Lagrange-projection scheme for two-phase flows 1. Applications to realistic test cases

We are concerned with the numerical simulation of two-phase flows representing oil transportation along a 1-D pipeline. More specifically, we wish to show that, for this kind of problems, it is highly recommended to make use of the Lagrange-Projection formalism. Indeed, this approach naturally splits the effects of fast (acoustic) and slow (kinematic) waves, thus enabling us to design numerical schemes with many desirable properties: explicit with respect to slow waves and implicit with respect to fast waves. The design of the overall explicit-implicit Lagrange-Projection method involves many interesting features, among which the positivity of the partial densities is of utmost interest to us. In this paper, the emphasis will be laid on the motivation and the general philosophy of the Lagrange-Projection method we propose. Extensive numerical results for realistic test cases are then presented in order to illustrate the capabilities of the method