 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.
 E. Audusse, F. Bouchut, M.O. Bristeau, R. Klein and B. Perthame, A fast and stable wellbalanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput. 25 (2004) 2050–2065 [CrossRef] [MathSciNet]
 J. Balbás and E. Tadmor, Nonoscillatory central schemes for one and twodimensional magnetohydrodynamics equations. ii: Highorder semidiscrete schemes. SIAM J. Sci. Comput. 28 (2006) 533–560.
 A. Bermudez and M.E. Vazquez, Upwind methods for hyperbolic conservation laws with source terms. Comput. Fluids 23 (1994) 1049–1071. [CrossRef] [MathSciNet]
 F. Bouchut, Nonlinear stability of finite volume methods for hyperbolic conservation laws and wellbalanced schemes for sources. Birkhauser, Basel, Switzerland, Berlin (2004).
 M.J. Castro, J. Macias and C. Pares, A Qscheme for a class of systems of coupled conservation laws with source terms. Application to a twolayer 1d shallow water system. ESAIM: M2AN 35 (2001) 107–127.
 M.J. Castro, J.A. GarcíaRodríguez, J.M. GonzálezVida, J. Macías, C. Parés and M.E. VázquezCendón, Numerical simulation of twolayer shallow water flows through channels with irregular geometry. J. Comput. Phys. 195 (2004) 202–235. [CrossRef] [MathSciNet]
 N. ČrnjarićŽic, S. Vuković and L. Sopta, Balanced finite volume WENO and central WENO schemes for the shallow water and the openchannel flow equations. J. Comput. Phys. 200 (2004) 512–548. [CrossRef] [MathSciNet]
 S. Gottlieb, C.W. Shu and E. Tadmor, Strong stabilitypreserving highorder time discretization methods. SIAM Rev. 43 (2001) 89–112. [NASA ADS] [CrossRef] [MathSciNet]
 J.M. Greenberg and A.Y. Le Roux, Wellbalanced scheme for the processing of source terms in hyperbolic equations. SIAM J. Numer. Anal. 33 (1996) 1–16. [CrossRef] [MathSciNet]
 A. Harten, High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 49 (1983) 357–393. [NASA ADS] [CrossRef] [MathSciNet]
 S. Jin, A steadystate capturing method for hyperbolic systems with geometrical source terms. ESAIM: M2AN 35 (2001) 631–645. [CrossRef] [EDP Sciences]
 A. Kurganov and D. Levy, Centralupwind schemes for the SaintVenant system. ESAIM: M2AN 36 (2002) 397–425. [CrossRef] [EDP Sciences]
 A. Kurganov and G. Petrova, A secondorder wellbalanced positivity preserving centralupwind scheme for the SaintVenant system. Commun. Math. Sci. 5 (2007) 133–160. [MathSciNet]
 A. Kurganov and E. Tadmor, New highresolution central schemes for nonlinear conservation laws and convectiondiffusion equations. J. Comput. Phys. 160 (2000) 241–282. [NASA ADS] [CrossRef] [MathSciNet]
 A. Kurganov, S. Noelle and G. Petrova, Semidiscrete centralupwind schemes for hyperbolic conservation laws and HamiltonJacobi equations. SIAM J. Sci. Comput. 23 (2001) 707–740. [CrossRef] [MathSciNet]
 R.J. LeVeque, Balancing source terms and flux gradients in high resolution Godunov methods: the quasisteady wavepropagation algorithm. J. Comp. Phys. 146 (1998) 346–365. [NASA ADS] [CrossRef] [MathSciNet]
 H. Nessyahu and E. Tadmor, Nonoscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87 (1990) 408–463. [CrossRef] [MathSciNet]
 S. Noelle, N. Pankratz, G. Puppo and J.R. Natvig, Wellbalanced finite volume schemes of arbitrary order of accuracy for shallow water flows. J. Comput. Phys. 213 (2006) 474–499. [CrossRef] [MathSciNet]
 S. Noelle, Y. Xing, and C.W. Shu, Highorder wellbalanced finite volume WENO schemes for shallow water equation with moving water. J. Comput. Phys. 226 (2007) 29–58. [CrossRef] [MathSciNet]
 C. Pares and M. Castro, On the wellbalance property of Roe's method for nonconservative hyperbolic systems. Applications to shallowwater systems. ESAIM: M2AN 38 (2004) 821–852.
 B. Perthame and C. Simeoni, A kinetic scheme for the SaintVenant system with a source term. Calcolo 38 (2001) 201–231. [CrossRef] [MathSciNet]
 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 shockcapturing schemes. II. Comput. Phys. 83 (1989) 32–78.
 W.C. Thacker, Some exact solutions to the nonlinear shallowwater wave equations. Journal of Fluid Mechanics Digital Archive 107 (1981) 499–508.
 B. van Leer, Towards the ultimate conservative difference scheme. V. A secondorder sequel to Godunov's method. J. Comput. Phys. 135 (1997) 229–248.
 M.E. VázquezCendón, 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] [MathSciNet]
 S. Vuković and L. Sopta, Highorder 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.
Free access
Issue 
ESAIM: M2AN
Volume 43, Number 2, MarchApril 2009



Page(s)  333  351  
DOI  http://dx.doi.org/10.1051/m2an:2008050  
Published online  18 December 2008 