LAUSR

We consider a system of coupled equations modeling a shallow water flow with solute transport and introduce an artificial dissipation in order to improve the dissipation properties of the original cell-vertex central-upwind numerical scheme applied to these equations. Namely, a formulation is proposed which involves an artificial dissipation parameter and guarantees a consistency property between the continuity equation and the transport equation at the discrete level and, in addition, ensures the nonlinear stability and positivity of the scheme. A well-balanced positivity-preserving reconstruction is stated in terms of the conservative variable describing the concentration. We establish that constant-concentration states are preserved in space and time for any hydrodynamic flow field in the absence of source terms in the transport equation. Furthermore, we prove the maximum and minimum principles for the concentration. A suitable discretization of the diffusion term is used in combination with the proposed reconstruction procedure and artificial dissipation formulation and this allows us to prove the positivity of the concentration in the presence of diffusion effects. Finally, our numerical experiments confirm the well-balanced and positivity-preserving properties when the artificial dissipation is introduced in the central-upwind scheme, and the accuracy of the scheme for modeling surface water flows with transport processes.