A moving water equilibria preserving nonstaggered central scheme achieved via flux globalization for the Ripa model

In this paper, we propose a second-order moving-water equilibria preserving non-staggered central scheme to solve the Ripa model via flux globalization. To maintain the moving-water steady states, we use the discrete source terms proposed by Britton et al. (J Sci Comput, 2020, 82(2): Art 30) by incorporating the expression of the source terms as a whole into the flux gradient, which directly avoids the discrete complexity of the source terms in order to maintain the well-balanced properties of the scheme. In addition, since the non-staggered central scheme requires re-projecting the updated values of the nonstaggered cells onto the staggered cells, we modify the calculation of the global variables by constructing ghost cells and alternating the values of the global variables with the water depths obtained from the solution through the nonlinear relationship between the global flux and the water depth. In order to maintain the second-order accuracy of the scheme on the time scale, we incorporate a new Runge-Kutta type time discretization in the evolution of the numerical solution for the nonstaggered cells. Meanwhile, we introduce the “draining” time step technique to ensure that the water depth is positive and that it satisfies mass conservation. Numerical experiments verify that the scheme is well-balanced, positivity-preserving and robust.