The slop flux method for numerical balance in using roe's approximate riemann solver

Imbalance arises when the Roe's method is directly applied in the shallow water simulation. The reasons are different for the continuity equation and the momentum equations. Based on the Roe's method, a partial surface method is proposed for a perfect balance for the continuity equation. In order to generate a mathematically hyperbolic formulation, the momentum equations are split, which causes incompatibility in the calculation of the momentum equations. In this article a numerical approach named the Slop Flux Method (SFM) is proposed to balance the source terms and the flux gradient based on the finite volume method. The method is first applied to shallow water equations. The model is verified by analytical results of classical test cases with good agreement. Finally the method is applied to a steady flow simulation over a practical complicated topography and the result shows good balance and conservation.