Well-Balanced Bottom Discontinuities Treatment for High-Order Shallow Water Equations WENO Scheme
Publication: Journal of Engineering Mechanics
Volume 135, Issue 7
Abstract
A finite volume well-balanced weighted essentially nonoscillatory (WENO) scheme, fourth-order accurate in space and time, for the numerical integration of shallow water equations with the bottom slope source term, is presented. The main novelty introduced in this work is a new method for managing bed discontinuities. This method is based on a suitable reconstruction of the conservative variables at the cell interfaces, coupled with a correction of the numerical flux based on the local conservation of total energy. Further changes regard the treatment of the source term, based on a high-order extension of the divergence form for bed slope source term method, and the application of an analytical inversion of the specific energy-depth relationship. Two ad hoc test cases, consisting of a steady flow over a step and a surge crossing a step, show the effectiveness of the method of treating bottom discontinuities. Several standard one-dimensional test cases are also used to verify the high-order accuracy, the C-property, and the good resolution properties of the resulting scheme, in the cases of both continuous and discontinuous bottoms. Finally, a comparison between the fourth-order scheme proposed here and a well-established second-order scheme emphasizes the improvement achieved using the higher-order approach.
Get full access to this article
View all available purchase options and get full access to this article.
References
Alcrudo, F., and Benkhaldoun, F. (2001). “Exact solutions to the Riemann problem of the shallow water equations with a bottom step.” Comput. Fluids, 30(6), 643–671.
Audusse, E., cois Bouchut, F., Bristeau, M.-O., Klein, R., and t Perthame, B. (2004). “A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows.” SIAM J. Sci. Comput. (USA), 25(6), 2050–2065.
Bermudez, A., and Vázquez-Cendón, M. E. (1994). “Upwind methods for hyperbolic conservation laws with source terms.” Comput. Fluids, 23(8), 1049–1071.
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.
Bradford, S. F., and Sanders, B. F. (2005). “Performance of high-resolution, nonlevel bed, shallow-water models.” J. Eng. Mech., 131(10), 1073–1081.
Butcher, J. C. (1987). The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods, Wiley, New York.
Caleffi, V., and Valiani, A. (2006). “Fourth order balanced WENO schemes for shallow water equations.” River Flow 2006—Int. Conf. on Fluvial Hydraulics, Vol. 1, R. Ferreira, E. Alves, J. Leal, and A. Cardoso, eds., Laboratório Nacional de Engenharia Civil—Lisbon, Lisbon, Portugal, Taylor & Francis, London, 281–290.
Caleffi, V., Valiani, A., and Bernini, A. (2006). “Fourth-order balanced source term treatment in central WENO schemes for shallow water equations.” J. Comput. Phys., 218(1), 228–245.
Caleffi, V., Valiani, A., and Bernini, A. (2007). “High-order balanced CWENO scheme for movable bed shallow water equations.” Adv. Water Resour., 30, 730–741.
Caleffi, V., Valiani, A., and Zanni, A. (2003). “Finite volume method for simulating extreme flood events in natural channels.” J. Hydraul. Res., 41(2), 167–177.
Chow, V. T. (1959). Open-channel hydraulics, McGraw-Hill Inc., New York.
Črnjarić-Žic, N., Vuković, 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.
Galland, J., Goutal, N., and Hervouet, J. (1991). “TELEMAC: A new numerical model for solving shallow water equations.” Adv. Water Resour., 14(3), 138–148.
Godunov, S. K. (1959). “A difference scheme for numerical computation of discontinuous solution of hydrodynamic equations.” Math. USSR. Sb., 43, 271–306.
Goutal, N., and Maurel, F. (1997). Proc., 2nd Workshop on Dam-Break Wave Simulation, Electricit de France, Direction des Etudes et Recherches, Rep. HE43/97/016/B.
Harten, A., Lax, P. D., and Leer, B. V. (1983). “On upstream differencing and Godunov-type schemes for hyperbolic conservation laws.” SIAM Rev., 25(1), 35–61.
Hu, K., Mingham, C., and Causon, D. (2000). “Numerical simulation of wave overtopping of coastal structures using the non-linear shallow water equations.” Coastal Eng., 41(4), 433–465.
Hubbard, M. E., and Garcia-Navarro, P. (2000). “Flux difference splitting and the balancing of source terms and flux gradients.” J. Comput. Phys., 165(1), 89–125.
Jiang, G.-S., Levy, D., Lin, C.-T., Osher, S., and Tadmor, E. (1998). “High-resolution nonoscillatory central schemes with nonstaggered grids for hyperbolic conservation laws.” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal., 35(6), 2147–2168.
