A numerical solution for moving boundary shallow water flow above parabolic bottom topography

Joe John Sampson


A numerical method has been applied to the nonlinear shallow water wave equations for unforced linear frictional flow above parabolic bottom topography and is found to be accurate. This solution involves moving shorelines. The motion decays over time. The numerical scheme used is adapted from the Selective Lumped Mass numerical scheme. The wetting and drying algorithm used in the numerical scheme is different to that in the Selective Lumped Mass scheme. The numerical scheme is finite element in space, using fixed triangular elements, finite difference in time and is explicit. The numerical solution compares well with an analytical solution.

  • Balzano, A., Evaluation of methods for numerical simulation of wetting and drying in shallow water flow models, Coastal Engineering, 34, 1998, 83--107.
  • Goraya, S., A study of finite element tidal models, PhD. thesis, Swinburne University of Technology, Melbourne, Australia, 2001.
  • Holdahl, R., Holden, H., and Lie,K-A., Unconditionally Stable Splitting Methods For the Shallow Water Equations, BIT, 39, 1998, 451--472.
  • Kawahara, M., Hirano, H., and Tsubota, K., Selective lumping finite element method for shallow water flow, International Journal for Numerical Methods in Fluids, 2, 1982, 89--112.
  • Lewis, C. H. III and Adams, W. M., Development of a tsunami-flooding model having versatile formation of moving boundary conditions,The Tsunami Society Monograph Series, 1983, No. 1, 128 pp.
  • Parker, B. B., Frictional Effects on the Tidal Dynamics of a Shallow Estuary, PhD thesis, The John Hopkins University, 1984.
  • Peterson P., Hauser J., Thacker W. C., Eppel D., An Error-Minimizing Algorithm for the Non-Linear Shallow-Water Wave Equations with Moving Boundaries. In Numerical Methods for Non-Linear Problems, editors C. Taylor, E. Hinton, D. R. J. Owen and E. Onate, 2, Pineridge Press, 1984, 826--836, http://www.cle.de/hpcc/publications/
  • Sampson, J., Easton, A., and Singh, M., Moving boundary shallow water flow in parabolic bottom topography, ANZIAM Journal, 47 (EMAC2005), C373-387, 2006, http://anziamj.austms.org.au/ojs/index.php/ANZIAMJ/article/view/1050.
  • Sampson, Joe, Easton, Alan and Singh, Manmohan, A New Moving Boundary Shallow Water Wave Numerical Model, Australian and New Zealand Industrial and Applied Mathematics Journal, 48 (CTAC2006), C605--C617, 2007, http://anziamj.austms.org.au/ojs/index.php/ANZIAMJ/article/view/78.
  • Sampson, Joe, Easton, Alan and Singh, Manmohan, Moving boundary shallow water flow in a region with quadratic bathymetry,Australian and New Zealand Industrial and Applied Mathematics Journal, 49 (EMAC2007), C666--C680, 2008. http://anziamj.austms.org.au/ojs/index.php/ANZIAMJ/article/view/306.
  • Thacker, W. C., Some exact solutions to the nonlinear shallow-water wave equations,J. Fluid. Mech., 107, 1981, 499--508.
  • Vreugdenhil, C. B., Numerical Methods for Shallow-Water Flow, Kluwer Academic Publishers, 1998.
  • Yoon S., B., and Cho J. H., Numerical simulation of Coastal Inundation over Discontinuous Topography, Water Engineering Research, 2(2), 2001, 75--87.

Full Text:


DOI: http://dx.doi.org/10.21914/anziamj.v50i0.1351

Remember, for most actions you have to record/upload into this online system
and then inform the editor/author via clicking on an email icon or Completion button.
ANZIAM Journal, ISSN 1446-8735, copyright Australian Mathematical Society.