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A First-Order Linear Energy Stable Scheme for the Cahn-Hilliard Equation with Dynamic Boundary Conditions under the Effect of Hyperbolic Relaxation

Minghui Yu, Rui Chen

TL;DR

This work analyzes a hyperbolic Cahn-Hilliard equation with dynamic boundary conditions by introducing inertial terms $\beta_1 \phi_{tt}$ and $\beta_2 \psi_{tt}$. It develops a linear, first-order in time, energy-stable scheme using stabilization ($s_1,s_2$), and provides a rigorous error analysis showing first-order temporal accuracy. Theoretical results are complemented by numerical experiments confirming energy stability, mass conservation, and that the hyperbolic terms slow energy decay and spinodal coarsening. Overall, the paper offers a stable, efficient framework for simulating diffuse-interface dynamics with hyperbolic relaxation and dynamic boundary effects, with potential impact on modeling fast phase transitions and moving contact lines.

Abstract

In this paper we focus on the Cahn-Hilliard equation with dynamic boundary conditions, by adding two hyperbolic relaxation terms to the system. We verify that the energy of the total system is decreasing with time. By adding two stabilization terms, we have constructed a first-order temporal accuracy numerical scheme, which is linear and energy stable. Then we prove that the scheme is of first-order in time by the error estimates. At last we carry out enough numerical results to validate the the temporal convergence and the energy stability of such scheme. Moreover, we have present the differences of the numerical results with and without the hyperbolic terms, which show that the hyperbolic terms can help the total energy decreasing slowly.

A First-Order Linear Energy Stable Scheme for the Cahn-Hilliard Equation with Dynamic Boundary Conditions under the Effect of Hyperbolic Relaxation

TL;DR

This work analyzes a hyperbolic Cahn-Hilliard equation with dynamic boundary conditions by introducing inertial terms and . It develops a linear, first-order in time, energy-stable scheme using stabilization (), and provides a rigorous error analysis showing first-order temporal accuracy. Theoretical results are complemented by numerical experiments confirming energy stability, mass conservation, and that the hyperbolic terms slow energy decay and spinodal coarsening. Overall, the paper offers a stable, efficient framework for simulating diffuse-interface dynamics with hyperbolic relaxation and dynamic boundary effects, with potential impact on modeling fast phase transitions and moving contact lines.

Abstract

In this paper we focus on the Cahn-Hilliard equation with dynamic boundary conditions, by adding two hyperbolic relaxation terms to the system. We verify that the energy of the total system is decreasing with time. By adding two stabilization terms, we have constructed a first-order temporal accuracy numerical scheme, which is linear and energy stable. Then we prove that the scheme is of first-order in time by the error estimates. At last we carry out enough numerical results to validate the the temporal convergence and the energy stability of such scheme. Moreover, we have present the differences of the numerical results with and without the hyperbolic terms, which show that the hyperbolic terms can help the total energy decreasing slowly.

Paper Structure

This paper contains 8 sections, 4 theorems, 77 equations, 13 figures.

Table of Contents

  1. Introduction
  2. The governing equations
  3. A first-order energy stable scheme
  4. Error estimates
  5. Numerical experiments
  6. Temporal accuracy test
  7. The effect of the hyperbolic term
  8. Conclusion

Key Result

Theorem 2.1

The model 2.9-2.13 is energy stable in the sense that where the energy $\mathcal{E}(\phi,\psi)=E^{total}(\phi,\psi)+\frac{\beta_1}{2M_1}||\nabla\Delta^{-1}\phi_t||^2+\frac{\beta_2}{2M_2}||\nabla_{\Gamma}\Delta_{\Gamma}^{-1}\psi_t||_{\Gamma}^2$.

Figures (13)

  • Figure 5.1: The $L^2$ numerical errors for $\phi$ and $\psi$ at $T = 1.0$.
  • Figure 5.2: The initial data of Case 1.
  • Figure 5.3: The energy evolution and the mass evolutions of Case 1.
  • Figure 5.4: Case 1: Snapshots of the numerical approximation are taken at $T= 0.015$, $0.045$, $0.15$, $0.3$, and $0.6$ with different $\beta$. Left: $\beta=1$; Middle: $\beta=0.1$; Right: $\beta=0$.
  • Figure 5.5: The initial data of Case 2.
  • ...and 8 more figures

Theorems & Definitions (8)

  • Remark 2.1
  • Theorem 2.1
  • proof
  • Theorem 3.1
  • proof
  • Lemma 4.1
  • Theorem 4.2
  • proof