Journal of Scientific Computing

, Volume 22, Issue 1–3, pp 245–267 | Cite as

A Discontinuous Galerkin Method for Three-Dimensional Shallow Water Equations

  • Clint  Dawson
  • Vadym  Aizinger


We describe the application of a local discontinuous Galerkin method to the numerical solution of the three-dimensional shallow water equations. The shallow water equations are used to model surface water flows where the hydrostatic pressure assumption is valid. The authors recently developed a DG\linebreak method for the depth-integrated shallow water equations. The method described here is an extension of these ideas to non-depth-integrated models. The method and its implementation are discussed, followed by numerical examples on several test problems.


Shallow water equations discontinuous Galerkin method free surface 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Aizinger, A. (2004). A Discontinuous Galerkin Method for Two- and Three-Dimensional Shallow-Water Equations, Ph.D. Dissertation, University of Texas at Austin. Google Scholar
  2. Aizinger, A., Dawson, C. 2002A discontinuous Galerkin method for two-dimensional flow and transport in shallow waterAdv. Water Resour.256784CrossRefGoogle Scholar
  3. Casulli, V., Walters, R. A. 2000An unstructured grid, three- dimensional model based on the shallow water equationsInt. J. Numer. Meth. Fluids32331348CrossRefGoogle Scholar
  4. Chippada, S., Dawson, C. N., Martinez, M., Wheeler, M. F. 1998A Godunov-type finite volume method for the system of shallow water equationsComput. Meth. Appl. Mech. Eng.151105129CrossRefMathSciNetGoogle Scholar
  5. Cockburn, B., Shu, C.-W. 1998The local discontinuous Galerkin finite element method for convection-diffusion systemsSIAM J. Numer. Anal.3524402463CrossRefGoogle Scholar
  6. Cockburn, B., Shu, C.-W. 1989TVB Runge-Kutta local projection discontinuous Galerkin finite element method for scalar conservation laws II: General frameworkMath. Comput.52411435Google Scholar
  7. Cockburn, B., Karniadakis, G., Shu , C.-W. 2000The development of discontinuous Galerkin methods.Cockburn, B.Karniadakis, G.Shu , C.-W. eds. Discontinuous Galerkin Methods Theory, Computation and Applications, Lecture Notes in Computational Science and Engineering, volume 11, Part I: OverviewSpringer-VerlagBerlin350Google Scholar
  8. Davies, A. M. 1986A three-dimensional model of the Northwest European continental shelf, with application to the M4 tideJ. Phys. Oceanogr.16797813CrossRefGoogle Scholar
  9. Ippen, A. T. 1951Mechanics of supercritical flowTrans. ASCE116268295Google Scholar
  10. Johnson, B. H., Kim, K. W., Heath, R. E., Hsieh, B. B. and Butler, H. L. (1991). Development and verification of a three-dimensional numerical hydrodynamic, salinity and temperature model of Chesapeake Bay, Technical report HL-91-7, U.S. Army Engineer Waterways Experiment Station, Vicksburg, MS.Google Scholar
  11. Le Veque, R. J. (1992). Numerical Methods for Conservation Laws, Basel, Birkhäuser. Google Scholar
  12. Luettich, R. A., Westerink, J. J., and Scheffner, N. W. (1991). ADCIRC: An Advanced Three-Dimensional Circulation Model for Shelves, Coasts and Estuaries, Report 1, U.S. Army Corps of Engineers, Washington, D.C. 20314-1000. Google Scholar
  13. Lynch, D. R., Werner, F. E. 1991Three-dimensional hydrodynamics on finite elements Part II: Non-linear time-stepping modelInt. J. Numer. Meth. Fluids12507533CrossRefGoogle Scholar
  14. Roe, P. L. 1981Approximate Riemann solvers, parameter vectors, and difference schemesJ. Comput. Phys.43357372CrossRefGoogle Scholar
  15. Toro, E. F. (2001). Shock-Capturing Methods for Free-Surface Shallow Flows, Wiley, Chichester.Google Scholar
  16. Vreugdenhil, C. B. 1994Methods for Shallow-Water FlowKluwer Academic PublishersDordrechtGoogle Scholar
  17. Weiyan, T. (1992). Shallow Water Hydrodynamics, Elsevier Oceanography Series, 55, Elsevier, Amsterdam.Google Scholar
  18. Zienkiewicz, O. C., Ortiz, P. 1995A split-characteristic based finite element model for the shallow water equationsInt. J. Numer. Meth. Fluids2010611080CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Center for Subsurface Modeling - C0200, Institute for Computational Engineering and Sciences (ICES)The University of Texas at AustinAustin USA

Personalised recommendations