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A Performance Comparison of Continuous and Discontinuous Finite Element Shallow Water Models

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Abstract

We present a comparative study of two finite element shallow water equation (SWE) models: a generalized wave continuity equation based continuous Galerkin (CG) model—an approach used by several existing SWE models—and a recently developed discontinuous Galerkin (DG) model. While DG methods are known to possess a number of favorable properties, such as local mass conservation, one commonly cited disadvantage is the larger number of degrees of freedom associated with the methods, which naturally translates into a greater computational cost compared to CG methods. However, in a series of numerical tests, we demonstrate that the DG SWE model is generally more efficient than the CG model (i) in terms of achieving a specified error level for a given computational cost and (ii) on large-scale parallel machines because of the inherently local structure of the method. Both models are verified on a series of analytic test cases and validated on a field-scale application.

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References

  1. Aizinger, V., Dawson, C.: A discontinuous Galerkin method for two-dimensional flow and transport in shallow water. Adv. Water Resour. 25, 67–84 (2002)

    Article  Google Scholar 

  2. Atkinson, J.H., Westerink, J.J., Luettich, R.A.: Two-dimensional dispersion analyses of finite element approximations to the shallow water equations. Int. J. Numer. Methods Fluids 45, 715–749 (2004)

    Article  MATH  Google Scholar 

  3. Bey, K.S., Patra, A., Oden, J.T.: hp-version discontinuous Galerkin methods for hyperbolic conservation laws—a parallel adaptive strategy. Int. J. Numer. Methods Eng. 38, 3889–3908 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  4. Biswas, R., Devine, K.D., Flaherty, J.E.: Parallel, adaptive finite element methods for conservation laws. Appl. Numer. Math. 14, 255–283 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  5. Blain, C.A., Massey, T.C.: Application of a coupled discontinuous–continuous Galerkin finite element shallow water model to coastal ocean dynamics. Ocean Model. 10, 283–315 (2005)

    Article  Google Scholar 

  6. Buffa, A., Hughes, T.J.R., Sangalli, G.: Analysis of a multiscale discontinuous galerkin method for convection-diffusion problems. SIAM J. Numer. Anal. 44, 1420–1440 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  7. Dawson, C., Sun, S., Wheeler, M.F.: Compatible algorithms for coupled flow and transport. Comput. Methods Appl. Mech. Eng. 193, 2565–2580 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  8. Dawson, C., Westerink, J.J., Feyen, J.C., Pothina, D.: Continuous, discontinuous and coupled discontinuous–continuous Galerkin finite element methods for the shallow water equations. Int. J. Numer. Methods Fluids 52, 63–88 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  9. Eskilsson, C., Sherwin, S.J.: A triangular spectral/hp discontinuous Galerkin method for modelling 2D shallow water equations. Int. J. Numer. Methods Fluids 45, 605–623 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  10. Foreman, M.G.G.: A comparison of tidal models for the southwest coast of Vancouver Island. In: Celia, M. (ed.) Computational Methods in Water Resources: Proceedings of the VII International Conference. Elsevier, Amsterdam (1988)

    Google Scholar 

  11. Gray, W.G.: A finite element study of tidal flow data for the North Sea and English Channel. Adv. Water Resour. 12, 143–154 (1989)

    Article  Google Scholar 

  12. Gray, W.G., Lynch, D.R.: Time–stepping schemes for finite element tidal model computations. Adv. Water Resour. 1, 83–95 (1977)

    Article  Google Scholar 

  13. Hughes, T.J.R., Scovazzi, G., Bochev, P.B., Buffa, A.: A multiscale discontinuous Galerkin method with the computational structure of a continuous Galerkin method. Comput. Methods Appl. Mech. Eng. 195, 2761–2787 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  14. Kinnmark, I.P.E.: The Shallow Water Wave Equations: Formulations, Analysis and Application. Lecture Notes in Engineering, vol. 15. Springer, Berlin (1986)

    Google Scholar 

  15. Kolar, R.L., Westerink, J.J., Cantekin, M.E., Blain, C.A.: Aspects of nonlinear simulations using shallow-water models based on the wave continuity equation. Comput. Fluids 23, 523–528 (1994)

    Article  MATH  Google Scholar 

  16. Kolar, R.L., Gray, W.G., Westerink, J.J.: Boundary conditions in shallow water models–an alternative implementation for finite element codes. Int. J. Numer. Methods Fluids 22, 603–618 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  17. Kubatko, E.J., Westerink, J.J., Dawson, C.: hp discontinuous Galerkin methods for advection dominated problems in shallow water flow. Comput. Methods Appl. Mech. Eng. 196, 437–451 (2006)

