Modelling Shallow Water Flow And Pollutant Transport Using A Well-Balanced Discontinuous Galerkin Method

In the present paper, a well-balanced second-order Runge-Kutta discontinuous Galerkin (RKDG) method is presented for modelling shallow water flows and pollutant transport over unstructured triangular meshes. The model is designed to handle arbitrary topography and moving wetting and drying fronts. The mathematical model describing these flows is derived from the shallow water equations (SWEs), which are coupled with the convection equation. The fluxes across cells interfaces are evaluated using the HLL Riemann solver. Treatment of topography source term is built in the DG approximation to guaranty the well-balanced property of the SWEs. The friction source term is calculated implicitly via a splitting integration approach. To improve the model, we adapt and refine positivity preserving and wetting and drying techniques originally designed for the finite volume schemes, ensuring its effective integration within the RKDG framework. The resulting method is implemented and systematically assessed for accuracy and robustness on a carefully selected suite of test problems including wetting and drying, irregular topography, and discontinuous flows. Numerical results are provided, showcasing the effectiveness of the proposed approach in accurately predicting both water depth and pollutant concentration.