Journal of Scientific Computing

, Volume 22, Issue 1–3, pp 269–288 | Cite as

Discontinuous Galerkin Spectral/hp Element Modelling of Dispersive Shallow Water Systems

  • C. Eskilsson
  • S. J. Sherwin


Two-dimensional shallow water systems are frequently used in engineering practice to model environmental flows. The benefit of these systems are that, by integration over the water depth, a two-dimensional system is obtained which approximates the full three-dimensional problem. Nevertheless, for most applications the need to propagate waves over many wavelengths means that the numerical solution of these equations remains particularly challenging. The requirement for an accurate discretization in geometrically complex domains makes the use of spectral/hp elements attractive. However, to allow for the possibility of discontinuous solutions the most natural formulation of the system is within a discontinuous Galerkin (DG) framework. In this paper we consider the unstructured spectral/hp DG formulation of (i) weakly nonlinear dispersive Boussinesq equations and (ii) nonlinear shallow water equations (a subset of the Boussinesq equations). Discretization of the Boussinesq equations involves resolving third order mixed derivatives. To efficiently handle these high order terms a new scalar formulation based on the divergence of the momentum equations is presented. Numerical computations illustrate the exponential convergence with regard to expansion order and finally, we simulate solitary wave solutions.


Boussinesq equations shallow water equations spectral/hp discontinuous Galerkin method 


