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

- Volume: 43, Issue: 2, page 333-351
- ISSN: 0764-583X

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topBalbás, Jorge, and Karni, Smadar. "A central scheme for shallow water flows along channels with irregular geometry." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 43.2 (2009): 333-351. <http://eudml.org/doc/246108>.

@article{Balbás2009,

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.},

author = {Balbás, Jorge, Karni, Smadar},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {hyperbolic systems of conservation and balance laws; semi-discrete schemes; Saint-Venant system of shallow water equations; non-oscillatory reconstructions; channels with irregular geometry; Saint-Venant equations},

language = {eng},

number = {2},

pages = {333-351},

publisher = {EDP-Sciences},

title = {A central scheme for shallow water flows along channels with irregular geometry},

url = {http://eudml.org/doc/246108},

volume = {43},

year = {2009},

}

TY - JOUR

AU - Balbás, Jorge

AU - Karni, Smadar

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

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2009

PB - EDP-Sciences

VL - 43

IS - 2

SP - 333

EP - 351

AB - 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.

LA - eng

KW - hyperbolic systems of conservation and balance laws; semi-discrete schemes; Saint-Venant system of shallow water equations; non-oscillatory reconstructions; channels with irregular geometry; Saint-Venant equations

UR - http://eudml.org/doc/246108

ER -

## References

top- [1] 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. Zbl05258327
- [2] E. Audusse, F. Bouchut, M.-O. Bristeau, R. Klein and B. Perthame, A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput. 25 (2004) 2050–2065 Zbl1133.65308MR2086830
- [3] 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. Zbl1136.65340MR2231720
- [4] A. Bermudez and M.E. Vazquez, Upwind methods for hyperbolic conservation laws with source terms. Comput. Fluids 23 (1994) 1049–1071. Zbl0816.76052MR1314237
- [5] F. Bouchut, Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources. Birkhauser, Basel, Switzerland, Berlin (2004). Zbl1086.65091MR2128209
- [6] 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. Zbl1094.76046
- [7] M.J. Castro, J.A. García-Rodríguez, J.M. González-Vida, J. Macías, C. Parés and M.E. Vázquez-Cendón, Numerical simulation of two-layer shallow water flows through channels with irregular geometry. J. Comput. Phys. 195 (2004) 202–235. Zbl1087.76077
- [8] N. Črnjarić-Žic, S. Vuković and L. Sopta, 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. Zbl1115.76364
- [9] S. Gottlieb, C.-W. Shu and E. Tadmor, Strong stability-preserving high-order time discretization methods. SIAM Rev. 43 (2001) 89–112. Zbl0967.65098
- [10] J.M. Greenberg and A.Y. Le Roux, Well-balanced scheme for the processing of source terms in hyperbolic equations. SIAM J. Numer. Anal. 33 (1996) 1–16. Zbl0876.65064
- [11] A. Harten, High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 49 (1983) 357–393. Zbl0565.65050MR701178
- [12] S. Jin, A steady-state capturing method for hyperbolic systems with geometrical source terms. ESAIM: M2AN 35 (2001) 631–645. Zbl1001.35083MR1862872
- [13] A. Kurganov and D. Levy, Central-upwind schemes for the Saint-Venant system. ESAIM: M2AN 36 (2002) 397–425. Zbl1137.65398MR1918938
- [14] A. Kurganov and G. Petrova, A second-order well-balanced positivity preserving central-upwind scheme for the Saint-Venant system. Commun. Math. Sci. 5 (2007) 133–160. Zbl1226.76008MR2310637
- [15] A. Kurganov and E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations. J. Comput. Phys. 160 (2000) 241–282. Zbl0987.65085MR1756766
- [16] A. Kurganov, S. Noelle and G. Petrova, Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton-Jacobi equations. SIAM J. Sci. Comput. 23 (2001) 707–740. Zbl0998.65091MR1860961
- [17] R.J. LeVeque, Balancing source terms and flux gradients in high resolution Godunov methods: the quasi-steady wave-propagation algorithm. J. Comp. Phys. 146 (1998) 346–365. Zbl0931.76059MR1650496
- [18] H. Nessyahu and E. Tadmor, Nonoscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87 (1990) 408–463. Zbl0697.65068MR1047564
- [19] S. Noelle, N. Pankratz, G. Puppo and J.R. Natvig, Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows. J. Comput. Phys. 213 (2006) 474–499. Zbl1088.76037MR2207248
- [20] S. Noelle, Y. Xing, and C.-W. Shu, High-order well-balanced finite volume WENO schemes for shallow water equation with moving water. J. Comput. Phys. 226 (2007) 29–58. Zbl1120.76046MR2356351
- [21] 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. Zbl1130.76325MR2104431
- [22] B. Perthame and C. Simeoni, A kinetic scheme for the Saint-Venant system with a source term. Calcolo 38 (2001) 201–231. Zbl1008.65066MR1890353
- [23] 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. MR1871169
- [24] C.-W. Shu and S. Osher, Efficient implementation of essentially nonoscillatory shock-capturing schemes. II. Comput. Phys. 83 (1989) 32–78. Zbl0674.65061MR1010162
- [25] W.C. Thacker, Some exact solutions to the nonlinear shallow-water wave equations. Journal of Fluid Mechanics Digital Archive 107 (1981) 499–508. Zbl0462.76023MR623361
- [26] 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. Zbl0939.76063MR1486274
- [27] M.E. Vázquez-Cendón, 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. Zbl0931.76055MR1669644
- [28] 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. Zbl1017.65071MR1966882

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