two-layer shallow water formula with slope and uneven bottom solved by finite volume method

This paper proposes a numerical approach to solve a two-layer shallow water formula with a slope and uneven bottom. the finite volume method (fvm) is applied to solve the shallow water model because the method is suitable for computational fluid dynamics problems. rather than pointwise approximations at grid points, the fvm breaks the domain into grid cells and approximates the total integral over grid cells. the shallow water model is examined in two cases, the shallow water model in the steady state and the unsteady state. the quadratic upstream interpolation for convective kinetics (quick) is chosen to get the discretization of the space domain since it is a third-order scheme, which provides good accuracy, and this scheme proves its numerical stability. the advantage of the quick method is that the main coefficients are positive and satisfy the requirements for conservativeness, boundedness, and transportation. an explicit scheme is used to get the discretization of the time domain. finally, the numerical solution of the steady state model shows that the flow remains unchanged. an unsteady-state numerical solution produces instability (wavy at the bottom layer). moreover, the larger slope results in higher velocity and higher depth at the second layer.