  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.
