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

Aaron Brunk, Marco F. P. ten Eikelder, Marvin Fritz, Dennis Höhn, Dennis Trautwein

TL;DR

This review analyzes the thermodynamic structure of Cahn–Hilliard-type models (CH, CHD, CHNS) and surveys structure-preserving discretisations that conserve mass and ensure energy dissipation. It details finite-element based discretisations with time-discrete energy laws, including a time-averaged potential derivative approach and fully implicit interfacial treatment, to maintain stability in long-time simulations. The paper compares CH, CHD, and CHNS in terms of conservation, momentum balance, and dissipation, and demonstrates these properties through 2D numerical experiments. The findings highlight the trade-offs between accuracy, efficiency, and structure preservation, and emphasize how discrete energy laws guide robust large-scale simulations of multiphase flows.

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.

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

TL;DR

This review analyzes the thermodynamic structure of Cahn–Hilliard-type models (CH, CHD, CHNS) and surveys structure-preserving discretisations that conserve mass and ensure energy dissipation. It details finite-element based discretisations with time-discrete energy laws, including a time-averaged potential derivative approach and fully implicit interfacial treatment, to maintain stability in long-time simulations. The paper compares CH, CHD, and CHNS in terms of conservation, momentum balance, and dissipation, and demonstrates these properties through 2D numerical experiments. The findings highlight the trade-offs between accuracy, efficiency, and structure preservation, and emphasize how discrete energy laws guide robust large-scale simulations of multiphase flows.

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.
Paper Structure (8 sections, 1 theorem, 15 equations, 3 figures, 1 table)

This paper contains 8 sections, 1 theorem, 15 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Models
  3. Discretisation
  4. Cahn--Hilliard system
  5. Cahn--Hilliard--Darcy system
  6. Cahn--Hilliard--Navier--Stokes system
  7. Mass balance and energy dissipation
  8. Experiments

Key Result

theorem 1

Solutions of all schemes conserve mass $\langle \phi_h^{n+1},1 \rangle = \langle \phi_h^n,1 \rangle$.

Figures (3)

  • Figure 1: Snapshots of $\phi$ for CH, CHD, CHNS from top to bottom.
  • Figure 2: Comparison of mass conservation and energy evolution for CH, CHNS, CHD.
  • Figure 3: Comparison of local divergence-freedom for CHD and CHNS on logarithmic scale (left) and conservation errors of linear and angular momentum for CHNS (right).

Theorems & Definitions (2)

  • theorem 1
  • proof