A Godunov-type scheme for the isentropic model of a fluid flow in a nozzle with variable cross-section

We present a Godunov-type scheme for the isentropic model of a fluid flow in a nozzle with variable cross-section. The model is of nonconservative form, making it hard for standard numerical discretizations of the nonconservative term. In particular, the error for a standard numerical scheme with a usual discretization of the nonconservative term may become larger as the mesh size gets smaller. We first re-investigate the Riemann problem of the model, pointing out several interesting properties of the wave curves, and establishing specific existence domain for each type of solutions. Then, we incorporate local Riemann solutions to build a Godunov-type scheme for the model. The scheme is constructed in subsonic and supersonic regions, where the system is strictly hyperbolic. Tests show that our scheme can capture standing waves, so that it is well-balanced. Furthermore, tests also show that our Godunov-type scheme can give a good accuracy for numerical approximations of exact solutions. Our Godunov-type scheme can resolve the difficulty of other existing schemes for similar models of fluid flows with nonconservative terms.