A second-order structure-preserving discretization for the Cahn-Hilliard/Allen-Cahn system with cross-kinetic coupling

TL;DR

This paper introduces a novel second-order discretization method for the coupled Cahn-Hilliard/Allen-Cahn system that preserves the gradient flow structure and ensures energy dissipation at the discrete level.

Contribution

It proposes a new fully discrete scheme that maintains the system's structure, guarantees existence and uniqueness, and achieves optimal convergence rates.

Findings

The scheme preserves the gradient flow and energy dissipation.

Optimal convergence rates are proven under minimal assumptions.

Numerical tests confirm theoretical results and method viability.

Abstract

We study the numerical solution of a Cahn-Hilliard/Allen-Cahn system with strong coupling through state and gradient dependent non-diagonal mobility matrices. A fully discrete approximation scheme in space and time is proposed which preserves the underlying gradient flow structure and leads to dissipation of the free-energy on the discrete level. Existence and uniqueness of the discrete solution is established and relative energy estimates are used to prove optimal convergence rates in space and time under minimal smoothness assumptions. Numerical tests are presented for illustration of the theoretical results and to demonstrate the viability of the proposed methods.