Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Aaron Brunk, Herbert Egger, Oliver Habrich

TL;DR

The paper addresses numerical approximation for a coupled Cahn-Hilliard/Allen-Cahn system with cross-kinetic coupling and gradient-dependent non-diagonal mobilities within a periodic domain. It proposes a fully discrete, variational discretization that preserves mass conservation and energy dissipation at the discrete level, and proves existence, stability via a relative-energy framework, and optimal convergence rates of order $O(h^2+\tau^2)$ under minimal regularity. The key contributions include the first complete error analysis for a second-order scheme with cross-kinetic coupling, a rigorous discrete stability theory, and a demonstration of the method’s robustness through numerical tests that confirm energy decay and mass conservation. The results have practical impact for reliable, structure-preserving simulations of complex multicomponent phase-field models, with potential extensions to higher-order schemes and more complex multiphase settings.

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.

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

TL;DR

The paper addresses numerical approximation for a coupled Cahn-Hilliard/Allen-Cahn system with cross-kinetic coupling and gradient-dependent non-diagonal mobilities within a periodic domain. It proposes a fully discrete, variational discretization that preserves mass conservation and energy dissipation at the discrete level, and proves existence, stability via a relative-energy framework, and optimal convergence rates of order under minimal regularity. The key contributions include the first complete error analysis for a second-order scheme with cross-kinetic coupling, a rigorous discrete stability theory, and a demonstration of the method’s robustness through numerical tests that confirm energy decay and mass conservation. The results have practical impact for reliable, structure-preserving simulations of complex multicomponent phase-field models, with potential extensions to higher-order schemes and more complex multiphase settings.

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.
Paper Structure (26 sections, 12 theorems, 78 equations, 3 figures, 1 table)

This paper contains 26 sections, 12 theorems, 78 equations, 3 figures, 1 table.

Table of Contents

  1. Preliminaries
  2. Notation
  3. Assumptions on the parameters
  4. Variational characterization
  5. Basic properties
  6. Discretization scheme and main results
  7. Space discretization
  8. Time discretization
  9. Numerical method
  10. Main results
  11. Discrete stability
  12. Relative energy
  13. Relative energy estimate
  14. Proof of Theorem \ref{['thm:fulldisk']}
  15. Error splitting and projection errors
  16. ...and 11 more sections

Key Result

Theorem 2

$$ Let (A0)--(A5) hold. Then for any $h,\tau>0$ and any choice $\rho_{h,0},\eta_{h,0} \in \mathcal{V}_h$ of initial values, Problem prob:pg has at least one solution. Moreover, any such discrete solution satisfies i.e. mass is conserved and energy is dissipated for all time steps $1 \le n \le N$.

Figures (3)

  • Figure 1: Snapshots of the conserved phase field $\rho$ for simulations with $h=2^{-6}$ and $\tau=0.001\cdot h$.
  • Figure 2: Snapshots of the non-conserved phase field $\eta$ for simulations with $h=2^{-6}$ and $\tau=0.001\cdot h$.
  • Figure 3: Evolution of the energy $\mathcal{E}(\rho,\eta)$ (left) as well as validity of discrete energy-dissipation and mass conservation identities (right) for simulations obtained with $h=2^{-6}$ and $\tau=0.001\cdot h$.

Theorems & Definitions (20)

  • Theorem 2: Existence and properties of discrete solutions
  • proof
  • Theorem 3: Convergence rates and uniqueness
  • Remark 4
  • Lemma 5
  • proof
  • Theorem 6
  • proof
  • Lemma 7: Projection error
  • Lemma 8
  • ...and 10 more