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

Time discretization of an initial value problem for a simultaneous abstract evolution equation applying to parabolic-hyperbolic phase-field systems