Two Interface-Type Numerical Methods for Computing Hyperbolic Systems with Geometrical Source Terms Having Concentrations

We propose two simple well-balanced methods for hyperbolic system with geometrical source terms having concentrations. Physical problems under consideration include the shallow water equations with topography, and the quasi one-dimensional isothermal nozzle flows. These two methods use the numerical fluxes already obtained from the corresponding homogeneous systems in the source terms, and one only needs a black-box (approximate) Riemann solver for the homogeneous system. Compared to our previous method developed in [17], these methods avoid the Newton iterations in the evaluation of the source term. Numerical experiments demonstrate that both methods give good numerical approximations to the suband super-critical flows. With a transonic fix, both methods also capture with a high resolution the transonic flows over the concentration. These methods are applicable to both unsteady and steady state computations. Research supported in part by NSF grant No. DMS-0305080, NSFC under the Project 10228101 and the Basic Research Projects of Tsinghua University under the Project JC2002010. Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA, and Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P.R. China. Email address: [email protected]. Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P.R. China. Email address: [email protected].