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We develop a new finite volume method using unstructured mesh-vertex gridsfor coupled systems modeling shallow water flows and solute transport overcomplex bottom topography. Novel well-balanced positivity preservingdiscretization techniques are proposed for the water surface elevation and theconcentration of the pollutant. For the hydrodynamic system, the proposedscheme preserves the steady state of a lake at rest and the positivity of thewater depth. For the scalar transport equation, the proposed method guaranteesthe positivity and a perfect balance of the scalar concentration. Theconstant-concentration states are preserved in space and time for anyhydrodynamic field and complex topography in the absence of source terms of thepassive pollutant. Importantly and this is one of the main features of ourapproach is that the novel reconstruction techniques proposed for the watersurface elevation and concentration satisfy the discrete maximum-minimumprinciple for the solute concentration. We demonstrate, in a series ofnumerical tests, the well-balanced and positivity properties of the proposedmethod and the accuracy of our techniques and their potential advantages inpredicting the solutions of the shallow water-transport model.

A well-balanced positivity preserving cell-vertex finite volume method  satisfying the discrete maximum-minimum principle for coupled models of  surface water flow and scalar transport