Well-balanced unstaggered central schemes for one and two-dimensional shallow water equation systems

We propose a new well-balanced unstaggered central finite volume scheme for hyperbolic balance laws with geometrical source terms. In particular we construct a new one and two-dimensional finite volume method for the numerical solution of shallow water equations on flat/variable bottom topographies. The proposed scheme evolves a non-oscillatory numerical solution on a single grid, avoids the time consuming process of solving Riemann problems arising at the cell interfaces, and is second-order accurate both in space and time. Furthermore, the numerical scheme follows a well-balanced discretization that first discretizes the geometrical source term according to the discretization of the flux terms, and then mimics the surface gradient method and discretizes the water height according to the discretization of the water level. The resulting scheme exactly satisfies the C-property at the discrete level. The proposed scheme is then applied and classical one and two-dimensional shallow water equation problems with flat or variable bottom topographies are successfully solved. The obtained numerical results are in good agreement with corresponding ones appearing in the recent literature, thus confirming the potential and efficiency of the proposed method.