On a rough AUSM scheme for a one-dimensional two-phase model

We are interested in exploring Advection Upstream Splitting Method (AUSM) schemes for hyperbolic systems of conservation laws which do not allow any analytical calculation of the Jacobian. For this purpose, we consider a two-phase model which has been used for modeling of unsteady compressible flow of oil and gas in pipes. The model consists of two mass conservation equations, one for each phase, and a common momentum equation. Since no analytical Jacobian can be obtained it is more difficult to use classical schemes such as Roe- and Godunov-type schemes. We propose an AUSM scheme for the current two-phase model obtained through natural generalizations of ideas described in M.-S. Liou [J. Comput. Phys. 129(2) (1996) 364]. A main feature of AUSM is simplicity and efficiency since it does not require the Jacobian. In particular, we prove that the proposed AUSM type scheme preserves the positivity of scalar quantities such as pressure, fluid densities and volume fractions. This guarantees that the scheme can handle the important and delicate case of transition from two-phase to single-phase flow without introducing negative masses. Many numerical results are included to confirm the accuracy and robustness of the proposed AUSM scheme. In particular, it is demonstrated that the AUSM scheme gives low numerical dissipation at volume fraction contact discontinuities and is able to produce stable and non-oscillatory solutions, also when more complex slip relations are used, that is, when the relative motion of one phase with respect to the other is more or less complicated. This makes the scheme suitable for simulations of many important two-phase flow processes. (C) 2003 Elsevier Science Ltd. All rights reserved.