A new well-balanced non-oscillatory central scheme for the shallow water equations on rectangular meshes

This paper is concerned with the development of high-order well-balanced central schemes to solve the shallow water equations in two spatial dimensions. A Runge–Kutta scheme with a natural continuous extension is applied for time discretization. A Gaussian quadrature rule is used to evaluate time integrals and a three-degree polynomial which calculates point-values from cell averages or flux values by avoiding the increase in the number of solution extrema at the interior of each cell is used as a reconstruction operator. That polynomial also guarantees that the number of extrema does not exceed the initial number of extrema and thus it avoids spurious numerical oscillations in the computed solution. A new procedure has been defined to evaluate the flux integrals and to approach the 2D source term integrals in order to verify the exact CC-property, using the water surface elevation instead of the water depth as a variable. Numerical experiments have confirmed the high-resolution properties of our numerical scheme in 2D test problems. The well-balanced property of the resulting scheme has also been investigated.