We present a new high-resolution, non-oscillatory semi-discrete central scheme for one-dimensional shallow water flows along channels with non-uniform rectangular cross sections and bottom topography. The scheme extends an existing central semi-discrete formulation for hyperbolic conservation laws and it enjoys two properties crucial for the accurate simulation of shallow water flows: it preserves the positivity of the water height, and it is well balanced, {\it i.e.}, the source terms arising from the geometry of the channel are discretized so as to balance the non-linear hyperbolic fluxes --a condition necessary to correctly approximate steady-state solutions. Along with a detailed description of the scheme and its properties, we present several numerical experiments --including the approximation of exact equilibrium solutions-- that demonstrate the robustness --and simplicity-- of the numerical algorithm.
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