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A central scheme for shallow water flows along channels with irregular geometry

Published online by Cambridge University Press:  18 December 2008

Jorge Balbás  and
Smadar Karni
Jorge Balbás
Affiliation:
Department of Mathematics, California State University, Northridge, CA 91330, USA. [email protected]
Smadar Karni
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA. [email protected]
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Abstract

We present a new semi-discrete central scheme for one-dimensional shallow water flows along channels with non-uniform rectangular cross sections and bottom topography. The scheme preserves the positivity of the water height, and it is preserves steady-states of rest (i.e., it is well-balanced). Along with a detailed description of the scheme, numerous numerical examples are presented for unsteady and steady flows. Comparison with exact solutions illustrate the accuracy and robustness of the numerical algorithm.

Keywords

Hyperbolic systems of conservation and balance lawssemi-discrete schemesSaint-Venant system of shallow water equationsnon-oscillatory reconstructionschannels with irregular geometry.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 43 , Issue 2 , March 2009 , pp. 333 - 351
Copyright
© EDP Sciences, SMAI, 2009

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References

R. Abgrall and S. Karni, A relaxation scheme for the two layer shallow water system, in Proceedings of the 11th International Conference on Hyperbolic Problems (Lyon, 2006), Springer (2008) 135–144.
Audusse, E., Bouchut, F., Bristeau, M.-O., Klein, R. and Perthame, B., A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput. 25 (2004) 2050–2065 CrossRef
J. Balbás and E. Tadmor, Nonoscillatory central schemes for one- and two-dimensional magnetohydrodynamics equations. ii: High-order semidiscrete schemes. SIAM J. Sci. Comput. 28 (2006) 533–560.
Bermudez, A. and Vazquez, M.E., Upwind methods for hyperbolic conservation laws with source terms. Comput. Fluids 23 (1994) 1049–1071. CrossRef
F. Bouchut, Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources. Birkhauser, Basel, Switzerland, Berlin (2004).
M.J. Castro, J. Macias and C. Pares, A Q-scheme for a class of systems of coupled conservation laws with source terms. Application to a two-layer 1-d shallow water system. ESAIM: M2AN 35 (2001) 107–127.
Castro, M.J., García-Rodríguez, J.A., González-Vida, J.M., Macías, J., Parés, C. and Vázquez-Cendón, M.E., Numerical simulation of two-layer shallow water flows through channels with irregular geometry. J. Comput. Phys. 195 (2004) 202–235. CrossRef
Črnjarić-Žic, N., Vuković, S. and Sopta, L., Balanced finite volume WENO and central WENO schemes for the shallow water and the open-channel flow equations. J. Comput. Phys. 200 (2004) 512–548. CrossRef
Gottlieb, S., Shu, C.-W. and Tadmor, E., Strong stability-preserving high-order time discretization methods. SIAM Rev. 43 (2001) 89–112. CrossRef
Greenberg, J.M. and Le Roux, A.Y., Well-balanced scheme for the processing of source terms in hyperbolic equations. SIAM J. Numer. Anal. 33 (1996) 1–16. CrossRef
Harten, A., High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 49 (1983) 357–393. CrossRef
Jin, S., A steady-state capturing method for hyperbolic systems with geometrical source terms. ESAIM: M2AN 35 (2001) 631–645. CrossRef
Kurganov, A. and Levy, D., Central-upwind schemes for the Saint-Venant system. ESAIM: M2AN 36 (2002) 397–425. CrossRef
Kurganov, A. and Petrova, G., A second-order well-balanced positivity preserving central-upwind scheme for the Saint-Venant system. Commun. Math. Sci. 5 (2007) 133–160. CrossRef
Kurganov, A. and Tadmor, E., New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations. J. Comput. Phys. 160 (2000) 241–282. CrossRef
Kurganov, A., Noelle, S. and Petrova, G., Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton-Jacobi equations. SIAM J. Sci. Comput. 23 (2001) 707–740. CrossRef
LeVeque, R.J., Balancing source terms and flux gradients in high resolution Godunov methods: the quasi-steady wave-propagation algorithm. J. Comp. Phys. 146 (1998) 346–365. CrossRef
Nessyahu, H. and Tadmor, E., Nonoscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87 (1990) 408–463. CrossRef
Noelle, S., Pankratz, N., Puppo, G. and Natvig, J.R., Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows. J. Comput. Phys. 213 (2006) 474–499. CrossRef
Noelle, S., Xing, Y., and Shu, C.-W., High-order well-balanced finite volume WENO schemes for shallow water equation with moving water. J. Comput. Phys. 226 (2007) 29–58. CrossRef
C. Pares and M. Castro, On the well-balance property of Roe's method for nonconservative hyperbolic systems. Applications to shallow-water systems. ESAIM: M2AN 38 (2004) 821–852.
Perthame, B. and Simeoni, C., A kinetic scheme for the Saint-Venant system with a source term. Calcolo 38 (2001) 201–231. CrossRef
G. Russo, Central schemes for balance laws, in Hyperbolic problems: theory, numerics, applications, Vols. I, II (Magdeburg, 2000), Internat. Ser. Numer. Math. 140, Birkhäuser, Basel (2001) 821–829.
C.-W. Shu and S. Osher, Efficient implementation of essentially nonoscillatory shock-capturing schemes. II. Comput. Phys. 83 (1989) 32–78.
Thacker, W.C., Some exact solutions to the nonlinear shallow-water wave equations. Journal of Fluid Mechanics Digital Archive 107 (1981) 499–508. CrossRef
B. van Leer, Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov's method. J. Comput. Phys. 135 (1997) 229–248.
Vázquez-Cendón, M.E., Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry. J. Comput. Phys. 148 (1999) 497–526. CrossRef
S. Vuković and L. Sopta, High-order ENO and WENO schemes with flux gradient and source term balancing, in Applied mathematics and scientific computing (Dubrovnik, 2001), Kluwer/Plenum, New York (2003) 333–346.