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

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 wellbalanced). 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. Mathematics Subject Classification. 65M99, 35L65. Received September 25, 2007. Revised June 27, 2008. Published online December 17, 2008. 1. The shallow-water model We consider the shallow water equations along channels with non-uniform rectangular cross sections and bottom topography. The model describes flows that are nearly horizontal and can be obtained by averaging the Euler equations over the channel cross section [7], resulting in the balance law ∂A ∂t + ∂Q ∂x = 0 (1.1a) ∂Q ∂t + ∂ ∂x (Q2 A + 1 2 gσh ) = 1 2 gh2σ′ − gσhB′, (1.1b) where h and σ(x) are, respectively, the height of the fluid above the bottom of the channel, and the channel breadth, A = σh is the wet cross-section, Q = Au is the discharge, with u denoting the (depth average) fluid velocity, B(x) describes the bottom topography of the channel, and g is the acceleration of gravity (see Fig. 1).