Non-oscillatory central schemes for the Saint-Venant system

The research aims to develop a well-balanced numerical method for solving the shallow water equations, which account for the balance laws and the source term related to the seabed slope. The proposed method combines a Runge-Kutta scheme for accurate time integration and the natural continuous extension method for spatial discretization. To achieve high-order spatial accuracy, the method employs central non-staggered (CNS) reconstructions of the conservative variables and the water surface elevation. This is achieved through two key steps. The initial step involves determining the specific values of the flux derivative and the bed slope source term at individual points. The subsequent step entails integrating the source term spatially. Both of these steps are designed to preserve the C-property, which ensures the exact preservation of the quiescent flow solution. The method is verified using a variety of standard one-dimensional test cases, including smooth and discontinuous solutions, to demonstrate its accuracy and resolution properties.