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We consider the shallow water equations along channels with non-uniform rectangular cross sections with source terms due to bottom topography, channel width, and friction factor. The system of equations consists of the mass and momentum conservation equations. We have two main goals in this paper. The first is to develop a numerical method for solving the model of shallow water equations involving those source terms. The second is to investigate effects of friction in water flows governed by the model. We limit our research to the flows of one-dimensional problems. The friction uses the Manning's formula. The mathematical model is solved numerically using a finite volume numerical method on staggered grids. We propose the use of this method, because the computation is cheap due to that no Riemann solver is needed in the flux calculation. Along with a detailed description of the scheme, in this paper, we show a strategy to include the discretization of the friction term in the staggered-grid finite volume method. Simulation results indicate that our strategy is successful in solving the problems. Furthermore, an obvious effect of friction is that it slows down water flows. We obtain that great friction values lead to slow motion of water, and at the same time, large water depth. Small friction values result in fast motion of water and small water depth.

A Staggered Method for Simulating Shallow Water Flows along Channels with Irregular Geometry and Friction