    Article  MATH  Google Scholar 

  18. Karypis, G., Kumar, V.: metis: A Software Package for Partitioning Unstructured Graphs, Partitioning Meshes, and Computing Fill-Reducing Orderings of Sparse Matrices. University of Minnesota Department of Computer Science/Army HPC Research Center, Minneapolis (1998)

    Google Scholar 

  19. Le Provost, C., Vincent, P.: Finite element for modeling ocean tides. In: Parker, B. (ed.) Tidal Hydrodynamics, pp. 41–60. Wiley, New York (1991)

    Google Scholar 

  20. Le Roux, D.Y., Lin, C.A., Staniforth, A.: A semi-implicit semi-Lagrangian finite-element shallow-water ocean model. Mon. Weather Rev. 128, 1384–1401 (2000)

    Article  Google Scholar 

  21. Li, H., Liu, R.: The discontinuous Galerkin finite element method for the 2D shallow water equations. Math. Comput. Simul. 56, 223–233 (2001)

    Article  MATH  Google Scholar 

  22. Luettich, R.A., Westerink, J.J., Scheffner, N.W.: ADCIRC: An advanced three-dimensional circulation model for shelves, coasts and estuaries, Report 1: Theory and methodology of ADCIRC-2DDI and ADCIRC-3DL. In: Dredging Research Program Technical Report DRP-92-6, US Army Engineers Waterways Experiment Station, Vicksburg, MS (1992)

  23. Lynch, D.R., Gray, W.G.: Analytic solutions for computer flow model testing. J. Hydraul. Div. 104, 1409–1428 (1978)

    Google Scholar 

  24. Lynch, D.R., Gray, W.G.: A wave equation model for finite element tidal computations. Comput. Fluids 7, 207–228 (1979)

    Article  MATH  Google Scholar 

  25. Lynch, D.R., Werner, F.E., Molines, J.M., Fornerino, M.: Tidal dynamics in a coupled ocean/lake system. Estuar. Coast. Shelf Sci. 31, 319–343 (1990)

    Article  Google Scholar 

  26. Mukai, A.Y., Westerink, J.J., Luettich, R.A., Mark, D.: Eastcoast 2001, a tidal constituent database for Western North Atlantic, Gulf of Mexico, and Caribbean Sea. TR ERDC 01-x, US Army Engineer, Engineer Research and Development Center, Vicksburg, MS (2001)

  27. National Oceanic and Atmospheric Administration: Tides and currents database. Online, available: http://tidesandcurrents.noaa.gov/

  28. Schwanenberg, D., Kiem, R., Kongeter, J.: Discontinuous Galerkin method for the shallow water equations. In: Cockburn, B., Karniadakis, G.E., Shu, C.-W. (eds.) Discontinuous Galerkin Methods, pp. 289–309. Springer, Heidelberg (2000)

    Google Scholar 

  29. Walters, R.A.: A model for tides and currents in the English Channel and southern North Sea. Adv. Water Resour. 10, 138–148 (1987)

    Article  Google Scholar 

  30. Werner, F.E., Lynch, D.R.: Harmonic structure of English Channel/Southern Bight tides from a wave equation simulation. Adv. Water Resour. 12, 121–142 (1989)

    Article  Google Scholar 

  31. Westerink, J.J., Luettich, R.A., Baptista, A.M., Scheffner, N.W., Farrar, P.: Tide and storm surge predictions in the Gulf of Mexico using a wave-continuity equation finite element model. In: Spaulding, M.L. (ed.) Estuarine and Coastal Modeling: Proceedings of the 2nd International Conference. American Society of Civil Engineers, New York (1992)

    Google Scholar 

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Authors and Affiliations

  1. Department of Civil and Environmental Engineering and Geodetic Science, The Ohio State University, Columbus, OH, 43210, USA

    Ethan J. Kubatko

  2. Department of Systems Innovation, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo, 113-8656, Japan

    Shintaro Bunya

  3. Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX, 78712, USA

    Clint Dawson & Chris Mirabito

  4. Department of Civil Engineering and Geological Sciences, University of Notre Dame, Notre Dame, IN, 46556, USA

    Joannes J. Westerink

Authors
  1. Ethan J. Kubatko
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  2. Shintaro Bunya
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  3. Clint Dawson
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  4. Joannes J. Westerink
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  5. Chris Mirabito
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Correspondence to Ethan J. Kubatko.

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Kubatko, E.J., Bunya, S., Dawson, C. et al. A Performance Comparison of Continuous and Discontinuous Finite Element Shallow Water Models. J Sci Comput 40, 315–339 (2009). https://doi.org/10.1007/s10915-009-9268-2

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  • DOI: https://doi.org/10.1007/s10915-009-9268-2

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