A central scheme for shallow water flows along channels with irregular geometry
ESAIM: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 2, pp. 333-351.

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.

DOI : 10.1051/m2an:2008050
Classification : 65M99, 35L65
Mots-clés : hyperbolic systems of conservation and balance laws, semi-discrete schemes, Saint-Venant system of shallow water equations, non-oscillatory reconstructions, channels with irregular geometry 
@article{M2AN_2009__43_2_333_0,
     author = {Balb\'as, Jorge and Karni, Smadar},
     title = {A central scheme for shallow water flows along channels with irregular geometry},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {333--351},
     publisher = {EDP-Sciences},
     volume = {43},
     number = {2},
     year = {2009},
     doi = {10.1051/m2an:2008050},
     mrnumber = {2512499},
     zbl = {1159.76026},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an:2008050/}
}
TY  - JOUR
AU  - Balbás, Jorge
AU  - Karni, Smadar
TI  - A central scheme for shallow water flows along channels with irregular geometry
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2009
SP  - 333
EP  - 351
VL  - 43
IS  - 2
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an:2008050/
DO  - 10.1051/m2an:2008050
LA  - en
ID  - M2AN_2009__43_2_333_0
ER  - 
%0 Journal Article
%A Balbás, Jorge
%A Karni, Smadar
%T A central scheme for shallow water flows along channels with irregular geometry
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2009
%P 333-351
%V 43
%N 2
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/m2an:2008050/
%R 10.1051/m2an:2008050
%G en
%F M2AN_2009__43_2_333_0
Balbás, Jorge; Karni, Smadar. A central scheme for shallow water flows along channels with irregular geometry. ESAIM: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 2, pp. 333-351. doi : 10.1051/m2an:2008050. http://archive.numdam.org/articles/10.1051/m2an:2008050/

[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.

[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 | MR | Zbl

[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. | MR | Zbl

[4] A. Bermudez and M.E. Vazquez, Upwind methods for hyperbolic conservation laws with source terms. Comput. Fluids 23 (1994) 1049-1071. | MR | Zbl

[5] F. Bouchut, Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources. Birkhauser, Basel, Switzerland, Berlin (2004). | MR | Zbl

[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. | EuDML | Numdam | Zbl

[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. | Zbl

[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. | Zbl

[9] S. Gottlieb, C.-W. Shu and E. Tadmor, Strong stability-preserving high-order time discretization methods. SIAM Rev. 43 (2001) 89-112. | Zbl

[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. | Zbl

[11] A. Harten, High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 49 (1983) 357-393. | MR | Zbl

[12] S. Jin, A steady-state capturing method for hyperbolic systems with geometrical source terms. ESAIM: M2AN 35 (2001) 631-645. | Numdam | MR | Zbl

[13] A. Kurganov and D. Levy, Central-upwind schemes for the Saint-Venant system. ESAIM: M2AN 36 (2002) 397-425. | Numdam | MR | Zbl

[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. | MR

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[18] H. Nessyahu and E. Tadmor, Nonoscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87 (1990) 408-463. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | Numdam | MR | Zbl

[22] B. Perthame and C. Simeoni, A kinetic scheme for the Saint-Venant system with a source term. Calcolo 38 (2001) 201-231. | MR | Zbl

[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. | MR

[24] C.-W. Shu and S. Osher, Efficient implementation of essentially nonoscillatory shock-capturing schemes. II. Comput. Phys. 83 (1989) 32-78. | MR | Zbl

[25] W.C. Thacker, Some exact solutions to the nonlinear shallow-water wave equations. Journal of Fluid Mechanics Digital Archive 107 (1981) 499-508. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

Cité par Sources :