  We develop a new finite volume method using unstructured mesh-vertex grids
for coupled systems modeling shallow water flows and solute transport over
complex bottom topography. Novel well-balanced positivity preserving
discretization techniques are proposed for the water surface elevation and the
concentration of the pollutant. For the hydrodynamic system, the proposed
scheme preserves the steady state of a lake at rest and the positivity of the
water depth. For the scalar transport equation, the proposed method guarantees
the positivity and a perfect balance of the scalar concentration. The
constant-concentration states are preserved in space and time for any
hydrodynamic field and complex topography in the absence of source terms of the
passive pollutant. Importantly and this is one of the main features of our
approach is that the novel reconstruction techniques proposed for the water
surface elevation and concentration satisfy the discrete maximum-minimum
principle for the solute concentration. We demonstrate, in a series of
numerical tests, the well-balanced and positivity properties of the proposed
method and the accuracy of our techniques and their potential advantages in
predicting the solutions of the shallow water-transport model.
