A two-dimensional well-balanced numerical model for shallow water equations

Modeling flow discontinuities, due to a numerical approach, often pose severe challenges. In this way, a number of techniques like artificial viscosity (particularly for finite difference methods), shock fitting and etc. have been proposed. These techniques usually require ad-hoc terms which need to be adjusted through calibration. In this study, an efficient numerical model based on a shallow water equation is developed. The model uses a first-order centered (FORCE) scheme, in combination with the Surface Gradient Method, (SGM) for spatial discretization, and the third order Runge–Kutta algorithm for time integration. At first, it is demonstrated that the model is well-balanced, then, through several classical examples, such as a 1D dam-break on both dry and wet beds, transcritical flow over a bump, with and without shock, sub-critical flow over a bump, circular dam-break, small perturbation propagation, dam-break on a dry bed channel with varying widths and right-angled channel junctions, it is shown that the model is capable of capturing flow discontinuities. Furthermore, the model can simulate dry bed conditions, and also presents smooth symmetric results.