  We deal with the reduced four-equation model for the dynamics of
heterogeneous compressible binary mixtures with the stiffened gas equations of
state. We study its further reduced form, with the excluded volume
concentrations, and with a quadratic equation for the common pressure of the
components; this form can be called a quasi-homogeneous form. We prove new
properties of the equation, derive simple formulas for the squared speed of
sound and present an alternative proof for a formula that relates it to the
squared Wood speed of sound; also, a short derivation of the pressure balance
equation is given. For the first time, we introduce regularizations of the
heterogeneous model (in the quasi-homogeneous form). Previously,
regularizations of such type were developed only for the homogeneous mixtures
of perfect polytropic gases, and it was unclear how to cover the case
considered here. In the 1D case, based on these regularizations, we construct
new explicit two-level in time and symmetric three-point in space
finite-difference schemes without limiters, and provide numerical results for
various flows with shock waves.
