Finite Volume Implementation of Non-Dispersive, Non-Hydrostatic Shallow Water Equations

  • Vincent Guinot
  • Didier Clamond
  • Denys Dutykh
Part of the Springer Hydrogeology book series (SPRINGERHYDRO)


A shock-capturing, finite volume implementation of recently proposed non-hydrostatic two-dimensional shallow water equations, is proposed. The discretization of the equations in conservation form implies the modification of the time derivative of the conserved variable, in the form of a mass/inertia matrix, and extra terms in the flux functions. The effect of this matrix is to slow down wave propagation in the presence of significant bottom slopes. The proposed model is first derived in conservation form using mass and momentum balance principles. Its finite volume implementation is then presented. The additional terms to the shallow water equations can be discretized very easily via a simple time-stepping procedure. Two application examples are presented. These examples seem to indicate that the proposed model does not exhibit strong differences with the classical hydrostatic shallow water model under steady-state conditions, but that its behavior is significantly different when transients are involved.


Shallow water model Non-hydrostatic pressure distribution Bottom acceleration 


  1. 1.
    Clamond, D., & Dutykh, D. (2012). Practical use of variational principles for modelling water waves. Physica D: Nonlinear Phenomena, 241, 25–36.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Radder, A. C. (1999). Hamiltonian dynamics of water waves. Advances in Coastal and Ocean Engineering, 4, 21–59.CrossRefGoogle Scholar
  3. 3.
    Salmon, R. (1988). Hamiltonian fluid mechanics. Annual Review of Fluid Mechanics, 20, 225–256.CrossRefGoogle Scholar
  4. 4.
    Dutykh, D., & Clamond, D. (2011). Shallow water equations for large bathymetry variations. Journal of Physics A: Mathematical and Theoretical, 44, 332001.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Stoker, J.J. (1957). Water waves. Interscience.Google Scholar
  6. 6.
    Finaud-Guyot, P., Delenne, C., Lhomme, J., Guinot, V., & Llovel, C. (2010). An approximate-state Riemann solver for the two-dimensional shallow water equations with porosity. International Journal for Numerical Methods in Fluids, 62, 1299–1331.MathSciNetzbMATHGoogle Scholar
  7. 7.
    Guinot, V., & Soares-Frazão, S. (2006). Flux and source term discretization for shallow water models with porosity on unstructured grids. International Journal for Numerical Methods in Fluids, 50, 309–345.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Guinot, V. (2012). Multiple porosity shallow water models for macroscopic modelling of urban floods. Advances in Water Resources, 37, 40–72.MathSciNetCrossRefGoogle Scholar
  9. 9.
    Guinot, V., Delenne, C., & Cappelaere, B. (2009). An approximate Riemann solver for sensitivity equations with discontinuous solutions. Advances in Water Resources, 32, 61–77.CrossRefGoogle Scholar
  10. 10.
    Van Leer, B. (1977). Toward the ultimate conservative difference scheme. IV. A new approach to numerical convection. Journal of Computational Physics, 23, 276–299.CrossRefzbMATHGoogle Scholar
  11. 11.
    Soares-Frazão, S., & Guinot, V. (2007). An eigenvector-based linear reconstruction scheme for the shallow water equations on two-dimensional unstructured meshes. International Journal for Numerical Methods in Fluids, 53, 23–55.MathSciNetCrossRefGoogle Scholar
  12. 12.
    Toro, E. F., Spruce, M., & Speares, W. (1994). Restoration of the contact surface in the HLL-Riemann solver. Shock Waves, 4, 25–34.CrossRefzbMATHGoogle Scholar
  13. 13.
    Guinot, V. (2010). Wave propagation in fluids. Models and numerical techniques. 2nd edition. Wiley-ISTE.Google Scholar

Copyright information

© Springer Science+Business Media Singapore 2014

Authors and Affiliations

  1. 1.Université Montpellier—HSM UMR 5569Place Eugène BataillonMontpellierFrance
  2. 2.Université de Nice-Sophial Antipolis—Laboratoire J. ADieudonné UMR 7351Nice CedexFrance
  3. 3.LAMA UMR 5127Campus ScientifiqueMarseilles CedexFrance

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