Adaptive Modelling of Two-Dimensional Shallow Water Flows with Wetting and Drying

  • Andreas Dedner
  • Dietmar Kröner
  • Nina Shokina
Part of the Notes on Numerical Fluid Mechanics and Multidisciplinary Design book series (NNFM, volume 115)

Abstract

The current work is done in the framework of the BMBF (Bundesministerium für Bildung und Forschung - the Federal Ministry of Education and Research) project AdaptHydroMod - Adaptive Hydrological Modelling with Application in Water Industry [1], which is devoted to the development of generic adaptive approach to modelling of coupled hydrological processes: surface and groundwater flows. The surface water flow is modelled by the two-dimensional shallow water equations and the surface flow - by the Richards equation. The implementation is based within DUNE - the Distributed and Unified Numerics Environment [14]. The surface flow, on which we focus in the presented paper, is numerically solved using the Runge-Kutta discontinuous Galerkin method [10] with modifications to render the scheme well-balanced and for handling correctly possible wetting and drying processes. The newly developed limiter [12] is used for the stabilization of the method. The validation of the code is done using several test problems with known exact solutions. The problem with a mass source term, which is a first step to the coupled simulation of the surface and groundwater flows, is solved numerically.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
  2. 2.
    Balzano, A.: Evaluation of methods for numerical simulation of wetting and drying in shallow water flow models. Coast. Eng. 34(1-2), 83–107 (1998)CrossRefGoogle Scholar
  3. 3.
    Bastian, P., Blatt, M., Dedner, A., Engwer, C., Klöfkorn, R., Ohlberger, M., Sander, O.: A generic grid interface for parallel and adaptive scientific computing. I: Abstract framework. Computing 82(2-3), 103–119 (2008)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bastian, P., Blatt, M., Dedner, A., Engwer, C., Klöfkorn, R., Kornhuber, R., Ohlberger, M., Sander, O.: A generic grid interface for parallel and adaptive scientific computing. II: Implementation and tests in DUNE. Computing 82(2-3), 121–138 (2008)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bates, P.D., Hervouet, J.-M.: A new method for moving-boundary hydrodynamic problems in shallow water. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 455, 3107–3128 (1999)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bokhove, O.: Flooding and drying in discontinuous Galerkin finite-element discretizations of shallow-water equations. Part 1: One dimension. J. Sci. Comput. 22-23, 47–82 (2005)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Bradford, S.F., Sanders, B.F.: Finite-volume model for shallow-water flooding of arbitrary topography. J. Hydraul. Eng. 128, 289–298 (2002)CrossRefGoogle Scholar
  8. 8.
    Bunya, S., Kubatko, E.J., Westerink, J.J., Dawson, C.: A wetting and drying treatment for the Runge-Kutta discontinuous Galerkin solution to the shallow water equations. Comput. Methods Appl. Mech. Engrg. 198, 1548–1562 (2009)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Burri, A., Dedner, A., Diehl, D., Klöfkorn, R., Ohlberger, M.: A general object oriented framework for discretizing nonlinear evolution equations. In: Shokin, Y.I., Resch, M., Danaev, N., Orunkhanov, M., Shokina, N. (eds.) Advances in High Performance Computing and Computational Sciences. The 1st Kazakh-German Advanced Research Workshop, Notes on Numerical Fluid Mechanics and Multidisciplinary Design (NNFM), Almaty, Kazakhstan, September 25-October 1, vol. 93, pp. 69–87 (2006)Google Scholar
  10. 10.
    Cockburn, B., Shu, C.-W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws V: Multidimensional Systems. J. Comput. Phys. 141, 199–224 (1998)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Dal Maso, G., LeFloch, P.G., Murat, F.: Definition and weak stability of nonconservative products. J. Math. Pures Appl. 74, 483–548 (1995)MATHMathSciNetGoogle Scholar
  12. 12.
    Dedner, A., Klöfkorn, R.: A generic stabilization approach for higher order Discontinuous Galerkin methods for convection dominated problems. Preprint no. 8 (submitted to SIAM Sci. Comput.), Mathematisches Institut, Unversität Freiburg (2008), http://www.mathematik.uni-freiburg.de/IAM/homepages/robertk/postscript/dedner_kloefkorn_limiter.pdf
  13. 13.
    Dedner, A., Klöfkorn, R., Nolte, M., Ohlberger, M.: A generic interface for parallel and adaptive discretization schemes. Abstraction principles and the dunefem module. Computing (to appear)Google Scholar
  14. 14.
  15. 15.
  16. 16.
  17. 17.
    Ern, A., Piperno, S., Djadel, K.: A well-balanced Runge-Kutta discontinuous Galerkin method for the shallow-water equations with flooding and drying. Int. J. Numer. Meth. Fluids 58, 1–25 (2008)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Kärnä, T., de Brye, B., Gourgue, O., Lambrechts, J., Comblen, R., Legat, V., Deleersnijder, E.: A fully implicit wetting-drying method for DG-FEM shallow water models, with an application to the Scheldt Estuary. Comp. Meth. Appl. Mech. Eng. (2010) (accepted manuscript, available online July 11, 2010)Google Scholar
  19. 19.
    Kröner, D.: Numerical Schemes for Conservation Laws. Verlag Wiley & Teubner, Stuttgart (1997)MATHGoogle Scholar
  20. 20.
    Kubatko, E.J., Bunya, S., Dawson, C., Westerink, J.J.: Dynamic p-adaptive Runge-Kutta discontinuous Galerkin methods for the shallow water equations. Comput. Methods Appl. Mech. Engrg. 198, 1766–1774 (2009)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Leveque, R.J.: Finite volume methods for hyperbolic problems. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002)MATHCrossRefGoogle Scholar
  22. 22.
    Lynch, D.R., Gray, W.G.: Finite element simulation of flow deforming regions. J. Comp. Phys. 36, 135–153 (1980)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Nielsen, C., Apelt, C.: Parameters affecting the performance of wetting and drying in a two-dimensional finite element long wave hydrodynamic model. J. Hydraul. Eng. 129, 628–636 (2003)CrossRefGoogle Scholar
  24. 24.
    Reid, R.O., Bodine, B.R.: Numerical model for storm surges in Galveston Bay. J. Waterways Harbors Division, ASCE 94(WW1), 33–57 (1968)Google Scholar
  25. 25.
    Thacker, W.C.: Some exact solutions to the nonlinear shallow-water wave equations. J. Flud Mech. 107, 499–608 (1981)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Vreugdenhil, C.B.: Numerical methods for shallow-water flow. Kluwer academic Publishers, Dordrecht (1994)Google Scholar
  27. 27.
    Xing, Y., Shu, C.-W.: High order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms. J. Comput. Phys. 214, 567–598 (2006)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Andreas Dedner
    • 1
  • Dietmar Kröner
    • 1
  • Nina Shokina
    • 1
  1. 1.Section of Applied MathematicsUniversity of FreiburgFreiburg i. Br.Germany

Personalised recommendations