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

## TL;DR

This paper develops a time discretization method for simultaneous abstract evolution equations, exemplified by parabolic-hyperbolic phase-field systems, and provides an error estimate for the approximation.

## Contribution

It introduces a novel time discretization approach for simultaneous evolution equations and establishes an error estimate, addressing a gap in existing research.

## Key findings

- Established a time discretization scheme for simultaneous evolution equations.
- Proved an error estimate between continuous and discrete solutions.
- Applied the method to a parabolic-hyperbolic phase-field system.

## Abstract

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.