A central-upwind scheme with artificial viscosity for shallow-water flows in channels

We develop a new high-resolution, non-oscillatory semi-discrete central-upwind scheme with artificial viscosity for shallow-water flows in channels with arbitrary geometry and variable topography. The artificial viscosity, proposed as an alternative to nonlinear limiters, allows us to use high-resolution reconstructions at a low computational cost. The scheme recognizes steady states at rest when a delicate balance between the source terms and flux gradients occurs. This balance in irregular geometries is more complex than that taking place in channels with vertical walls. A suitable technique is applied by properly taking into account the effects induced by the geometry. Incorporating the contributions of the artificial viscosity and an appropriate time step restriction, the scheme preserves the positivity of the water's depth. A description of the proposed scheme, its main properties as well as the proofs of well-balance and the positivity of the scheme are provided. Our numerical experiments confirm stability, well-balance, positivity-preserving properties and high resolution of the proposed method. Comparisons of numerical solutions obtained with the proposed scheme and experimental data are conducted, showing a good agreement. This scheme can be applied to shallow-water flows in channels with complex geometry and variable bed topography.