  We construct with full rigorous mathematical proof a family of approximate
solutions to the Cauchy problem for the standard system of two fluid flows with
energy equations and we pass to the limit by weak compactness to obtain Radon
measures that satisfy the equations in a natural weak sense. Our method
provides a convergent numerical method for the numerical calculation of these
Radon measures by reducing the system of partial differential equations in the
case of these approximate solutions to a system of ordinary differential
equations. We observe numerically on the standard Toumi shock tube problem that
the Radon measures from our method agree with the numerical solutions
previously obtained by other authors with various different numerical methods.
In a subsequent numerical paper, using a standard confident scheme with
splittings and vanishing viscosity (independent on the above construction), we
observe exactly the numerical solution given by our mathematical proof.
