Relaxation schemes for the calculation of two-phase flow in pipes

In this paper, we are interested in some basic investigations of properties of the relaxation schemes first introduced by Jin and Xin [1]. The main advantages of these schemes are that they neither require the use of Riemann solvers nor the computation of nonlinear flux Jacobians. This can be an important advantage when more complex models are considered where it is not possible to perform analytical calculations of Jacobians and/or when considering fluids with nonstandard equation of state. We apply the schemes (relaxing and relaxed) to a certain two-phase model where Jacobians cannot in general be calculated analytically. We first demonstrate that the original relaxation schemes of Jin and Xin produce a poor approximation for a typical mass transport example which involves transition from two-phase flow to single-phase flow. However, by introducing a slight modification of the original relaxation model by splitting the momentum flux into a mass and pressure part, we obtain some flux splitting relaxation schemes which for typical two-phase flow cases yield a more accurate and robust approximation.