[Paper Review] A second-order accurate structure-preserving scheme for the Cahn-Hilliard equation with a dynamic boundary condition

Makoto Okumura, T. Fukao+2 more|arXiv (Cornell University)|Jul 16, 2020

Solidification and crystal growth phenomena29 references4 citations

TL;DR

This paper proposes a second-order accurate, structure-preserving finite difference scheme for the Cahn–Hilliard equation with a dynamic boundary condition using the discrete variational derivative method (DVDM). By reformulating the energy discretization and applying a summation-by-parts formula, the scheme achieves second-order spatial accuracy, preserves mass conservation and energy dissipation, and ensures stability, existence, and uniqueness of solutions, validated through numerical examples.

ABSTRACT

We propose a structure-preserving finite difference scheme for the Cahn-Hilliard equation with a dynamic boundary condition using the discrete variational derivative method (DVDM). In this approach, it is important and essential how to discretize the energy which characterizes the equation. By modifying the conventional manner and using an appropriate summation-by-parts formula, we can use a standard central difference operator as an approximation of an outward normal derivative on the discrete boundary condition of the scheme. We show that our proposed scheme is second-order accurate in space, although the previous structure-preserving scheme by Fukao-Yoshikawa-Wada (Commun. Pure Appl. Anal. 16 (2017), 1915-1938) is first-order accurate in space. Also, we show the stability, the existence, and the uniqueness of the solution for the proposed scheme. Computation examples demonstrate the effectiveness of the proposed scheme. Especially through computation examples, we confirm that numerical solutions can be stably obtained by our proposed scheme.

Motivation & Objective

• To develop a high-order accurate numerical scheme for the Cahn–Hilliard equation with dynamic boundary conditions that preserves key physical structures.
• To overcome the first-order spatial accuracy limitation of prior structure-preserving schemes.
• To ensure numerical stability, existence, and uniqueness of solutions through structure-preserving discretization.
• To validate the scheme’s performance through long-time simulation examples showing stable and physically consistent behavior.

Proposed method

• The scheme employs the discrete variational derivative method (DVDM) to preserve the energy dissipation and mass conservation properties of the continuous problem.
• A modified energy discretization is introduced, combined with a specific summation-by-parts formula, to enable second-order accuracy in space.
• The outward normal derivative on the boundary is approximated using a standard central difference operator via careful treatment of the discrete boundary conditions.
• The scheme is formulated in a semi-implicit form, with the chemical potential $ P $ updated using a convex splitting of the free energy.
• Discrete boundary conditions enforce homogeneous Neumann conditions on both $ u $ and $ p $, ensuring consistency with the continuous problem.
• The scheme is proven to preserve mass and dissipate energy, with stability, existence, and uniqueness of the solution established via energy estimates.

Experimental results

Research questions

• RQ1 Can a structure-preserving finite difference scheme for the Cahn–Hilliard equation with dynamic boundary conditions achieve second-order spatial accuracy?
• RQ2 How can the discrete boundary treatment be designed to preserve the physical structure while achieving higher-order accuracy?
• RQ3 Does the proposed scheme maintain mass conservation and energy dissipation in long-time simulations?
• RQ4 What is the impact of the dynamic boundary condition on the long-time behavior of the solution compared to standard Neumann conditions?

Key findings

• The proposed scheme achieves second-order spatial accuracy, improving upon the first-order accuracy of the prior structure-preserving scheme by Fukao–Yoshikawa–Wada.
• The scheme preserves mass conservation to within 14 orders of magnitude in numerical experiments, confirming its structure-preserving nature.
• Energy dissipation is preserved numerically, with the discrete energy functional decreasing monotonically over time.
• The solution remains stable over long-time simulations, with no observed numerical blow-up or oscillations.
• Numerical results confirm that the dynamic boundary condition leads to different long-time behavior compared to the standard Neumann boundary condition.
• The discrete energy functional $ A_{ m d}^{(n)} $ is conserved to within 9 orders of magnitude, further validating the scheme’s structure preservation.