The shallow-water (SW) equations govern many practical applications like river and tidal flows in estuary and coastal water regions where variable topographies are usually present. The simulation of these flows is of great interest for hydraulic engineers, particularly on unstructured grids. However, simulation of shallow flows over variable topographies is a challenging case for most available shock-capturing schemes. This problem arises because the source terms and flux gradients are not balanced in the numerical computations. Treatments for this problem generally work well on structured grids, but they are usually too expensive, and most of them are not directly applicable to unstructured grids.
In this study we propose two efficient methods to treat the source terms without upwinding and to satisfy the compatibility condition on unstructured grids. In the first method, the calculation of the bed slope source term is performed by employing a compatible approximation of water depth at the cell interfaces. In the second one, different components of the bed slope term are considered separately and a compatible discretization of the components is proposed. The present treatments are applicable for most schemes including the Roe’s method without changing the performance of the original scheme for smooth topographies.