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  1. Abbott, M. B., McCowan, A. D., Warren, I. R. 1984Accuracy of short wave numerical modelsJ. Hydraul. Eng.11012871301Google Scholar
  2. Aizinger, V., Dawson, C. 2002A discontinuous Galerkin method for two-dimensional flow and transport in shallow waterAdv. Water Resour.256784CrossRefGoogle Scholar
  3. Ambrosi, D, Quartapelle, L 1998A Taylor–Galerkin method for simulating nonlinear dispersive water wavesJ. Comput. Phy146546569CrossRefGoogle Scholar
  4. Antunes Do Carmo, J.S, Seabra Santos, FJ 1993Surface waves propagation in shallow water: a finite element modelInt. J. Numer. Meth. Fluids16447459CrossRefGoogle Scholar
  5. Bassi, F, Rebay, S 1997A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier–Stokes equationsJ. Comput. Phy.131267279CrossRefGoogle Scholar
  6. Boyd, J. P. 1980Equatorial solitary waves Part 1: Rossby solitonsJ. Phys. Oceanogr1016991717CrossRefGoogle Scholar
  7. Boyd, J. P. 1985Equatorial solitary waves Part 3: Westward-traveling modonsJ. Phys. Oceanogr154654CrossRefGoogle Scholar
  8. Cockburn, B, Shu, CW 2001Runge–Kutta discontinuous Galerkin methods for convection-dominated problemsJ. Sci. Comput16173261CrossRefGoogle Scholar
  9. Dubiner, M. 1991Spectral methods on triangles and other domainsJ. Sci. Comput.6345390CrossRefGoogle Scholar
  10. Dupont, F. (2001). Comparison of Numerical Methods for Modelling Ocean Circulation in Basins with Irregular Coasts, Ph.D. Thesis, McGill University.Google Scholar
  11. Eskilsson, C, Sherwin, SJ 2002A discontinuous spectral element model for Boussinesq-type equationsJ. Sci. Comput17143152CrossRefMathSciNetGoogle Scholar
  12. Eskilsson, C., Sherwin, S. J. 2003An hp/spectral element model for efficient long-time integration of Boussinesq-type equationsCoast. Eng. J.45295320CrossRefGoogle Scholar
  13. Eskilsson, C., and Sherwin, S. J. A triangular spectral/hp discontinuous Galerkin method for modelling two-dimensional shallow water equations. Int. J. Numeric. Meth. Fluids, in press.Google Scholar
  14. Giraldo, F. X. (1998). A spectral element semi-Lagrangian method for the shallow water equations on unstructured grids. Proceeding of the Fourth World Congress on Computational Mechanics.Google Scholar
  15. Giraldo, F. X. 2001A spectral element shallow water model on spherical geodesic gridsInt. J. Numeric. Meth. Fluids35869901CrossRefGoogle Scholar
  16. Giraldo, F. X., Hesthaven, J. S., Warburton, T. 2002Nodal high-order discontinuous Galerkin methods for the spherical shallow water equationsJ. Comput. Phy.181499525CrossRefGoogle Scholar
  17. Gobbi, M. F., Kirby, J. T., Wei, G. 2000A fully nonlinear Boussinesq model for surface waves Part 2 Extension to \({\cal O}\)(kh)4J. Fluid Mech405181210CrossRefGoogle Scholar
  18. Iskandarani, M., Haidvogel, D. B., Boyd, J. P. 1995A staggered spectral element model with application to the oceanic shallow water equationsInt. J. Numeric. Meth. Fluids20393414CrossRefGoogle Scholar
  19. Karniadakis, GEm, Sherwin, SJ 1999Spectral/hp element methods for CFDOxford University PressNew YorkGoogle Scholar
  20. Katopodes, K. D., Wu, C. T. 1987Computation of finite amplitude dispersive wavesJ. Waterway Port Coast. Ocean Eng.113327346Google Scholar
  21. Koornwinder, T 1975Two-variable analogues of the classical orthogonal polynomialsAskey, RA eds. Theory and Application of Special FunctionsAcademic PressNew York435495Google Scholar
  22. Langtangen, H. P., Pedersen, G. 1998Computational models for weakley dispersive nonlinear water wavesComput. Meth. Appl. Mech. Eng.160337358CrossRefMathSciNetGoogle Scholar
  23. Li, H., Liu, R. X. 2001The discontinuous Galerkin finite element method for the 2d shallow water equationsMath. Comput. Simulation56171184CrossRefGoogle Scholar
  24. Ma, H. 1993A spectral element basin model for the shallow water equationsJ. Comput. Phy.109133149CrossRefGoogle Scholar
  25. Madsen, P.A., Murray, I. R., S\orensen, O.R. 1991A new form of the Boussinesq equations with improved linear dispersion characteristicsCoast. Eng.15371388CrossRefGoogle Scholar
  26. Madsen, P. A., Schäffer, H. A. 1998Higher-order Boussinesq-type equations for surface gravity waves: derivation and analysis.Philos. Trans. R. Soc.A35631233184CrossRefGoogle Scholar
  27. Madsen, P. A., Bingham, H. B., Liu, H. 2002A new Boussineq method for fully nonlinear waves from shallow to deep waterJ. Fluid Mech.462130CrossRefGoogle Scholar
  28. Nwogu, O. 1993Alternative form of Boussinesq equations for nearshore wave propagationJ. Waterway Port Coast. Ocean Eng.119618638Google Scholar
  29. Peregrine, D. H. 1967Long waves on a beachJ. Fluid Mech.27815827Google Scholar
  30. Proriol, J. 1957Sur une famile de polynomes á deux variables orthogonax dans un triangleC. R. Acad. Sci Paris24524592461Google Scholar
  31. Shi, F., Dalrymple, R. A., Kirby, J. T., Chen, Q., Kennedy, A. 2001A fully nonlinear Boussinesq model in generalized curvilinear coordintaesCoast. Eng.42337358CrossRefGoogle Scholar
  32. Schwanenberg, D., Köngeter, J. 2000A discontinuous Galerkin method for the shallow water equations with source termsCockburn, BKarniadakis, G. E.Shu, C.-W. eds. Discontinuous Galerkin MethodsSpringer Heidelberg289309Google Scholar
  33. Taylor, M., Tribbia, J., Iskandarani, M. 1997The spectral element method for the shallow water equations on a sphereJ. Comput. Phy.13092108CrossRefGoogle Scholar
  34. Toro, E.F. 2001Shock-Capturing Methods for Free-Surface Shallow FlowsWileyNew YorkGoogle Scholar
  35. Walkley, M. A. 1999A Numerical Method for Extended Boussinesq Shallow-Water Wave EquationsUniversity of LeedsUKPh.D. ThesisGoogle Scholar
  36. Wei, G., Kirby, J. T. 1995Time-dependent numerical code for extended Boussinesq equationsJ. Waterway Port Coast. Ocean Eng.121251261CrossRefGoogle Scholar
  37. Yan, J., Shu, C.-W. 2002Local discontinuous Galerkin methods for partial differential equations with higher order derivatives.J. Sci. Comput.172747CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Water Environment TransportChalmers University of TechnologyGöteborgSweden
  2. 2.Department of AeronauticsImperial College LondonLondonUK

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