The mR scheme to the shallow water equation with horizontal density gradients in one and two dimensions

In this work, we consider the model of shallow water equation with horizontal density gradients. We develop the modified Rusanov (mR) scheme to solve this model in one and two dimensions. Predictor and corrector are the two stages of the suggested scheme. The predictor stage is dependent on a local parameter $({\alpha }_{i+\frac{1}{2}}^n)$ that allows for diffusion control. The balance conservation equation is recovered in the corrector stage. The proposed approach is well-balanced, conservative, and straightforward. Several 1D and 2D test cases are produced after presenting the shallow water model and the numerical technique. In the 1D case, we compared the proposed scheme with the Rusanov scheme, mR with constant $\alpha$ and analytical solutions. The numerical simulation demonstrates the mR's great resolution and attests to its capacity to produce accurate simulations of the shallow water equation with horizontal density gradients. Our results demonstrate that the mR technique is a highly effective instrument for solving a variety of equations in applied science and developed physics.