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

A Godunov-type scheme for the model of a general fluid flow in a nozzle with variable cross-section is presented. The model has the form of a nonconservative system of balance laws, which poses many challenging questions for study. First, exact Riemann solvers in computing form are described in both subsonic and supersonic regions. Second, the computable exact solutions of local Riemann problem are incorporated into a Godunov-type scheme. Third, the scheme is shown to be well-balanced in the sense that it can capture exactly stationary waves. Finally, numerical tests for data belong to both subsonic and supersonic regions are presented. These tests show that the scheme has a very fine accuracy. Especially, the scheme can give very good approximations to the exact solutions even in the resonant phenomenon where the solution contains three waves of the same speed.