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

We propose two simple well-balanced methods for hyperbolic systems 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 needs only a black-box (approximate) Riemann solver for the homogeneous system. Compared with our previous method developed in [S. Jin and X. Wen, J. Comput. Math., 22 (2004), pp. 230--249], 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 subcritical and supercritical 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.