  This article deals with a simultaneous abstract evolution equation. This
includes a parabolic-hyperbolic phase-field system as an example which consists
of a parabolic equation for the relative temperature coupled with a semilinear
damped wave equation for the order parameter. Although a time discretization of
an initial value problem for an abstract evolution equation has been studied,
time discretizations of initial value problems for simultaneous abstract
evolution equations seem to be not studied yet. In this paper we focus on a
time discretization of a simultaneous abstract evolution equation applying to
parabolic-hyperbolic phase-field systems. Moreover, we can establish an error
estimate for the difference between continuous and discrete solutions.
