Skip to main content
SpringerLink
Log in

Hydrodynamic Instabilities in Well-Balanced Finite Volume Schemes for Frictional Shallow Water Equations. The Kinematic Wave Case

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

We report the developments of hydrodynamic instabilities in several well-balanced finite volume schemes that are observed during the computation of the temporal evolution of an out-balance flow which is essentially a kinematic wave. The numerical simulations are based on the one-dimensional shallow-water equations for a uniformly sloping bed with hydraulic resistance. Subsequently, we highlight the need of low dissipative high-order well-balanced filter schemes for non-equilibrium flows with variable cut-off wavenumber to compute the out-balance flow under consideration, i.e. the kinematic wave.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Lighthill, M., Whitham, G.: On kinematic waves. Part I. Flood movement in long rivers. Part II. Theory of traffic flow on long crowded roads. Proc. R. Soc. A (London) 229, 281–345 (1955)

    Article  MATH  MathSciNet  Google Scholar 

  2. Brock, R.R.: Development of roll-wave trains in open channels. J. Hydraul. Div. 95, 1401–1427 (1969)

    Google Scholar 

  3. Bohorquez, P.: Competition between kinematic and dynamic waves in floods on steep slopes. J. Fluid Mech. 645, 375–409 (2010)

    Article  MATH  Google Scholar 

  4. Castro, M., Gallardo, J.M., Parés, C.: High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow-water systems. Math. Comput. 75, 1103–1134 (2006)

    Article  MATH  Google Scholar 

  5. Noelle, S., Xing, Y., Shu, C.-W.: High-order well-balanced finite volume WENO schemes for shallow water equation with moving water. J. Comput. Phys. 226, 29–58 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  6. Dumbser, M., Enaux, C., Toro, E.F.: Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws. J. Comput. Phys. 227, 3971–4001 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  7. Wang, W., Yee, H.C., Sjögreen, B., Magin, T., Shu, C.-W.: Construction of low dissipative high-order well-balanced filter schemes for non-equilibrium flows. J. Comput. Phys. doi:10.1016/j.jcp.2010.04.033

  8. Burguete, J., García-Navarro, P.: Efficient construction of high-resolution TVD conservative schemes for equations with source terms: application to shallow water flows. Int. J. Numer. Methods Fluids 37(2), 209–248 (2001)

    Article  MATH  Google Scholar 

  9. Črnjarić Žic, N., Vuković, S., Sopta, L.: Balanced finite volume WENO and central WENO schemes for the shallow water and the open-channel flow equations. J. Comput. Phys. 200, 512–548 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  10. Needham, D.J., Merkin, J.H.: On roll waves down an open inclined channel. Proc. R. Soc. A (London) 394, 259–278 (1984)

    Article  MATH  Google Scholar 

  11. Gottlieb, S., Ketcheson, D.I., Shu, C.-W.: High order strong stability preserving time discretizations. J. Sci. Comput. 38(3), 251–289 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  12. Ketcheson, D.I., LeVeque, R.J.: WENOCLAW: a higher order wave propagation method. In: Benzoni-Gavage, S., Serre, D. (eds.) Hyperbolic Problems: Theory, Numerics, Applications, pp. 609–616. Springer, Berlin (2008)

    Chapter  Google Scholar 

  13. Kim, J.W.: High-order compact filters with variable cut-off wavenumber and stable boundary treatment. Comput. Fluids 39, 1168–1182 (2010)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Área de Mecánica de Fluidos, Departamento de Ingeniería Mecánica y Minera, Universidad de Jaén, Campus de las Lagunillas, 23071, Jaén, Spain

    Patricio Bohorquez

  2. School of Architecture, Civil and Environmental Engineering, École Polytechnique Fédérale de Lausanne, 1015, Lausanne, Switzerland

    Martin Rentschler

Authors
  1. Patricio Bohorquez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Martin Rentschler
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Patricio Bohorquez.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bohorquez, P., Rentschler, M. Hydrodynamic Instabilities in Well-Balanced Finite Volume Schemes for Frictional Shallow Water Equations. The Kinematic Wave Case. J Sci Comput 48, 3–15 (2011). https://doi.org/10.1007/s10915-010-9444-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-010-9444-4

Keywords

Search

Navigation