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.
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://www.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://www.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://www.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://www.numdam.org/articles/10.1051/m2an:2008050/
[1] 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.
and ,[2] A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput. 25 (2004) 2050-2065 | MR | Zbl
, , , and ,[3] 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
and ,[4] Upwind methods for hyperbolic conservation laws with source terms. Comput. Fluids 23 (1994) 1049-1071. | MR | Zbl
and ,[5] Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources. Birkhauser, Basel, Switzerland, Berlin (2004). | MR | Zbl
,[6] 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
, and ,[7] Numerical simulation of two-layer shallow water flows through channels with irregular geometry. J. Comput. Phys. 195 (2004) 202-235. | Zbl
, , , , and ,[8] 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
, and ,[9] Strong stability-preserving high-order time discretization methods. SIAM Rev. 43 (2001) 89-112. | Zbl
, and ,[10] Well-balanced scheme for the processing of source terms in hyperbolic equations. SIAM J. Numer. Anal. 33 (1996) 1-16. | Zbl
and ,[11] High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 49 (1983) 357-393. | MR | Zbl
,[12] A steady-state capturing method for hyperbolic systems with geometrical source terms. ESAIM: M2AN 35 (2001) 631-645. | Numdam | MR | Zbl
,[13] Central-upwind schemes for the Saint-Venant system. ESAIM: M2AN 36 (2002) 397-425. | Numdam | MR | Zbl
and ,[14] A second-order well-balanced positivity preserving central-upwind scheme for the Saint-Venant system. Commun. Math. Sci. 5 (2007) 133-160. | MR
and ,[15] New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations. J. Comput. Phys. 160 (2000) 241-282. | MR | Zbl
and ,[16] Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton-Jacobi equations. SIAM J. Sci. Comput. 23 (2001) 707-740. | MR | Zbl
, and ,[17] 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] Nonoscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87 (1990) 408-463. | MR | Zbl
and ,[19] Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows. J. Comput. Phys. 213 (2006) 474-499. | MR | Zbl
, , and ,[20] High-order well-balanced finite volume WENO schemes for shallow water equation with moving water. J. Comput. Phys. 226 (2007) 29-58. | MR | Zbl
, , and ,[21] 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
and ,[22] A kinetic scheme for the Saint-Venant system with a source term. Calcolo 38 (2001) 201-231. | MR | Zbl
and ,[23] 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] Efficient implementation of essentially nonoscillatory shock-capturing schemes. II. Comput. Phys. 83 (1989) 32-78. | MR | Zbl
and ,[25] Some exact solutions to the nonlinear shallow-water wave equations. Journal of Fluid Mechanics Digital Archive 107 (1981) 499-508. | MR | Zbl
,[26] 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] 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] 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
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