Analysis of a time discretization scheme for a nonstandard viscous Cahn-Hilliard system

TL;DR

This paper introduces a new time discretization scheme for a nonstandard viscous Cahn-Hilliard system, proving its convergence and providing an error estimate, which advances numerical analysis of phase field models.

Contribution

The paper develops and rigorously analyzes a novel time discretization scheme for a complex phase field system, establishing convergence and error bounds.

Findings

Proves convergence of the discretization scheme to the continuous solution.

Establishes an error estimate of order one with respect to the time step.

Provides uniform estimates and handles nonlinearities effectively.

Abstract

In this paper we propose a time discretization of a system of two parabolic equations describing diffusion-driven atom rearrangement in crystalline matter. The equations express the balances of microforces and microenergy; the two phase fields are the order parameter and the chemical potential. The initial and boundary-value problem for the evolutionary system is known to be well posed. Convergence of the discrete scheme to the solution of the continuous problem is proved by a careful development of uniform estimates, by weak compactness and a suitable treatment of nonlinearities. Moreover, for the difference of discrete and continuous solutions we prove an error estimate of order one with respect to the time step.