Abstract

In this article we present the approximation of the coupled model of the equations of motion of a fluid in shallow waters with the convection-diffusion-reaction (CDR) equation of pollutant transport. This approximation is carried out using high order finite elements and using stabilised variational sub-scale methods. We write the coupled system of equations, previously discretised in time and linearised, as a transient vector equation of CDR. The stabilised finite element methods used are the known ASGS and OSS sub-scale methods, the same ones that allow us to use the same interpolation for all unknowns, as well as to deal with dominant convection and reaction flows. We consider the possibility of non-linearity in both the convective and reaction terms. We will not consider the possible development of shocks in the solution. In order to examine the accuracy and robustness of the ASGS and OSS methods, we present four test cases: mesh convergence, transport of a pollutant in a square cavity, transport of a pollutant in the Gulf of Roses and at the river Guadalquivir mouth, and the predator-prey model, which can be written as a transient CDR vector equation with non-linearity in the reaction term.