Preserving bounded and conservative solutions of transport in one‐dimensional shallow‐water flow with upwind numerical schemes: Application to fertigation and solute transport in rivers

This work intends to show that conservative upwind schemes based on a separate discretization of the scalar solute transport from the shallow‐water equations are unable to preserve uniform solute profiles in situations of one‐dimensional unsteady subcritical flow. However, the coupled discretization of the system is proved to lead to the correct solution in first‐order approximations. This work is also devoted to show that, when using a coupled discretization, a careful definition of the flux limiter function in second‐order TVD schemes is required in order to preserve uniform solute profiles. The work shows that, in cases of subcritical irregular flow, the coupled discretization is necessary but nevertheless not sufficient to ensure concentration distributions free from oscillations and a method to avoid these oscillations is proposed. Examples of steady and unsteady flows in test cases, river and irrigation are presented.