Review of thermodynamic structures and structure-preserving discretisations of Cahn--Hilliard-type models

TL;DR

This review discusses the thermodynamic foundations of Cahn-Hilliard models and examines structure-preserving discretisation methods that ensure conservation, stability, and energy decay in numerical simulations.

Contribution

It provides a comprehensive comparison of thermodynamic structures and discretisation strategies for Cahn-Hilliard-type models, highlighting how to preserve physical properties numerically.

Findings

Comparison of thermodynamic structures and variational principles.

Analysis of discretisation methods preserving mass and energy.

Discussion of trade-offs in large-scale simulations.

Abstract

The Cahn-Hilliard equation and extensions, notably the Cahn-Hilliard-Darcy and Cahn-Hilliard-Navier-Stokes systems, provide widely used frameworks for coupling interfacial thermodynamics with flow. This review surveys the thermodynamic structures underlying these models, focusing on the formulation of free energy functionals, dissipation mechanisms, and variational principles. We compare structural properties, emphasizing how these models encode conservation laws and energy dissipation. A central theme is the translation of these thermodynamic structures into numerical practice by providing representative discretisation strategies that aim to preserve mass conservation, stability, and energy decay. Particular attention is paid to the trade-offs between accuracy, efficiency, and structure preservation in large-scale simulations.