Some recent finite volume schemes for one and two layers shallow water equations with variable density

We propose a class of finite volume algorithms that are both simple and efficient for solving numerically the shallow water equations with varying densities; shallow water flows in single and two layers are considered. In these flow regimes, variable horizontal or vertical density is taken into account. The shallow water equations for the hydraulic variables are coupled with a suspended sediment transport equation for the concentration variable to construct the model. To approximate the numerical solution of the models under consideration, a generalized Rusanov method is proposed; this method avoids solving Riemann problems during the time integration process, and it is simple and accurate. The proposed method is divided into two stages: predictor and corrector; the presented finite volume approach is well balanced, conservative, and simple. Non-oscillatory and suitable for shallow water equations when Riemann problems are challenging to solve. Several test problems for single-layer and two-layer shallow water flows are applied to the proposed method. The numerical results show that the proposed finite volume approach has high resolution and that it is capable of simulating shallow water equations with variable density in flow regimes with strong shocks accurately.