710 pp.

  • Rančić, M., R. J. Purser, and F. Messinger, 1996: A global shallow-water model using an expanded spherical cube: Gnomonic versus conformal coordinates. Quart. J. Roy. Meteor. Soc., 122, 959982.

    • Search Google Scholar
    • Export Citation
  • Remacle, J.-F., X. Li, M. S. Shephard, and J. E. Flaherty, 2005: Anisotropic adaptive simulation of transient flows using discontinuous Galerkin methods. Int. J. Numer. Methods Eng., 62 (7), 899923.

    • Search Google Scholar
    • Export Citation
  • Remacle, J.-F., S. S. Frazao, L. Xiangrong, and M. S. Shephard, 2006: An adaptive discretization of shallow-water equations based on discontinuous Galerkin methods. Int. J. Numer. Methods Fluids, 52 (8), 903992.

    • Search Google Scholar
    • Export Citation
  • Remaki, L., and W. G. Habashi, 2006: 3-d mesh adaptation on multiple weak discontinuities and boundary layers. SIAM J. Sci. Comput., 28, 13791397, doi:10.1137/S1064827503429879.

    • 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
  • Satoh, M., T. Matsuno, H. Tomita, H. Miura, T. Nasuno, and S. Iga, 2008: Nonhydrostatic icosahedral atmospheric model (NICAM) for global cloud resolving simulations. J. Comput. Phys., 227, 34863514, doi:10.1016/j.jcp.2007.02.006.

    • Search Google Scholar
    • Export Citation
  • Slingo, J., and Coauthors, 2009: Developing the next-generation climate system models: Challenges and achievements. Philos. Trans. Roy. Soc. A: Math. Phys. Eng. Sci., 367, 815831.

    • Search Google Scholar
    • Export Citation
  • St-Cyr, A., and D. Neckels, 2009: A fully implicit Jacobian-free high-order discontinuous Galerkin mesoscale flow solver. Proc. Ninth Int. Conf. on Computational Science, Baton Roue, LA, LSU Center for Computation and Technology, 243–252.

    • Search Google Scholar
    • Export Citation
  • St-Cyr, A., C. Jablonowski, J. M. Dennis, H. M. Tufo, and S. J. Thomas, 2008: A comparison of two shallow-water models with nonconforming adaptive grids. Mon. Wea. Rev., 136, 18981922.

    • Search Google Scholar
    • Export Citation
  • Stewart, J. R., and H. C. Edwards, 2002: SIERRA framework version 3: h-adaptivity design and use. Tech. Rep. SAND2002-4016, Sandia National Laboratories, Albuquerque, NM, 28 pp.

    • Search Google Scholar
    • Export Citation
  • Watson, R. T., M. C. Zinyowera, R. H. Moss, and D. J. Dokken, 1997: The Regional Impacts of Climate Change: An Assessment of Vulnerability. IPCC, 27 pp. [Available online at http://www.ipcc.ch/pdf/special-reports/spm/region-en.pdf.]

    • Search Google Scholar
    • Export Citation
  • Wehner, M., 2008: Towards ultra-high resolution models of climate and weather. Int. J. High Perform. Comput. Appl., 22 (2), 149156.

  • Weller, H., H. G. Weller, and A. Fournier, 2009: Voronoi, Delaunay, and block-structured mesh refinement for solution of the shallow-water equations on the sphere. Mon. Wea. Rev., 137, 42084224.

    • Search Google Scholar
    • Export Citation
  • Wheeler, M., and G. N. Kiladis, 1999: Convectively-coupled equatorial waves: Analysis of clouds and temperature in the wavenumber-frequency domain. J. Atmos. Sci., 56, 374399.

    • Search Google Scholar
    • Export Citation
  • Wilcox, L. C., G. Stadler, C. Burstedde, and O. Ghattas, 2010: A high-order discontinuous Galerkin method for wave propagation through coupled elastic-acoustic media. J. Comput. Phys., 229, 93739396.

    • Search Google Scholar
    • Export Citation
  • Williamson, D. L., J. B. Drake, J. J. Hack, R. Jakob, and P. N. Swarztrauber, 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
  • Wilson, C., 2009: Modelling multiple-material flows on adaptive unstructured meshes. Ph.D. thesis, Imperial College London, 217 pp.

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 1273 833 54
PDF Downloads 428 69 10