A primitive-variable Riemann method for solution of the shallow water equations with wetting and drying

  • Authors:

  • P. Sivakumar;D. G. Hyams;L. K. Taylor;W. R. Briley

  • Affiliations:

  • SimCenter: National Center for Computational Engineering, University of Tennessee at Chattanooga, Chattanooga, TN 37403, USA;SimCenter: National Center for Computational Engineering, University of Tennessee at Chattanooga, Chattanooga, TN 37403, USA;SimCenter: National Center for Computational Engineering, University of Tennessee at Chattanooga, Chattanooga, TN 37403, USA;SimCenter: National Center for Computational Engineering, University of Tennessee at Chattanooga, Chattanooga, TN 37403, USA

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2009

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Abstract

A Riemann flux that uses primitive variables rather than conserved variables is developed for the shallow water equations with nonuniform bathymetry. This primitive-variable flux is both conservative and well behaved at zero depth. The unstructured finite-volume discretization used is suitable for highly nonuniform grids that provide resolution of complex geometries and localized flow structures. A source-term discretization is derived for nonuniform bottom that balances the discrete flux integral both for still water and in dry regions. This primitive-variable formulation is uniformly valid in wet and dry regions with embedded wetting and drying fronts. A fully nonlinear implicit scheme and both nonlinear and time-linearized explicit schemes are developed for the time integration. The implicit scheme is solved by a parallel Newton-iterative algorithm with numerically computed flux Jacobians. A concise treatment of characteristic-variable boundary conditions with source terms is also given. Computed results obtained for the one-dimensional dam break on wet and dry beds and for normal-mode oscillations in a circular parabolic basin are in very close agreement with the analytical solutions. Other results for a forced breaking wave with friction interacting with a sloped bottom demonstrate a complex wave motion with wetting, drying and multiple interacting wave fronts. Finally, a highly nonuniform, coastline-conforming unstructured grid is used to demonstrate an unsteady simulation that models an artificial coastal flooding due to a forced wave entering the Gulf of Mexico.