Well-Balanced Scheme between Flux and Source Terms for Computation of Shallow-Water Equations over Irregular Bathymetry
Publication: Journal of Engineering Mechanics
Volume 134, Issue 4
Abstract
Although many numerical techniques such as approximate Riemann solvers can be used to analyze subcritical and supercritical flows modeled by hyperbolic-type shallow-water equations, there are some difficulties in practical applications due to the numerical unbalance between source and flux terms. In this study, a revised surface gradient method is proposed that balances source and flux terms. The new numerical model employs the MUSCL–Hancock scheme and the HLLC approximate Riemann solver. Several verifications are conducted, including analyses of transcritical steady-state flows, unsteady dam break flows on a wet and dry bed, and flows over an irregular bathymetry. The model consistently returns accurate and reasonable results comparable to those obtained through analytical methods and laboratory experiments. The revised surface gradient method may be a simple but robust numerical scheme appropriate for solving hyperbolic-type shallow-water equations over an irregular bathymetry.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
This work was supported by the Research Fund of Hanyang University, Grant No. UNSPECIFIEDHY-2007-I.
References
Bermudez, A., and Vazquez, M. E. (1994). “Upwind methods for hyperbolic conservation laws with source terms.” Comput. Fluids, 23(8), 1049–1071.
Birman, A., and Falcovitz, J. (2006). “Application of the GRP scheme to open channel flow equations.” J. Comput. Phys., in press.
Bradford, S. F., and Sanders, B. F. (2002). “Finite-volume model for shallow-water flooding of arbitrary topography.” J. Hydraul. Eng., 128(3), 289–298.
Chow, V. T. (1959). Open-channel hydraulics, McGraw-Hill, Kogakusha.
Garcia, P., and Vazquez, M. E. (2000). “On numerical treatment of the source terms in the shallow water equations.” Comput. Fluids, 29(8), 951–979.
Hu, K., Mingham, C. G., and Causon, D. M. (2000). “Numerical simulation of wave overtopping of coastal structures using the nonlinear shallow water equations.” Coastal Eng., 41(4), 433–465.
Hubbard, M. E., and Garcia, P. (2000). “Flux difference splitting and the balancing of source terms and flux gradients.” J. Comput. Phys., 165(1), 89–125.
Kim, D. H., Cho, Y. S., and Kim, W. G. (2004). “WAF-type scheme for shallow-water equations with fractional step method.” J. Eng. Mech., 130(2), 152–160.
Morris, M. (2000). “CADAM: Concerted action on dambreak modeling—Final report.” Rep. No. SR 571, HR Wallingford.
Morris, M. (2001). “CADAM-EU converted action on dam break modelling.” ⟨http://www.hrwallingford.co.uk/projects/CADAM/⟩.
Nelida, C. Z., Vukovic, S., and Sopta, L. (2004). “Balanced finite volume WENO and central WENO schemes for the shallow water and the open-channel flow equations.” J. Comput. Phys., 200(2), 512–548.
Synolakis, C. E. (1987). “The runup of solitary waves.” J. Electrocardiol., 185, 523–545.
Toro, E. F. (2001). Shock-capturing methods for free-surface shallow flows, Wiley, New York.
Vukovic, S., and Sopta, L. (2002). “ENO and WENO schemes with the exact conservation property for one-dimensional shallow water equations.” J. Comput. Phys., 179(2), 593–621.
Watson, G., Peregrine, D. H., and Toro, E. F. (1992). “Numerical solution of the shallow water equations on a beach using the weighted average flux method.” Computational fluid dynamics, C. Hirsch, ed., Vol. 1, Elsevier, New York.
Wen, X. (2006). “A steady state capturing and preserving method for computing hyperbolic systems with geometrical source terms having concentrations.” J. Comput. Phys., 219, 322–390.
Zanuttigh, B., and Lamberti, A. (2002). “Roll waves simulation using shallow water equations and weighted average flux method.” J. Hydraul. Res., 40(5), 610–622.
Zhou, J. G., Causon, D. M., Ingram, D. M., and Mingham, C. G. (2002). “Numerical solutions of the shallow water equations with discontinuous bed topography.” Int. J. Numer. Methods Fluids, 38, 769–788.
Zhou, J. G., Causon, D. M., Mingham, C. G., and Ingram, D. M. (2001). “The surface gradient method for the treatment of source terms in the shallow-water equations.” J. Comput. Phys., 168(1), 1–25.
Zhou, J. G., Causon, D. M., Mingham, C. G., and Ingram, D. M. (2004). “Numerical prediction of dam-break flows in general geometries with complex bed topography” J. Hydraul. Eng., 130(4), 332–340.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 134 • Issue 4 • April 2008
Pages: 277 - 290
Copyright
© 2008 ASCE.
History
Received: Jul 27, 2005
Accepted: Nov 6, 2007
Published online: Apr 1, 2008
Published in print: Apr 2008
Notes
Note. Associate Editor: Arif Masud
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.