A steady state capturing and preserving method for computing hyperbolic systems with geometrical source terms having concentrations

We propose a simple well-balanced method named the slope selecting method which is efficient in both steady state capturing and preserving for hyperbolic system with geometrical source terms having concentrations. Physical problems under consideration include the shallow water equations with discontinuous topography, and the quasi-one-dimensional nozzle flows with discontinuous cross-sectional area. This method is an extension from the interface type method developed in [S. Jin, X. Wen, An efficient method for computing hyperbolic systems with geometrical source terms having concentrations, J. Comput. Math. 22 (2004) 230–249]. The slope selecting method keeps two merits of the previous method. It can be applied when the homogeneous system solver is available and has efficient steady state capturing property. Compared with the previous method, the slope selecting method has two improvements. One is this method also has satisfactory steady state preserving property. The other is this method can be applied to any conservative scheme for the homogeneous system. Numerical examples provide strong evidence on the effectiveness of this slope selecting method for various unsteady, steady and quasi-steady state solutions calculations as well as the flexibility of this method of being applicable to any conservative scheme for the homogeneous system.