This paper is devoted to the numerical approximation of the solutions of a system of conservation laws arising 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 one. The severe nonlinearities encoded in these laws make the classical approximate Riemann solvers virtually intractable at a reasonable cost of evaluation. We propose a strategy for relaxing solely these two nonlinearities. The relaxation system we introduce is of course hyperbolic but all associated eigenfields are linearly degenerate. Such a property not only makes it trivial to solve the Riemann problem but also enables us to enforce some further stability requirements, in addition to those coming from a Chapman-Enskog analysis. The new method turns out to be fairly simple and robust while achieving desirable positivity properties on the density and the mass fractions. Extensive numerical evidences are provided.

Abstract. This paper could have been given the title: "How to positively and implicitly solve Euler equations using only linear scalar advections." The new relaxation method we propose is able to solve Euler-like systems-as well as initial and boundary value problems-with real state laws at very low cost, using a hybrid explicit-implicit time integration associated with the Arbitrary Lagrangian-Eulerian formalism. Furthermore, it possesses many attractive properties, such as: (i) the preservation of positivity for densities; (ii) the guarantee of min-max principle for mass fractions; (iii) the satisfaction of entropy inequality, under an expressible bound on the CFL ratio. The main feature that will be emphasized is the design of this optimal time-step, which takes into account data not only from the inner domain but also from the boundary conditions.

A domain decomposition technique is proposed for the computation of the acoustic wave equation in which the bulk modulus and density fields are allowed to be discontinuous at the interfaces. Inside each subdomain, the method presented coincides with the second-order finite difference schemes traditionally used in geophysical modeling. However, the possibility of assigning to each subdomain its own space-step makes numerical simulations much less expensive.Another interest of the method lies in the fact that its hybrid variational formulation naturally leads to exact equations for gridpoints on the interfaces. Transposing Babuška-Brezzi's formalism on mixed and hybrid finite elements provides a suitable functional framework for this domain decomposition formulation and shows that the inf-sup condition remains the basic requirement for convergence to occur.

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.

TRANSIENT SIMULATION OF TWO-PHASE FLOWS IN PIPESTransient simulation of two-phase gas-liquid flow in pipes requires considerable computational efforts. Until recently, most available commercial codes are based on two-fluid models which include one momentum conservation equation for each phase. However, in normal pipe flow, especially in oil and gas transport, the transient response of the system proves to be relatively slow. Thus, it is reasonable to think that simpler forms of the transport equations might suffice to represent transient phenomena. Furthermore, these types of models may be solved using less time-consuming numerical algorithms. SIMULACIîN TRANSITORIA DE LOS FLUJOS DIFçSICOS EN LOS CONDUCTOS.La simulaci-n transitoria de los flujos dif ‡sicos de gas l'quido en los conductos, precisa esfuerzos de c ‡lculo considerables. Hasta hace poco tiempo, la mayor parte de los c-digos de simulaci-n comercialmente disponibles se fundaban en modelos de dos fluidos que aplican una ecuaci-n de conservaci-n de la cantidad de movimiento para cada fase. No obstante, en los procesos normales de transporte de hidrocarburos por medio de conductos, y especialmente al tratarse del transporte de petr-leo y de gas, la respuesta transitoria del sistema se manifiesta ser de forma relativamente lenta. Por todo ello, parece razonable pensar que una forma simplificada de las ecuaciones de transporte, podr'a ser de forma suficiente para la reproducci-n de tales fen-menos transitorios. Adem ‡s, este tipo de modelo se podr'a resolver por v'a digital mediante algoritmos menos costosos en cuanto a tiempos de c ‡lculo.