  In this study, we focus on the modelling of coupled systems of shallow water
flows and solute transport with source terms due to variable topography and
friction effect. Our aim is to propose efficient and accurate numerical
techniques for modelling these systems using unstructured triangular grids. We
used a Riemann-solver free method for the hyperbolic shallow water system and a
suitable discretization technique for the bottom topography. The friction
source term is discretized using the techniques proposed by (Xia and Liang
2018). Our approach performs very well for stationary flow in the presence of
variable topography, and it is well-balanced for the concentration in the
presence of wet and dry zones. In our techniques, we used linear piecewise
reconstructions for the variables of the coupled system. The proposed method is
well-balanced, and we prove that it exactly preserves the nontrivial
steady-state solutions of the coupled system. Numerical experiments are carried
out to validate the performance and robustness of the proposed numerical
method. Our numerical results show that the method is stable, well-balanced and
accurate to model the coupled systems of shallow water flows and solute
transport.