Jiang, G.-S., and Shu, C.-W. (1996). “Efficient implementation of Weighted ENO schemes.” J. Comput. Phys., 126(1), 202–228.
Kurganov, A., and Levy, D. (2002). “Central-upwind schemes for the Saint-Venant system.” Math. Modell. Numer. Anal., 36, 397–425.
Kurganov, A., Noelle, S., and Petrova, G. (2001). “Semi-discrete central-upwind schemes for hyperbolic conservation laws and Hamilton-Jacobi equations.” SIAM J. Sci. Comput. (USA), 23, 707–740.
Lax, P. D. (1954). “Weak solutions of nonlinear hyperbolic equations and their numerical computations.” Commun. Pure Appl. Math., 44, 21–41.
LeVeque, R. J. (1998). “Balancing source terms and flux gradients in high-resolution Godunov methods: The quasi-steady wave-propagation algorithm.” J. Comput. Phys., 146(1), 346–365.
LeVeque, R. J. (2002). Finite volume methods for hyperbolic problems, Cambridge University Press, Cambridge.
Levy, D., Puppo, G., and Russo, G. (1999). “Central WENO schemes for hyperbolic systems of conservation laws.” Math. Modell. Numer. Anal., 33(3), 547–571.
Liu, X.-D., Osher, S., and Chan, T. (1994). “Weighted essentially non-oscillatory schemes.” J. Comput. Phys., 115(1), 200–212.
Nessyahu, H., and Tadmor, E. (1990). “Non-oscillatory central differencing for hyperbolic conservation laws.” J. Comput. Phys., 87(2), 408–463.
Noelle, S., Pankratz, N., Puppo, G., and Natvig, J. (2006). “Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows.” J. Comput. Phys., 213(2), 474–499.
Noelle, S., Xing, Y., and Shu, C.-W. (2007). “High order well balanced finite volume WENO schemes for shallow water equations with moving water.” J. Comput. Phys., 226, 29–58.
Nujic, M. (1995). “Efficient implementation of non-oscillatory schemes for the computation of free-surface flows.” J. Hydraul. Res., 33(1), 101–111.
Pareschi, L., Puppo, G., and Russo, G. (2005). “Central Runge-Kutta schemes for conservation laws.” SIAM J. Sci. Comput. (USA), 26(3), 979–999.
Qiu, J., and Shu, C.-W. (2002). “On the construction, comparison, and local characteristic decomposition for high-order central WENO schemes.” J. Comput. Phys., 183(1), 187–209.
Shu, C.-W. (1997). “Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws.” ICASE Technical Rep. 97-65, NASA, Langley Research Center, Hampton, Va.
Shu, C.-W. (2002). “A survey of strong stability preserving high-order time discretizations.” Collected lectures on the preservation of stability under discretization, D. Estep and S. Taverner, eds., SIAM, Philadelphia, 51–65.
Soulis, J. V. (1992). “Computation of two-dimensional dambreak flood flows.” Int. J. Numer. Methods Fluids, 14, 631–664.
Spiteri, R. J., and Ruuth, S. J. (2002). “A new class of optimal high-order strong-stability-preserving time discretization methods.” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal., 40, 469–491.
Valiani, A., and Begnudelli, L. (2006). “Divergence form for bed slope source term in shallow water equations.” J. Hydraul. Eng., 132(7), 652–665.
Valiani, A., and Begnudelli, L. (2008). “Closure to divergence form for bed slope source term in shallow water equations.” J. Hydraul. Eng., 134(5), 680–682.
Valiani, A., and Caleffi, V. (2008). “Depth-energy and depth-force relationships in open channel flows: Analytical findings.” Adv. Water Resour., 31(3), 447–454.
Valiani, A., Caleffi, V., and Zanni, A. (2002). “Case study: Malpasset dambreak simulation using a 2D finite volume method.” J. Hydraul. Eng., 128(5), 460–472.
Vázquez-Cendón, M. E. (1999). “Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry.” J. Comput. Phys., 148(2), 497–526.
Vuković, 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.
Xing, Y., and Shu, C.-W. (2005). “High order finite difference weno schemes with the exact conservation property for the shallow water equations.” J. Comput. Phys., 208(1), 206–227.
Zhou, J. G., Causon, D. M., Ingram, D. M., and Mingham, C. G. (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., 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(8), 769–788.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 135 • Issue 7 • July 2009
Pages: 684 - 696
Copyright
© 2009 ASCE.
History
Received: Oct 4, 2007
Accepted: Nov 25, 2008
Published online: Jun 15, 2009
Published in print: Jul 2009
Notes
Note. Associate Editor: Brett F. Sanders
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.