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Finite volume methods have proven themselves a powerful tool for finding solutions to the shallow water wave equations. They are based on the conservation laws for the mass and momentum, integrated over discrete finite volumes. These methods tend to do well at the difficult problem of capturing solutions involving shocks. However, one area that causes problems is the approximation of steady or near steady states when there is a sloping bed elevation. The problem arises due to a poor balance between the discretisation of the flux terms across the edge of a finite volume and the pressure terms due to the sloping bed. Methods that overcome these difficulties and reproduce the still lake steady state solution, are called well balanced. In this work we are interested in a well balanced scheme for the one dimensional shallow water wave equations but with a modification that allows for varying width in the transverse direction. Here a well balanced method developed by Audusse et al. for the constant width case is extended to the case of varying (possibly discontinuous) width. Numerical validation of this new method is provided.   References E. Audusse, F. Bouchut, M. Bristeau, R. Klein, and B. Perthame. A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM Journal of Scientific Computing , 25(6):2050--2065, 2004. doi:10.1137/S1064827503431090 A. Birman and J. Falcovitz. Application of the GRP scheme to open channel flow equations. Journal of Computational Physics , 222:131--154, 2007. doi:10.1016/j.jcp.2006.07.008 Sigal Gottlieb, Chi-Wang Shu, and Eitan Tadmor. Strong stability-preserving high-order time discretization methods. SIAM Rev. , 43(1):89--112 (electronic), 2001. doi:10.1137/S003614450036757X A. Kurganov, S. Noelle, and G. Petrova. Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton--Jacobi equations. SIAM Journal of Scientific Computing , 23(3):707--740, 2001. doi:10.1137/S1064827500373413 R. LeVeque. Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave propagation algorithm. Journal of Computational Physics , 146(1):346--365, 1998. doi:10.1006/jcph.1998.6058 Randall J. LeVeque. Finite volume methods for hyperbolic problems . Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2002. doi:10.1017/CBO9780511791253 B. Van Leer. Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov's method. Journal of Computational Physics , 32(1):101--136, 1979. doi:10.1016/0021-9991(79)90145-1 J. Zhou, D. Causon, C. Mingham, and D. Ingram. The surface gradient method for the treatment of source terms in the shallow-water equations. Journal of Computational Physics , 168:1--25, 2001. doi:10.1006/jcph.2000.6670 Christopher Zoppou and Stephen Roberts. Explicit schemes for dam-break simulations. ASCE, J. Hydraulic Engineering , 129(1):11--34, 2003. doi:10.1061/(ASCE)0733-9429(2003)129:1(11)

A well balanced scheme for the shallow water wave equations in open channels with (discontinuous) varying width and bed