A Semi-implicit Relaxation Scheme for Modeling Two-Phase Flow in a Pipeline

This work is devoted to the numerical approximation of the solutions to the system of conservation laws which arises in the modeling of two-phase flows in pipelines. The PDEs are closed by two highly nonlinear algebraic relations, namely, a pressure law and a hydrodynamic law. We have previously proposed an explicit relaxation scheme which allows us to cope with these nonlinearities. But the system considered has eigenvalues which are of very different orders of magnitude, which prevents the explicit scheme from being effective, since the time step has to be very small. In order to solve this effectiveness problem, we now proceed to construct a scheme which is explicit with respect to the small eigenvalues and linearly implicit with respect to the large eigenvalues. Numerical evidences are provided.