Positivity-preserving nonstaggered central difference schemes solving the two-layer open channel flows

This paper aims to propose positivity-preserving nonstaggered central difference schemes solving the two-layer shallow water equations with a nonflat bottom topography and geometric channels. One key step is to discretize the bed slope source term and the nonconservative product term due to the exchange of the momentum based on defining auxiliary variables. The second-order accuracy in space can be obtained by constructing nonoscillatory piecewise linear polynomials. In order to guarantee the strong-stability-preserving (SSP) property and second-order accuracy in time, we use a SSP Nonstaggered Runge-Kutta method. We prove that the designed nonstaggered central difference schemes can obtain a well-balanced property for the steady-state stationary solution. In particular, the positivity of layer heights is guaranteed by providing an appropriate CFL condition with the aid of a "draining" time-step technique. Finally, we conduct several classical problems of the twolayer open channel flows. Numerical results confirm that the current nonstaggered central difference scheme is efficient, accurate, and robust.