An efficient method for computing hyperbolic systems with geometrical source terms having concentrations ∗

We propose a simple numerical method for calculating both unsteady and steady state solution of hyperbolic system with geometrical source terms having concentrations. Physical problems under consideration include the shallow water equations with topography, and the quasi one-dimensional nozzle flows. We use the interface value, rather than the cell-averages, for the source terms, which results in a well-balanced scheme that can capture the steady state solution with a remarkable accuracy. This method approximates the source terms via the numerical fluxes produced by an (approximate) Riemann solver for the homogeneous hyperbolic systems with slight additional computation complexity using Newton’s iterations and numerical integrations. This method solves well the subor super-critical flows, and with a transonic fix, also handles well the transonic flows over the concentration. Numerical examples provide strong evidence on the effectiveness of this new method for both unsteady and steady state calculations. Research supported in part by U.S. National Science Foundation grant No. DMS-0305080, National Natural Science Foundation of China Project 10228101 and the Basic Research Projects of Tsinghua University under the Project JC2002010. Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P.R. China, and Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA. Email address: [email protected]. Department of Mathematical Science, Tsinghua University, Beijing 100084, P.R. China. Email address: [email protected].