A numerical approximation for the standard one pressure system of two fluid flows with energy equations

We study numerically the standard one pressure model of two fluid flows with energy equations. This system is not solved in time derivative. It has been transformed into an equivalent system solved in time derivative. We show that the scheme in this paper applies to both solved and nonsolved systems and gives same results. One usually adds a nonphysical term to render the system hyperbolic. However, explicit solutions and well posedness of the Cauchy problem for some nonlinear nonhyperbolic systems of physics have been obtained in some events by [B. Keyfitz et al.]. We also show that our scheme applies equally well to both versions, with and without the additional term, whether solved in time derivative or not, which provides four versions of the system. We observe that the nonhyperbolic and the hyperbolic systems give very close but slightly different results: the step values are always the same but peaks in gas and liquid velocities are observed in the nonhyperbolic model, which is typically observed in experimental results concerning the gas kick phenomenon though we are unable to say if this result is related or not. The numerical quality of the hyperbolic solved in time derivative system is better, therefore our results (same pressure and temperatures, and same step values in volume fraction and velocities besides the isolated peaks) provide a justification of the additional term that renders it hyperbolic. Another difficulty lies in that these systems are in nonconservative form and therefore its discontinuous solutions cannot make sense in the theory of distributions and it has been observed that different numerical schemes can lead to different discontinuous solutions. For the hyperbolic system this solution is identical to the main one in [S.T. Munkejord, S. Evje, T. Flatten] obtained from completely different methods.