[Paper Review] An energy stable fourth order finite difference scheme for the Cahn-Hilliard equation

Kelong Cheng, Wenqiang Feng+2 more|arXiv (Cornell University)|Dec 17, 2017

Solidification and crystal growth phenomena25 references3 citations

TL;DR

This paper proposes a second-order accurate, energy-stable finite difference scheme for the Cahn-Hilliard equation using a long-stencil fourth-order finite difference spatial discretization and a second-order BDF temporal scheme with artificial regularization. The key contribution is a rigorous discrete ℓ² truncation error analysis via Fourier analysis, reducing regularity requirements to H⁶ and enabling optimal convergence in the ℓ∞(0,T;ℓ²) ∩ ℓ²(0,T;Hₕ²) norm with proven unique solvability and energy stability.

ABSTRACT

In this paper we propose and analyze an energy stable numerical scheme for the Cahn-Hilliard equation, with second order accuracy in time and the fourth order finite difference approximation in space. In particular, the truncation error for the long stencil fourth order finite difference approximation, over a uniform numerical grid with a periodic boundary condition, is analyzed, via the help of discrete Fourier analysis instead of the the standard Taylor expansion. This in turn results in a reduced regularity requirement for the test function. In the temporal approximation, we apply a second order BDF stencil, combined with a second order extrapolation formula applied to the concave diffusion term, as well as a second order artificial Douglas-Dupont regularization term, for the sake of energy stability. As a result, the unique solvability, energy stability are established for the proposed numerical scheme, and an optimal rate convergence analysis is derived in the $\ell^\infty (0,T; \ell^2) \cap \ell^2 (0,T; H_h^2)$ norm. A few numerical experiments are presented, which confirm the robustness and accuracy of the proposed scheme.

Motivation & Objective

• To develop a high-order, energy-stable numerical scheme for the Cahn-Hilliard equation with improved spatial accuracy.
• To reduce the regularity requirement for the solution by introducing a discrete ℓ² truncation error estimate using Fourier analysis instead of Taylor expansion.
• To establish unique solvability and energy stability for a second-order temporal discretization with artificial regularization.
• To derive optimal convergence rates in the ℓ∞(0,T;ℓ²) ∩ ℓ²(0,T;Hₕ²) norm.
• To demonstrate the robustness and accuracy of the scheme through numerical experiments.

Proposed method

• A long-stencil fourth-order finite difference approximation is used for spatial discretization, with truncation error analyzed via discrete Fourier analysis instead of Taylor expansion.
• A second-order BDF temporal scheme is combined with second-order extrapolation for the concave diffusion term and an artificial Douglas-Dupont regularization term to ensure energy stability.
• The discrete ℓ² norm is used to estimate truncation error, leading to a reduced regularity requirement (H⁶) compared to classical C⁶ bounds.
• Fourier spectral analysis is applied in 2D and 3D to control aliasing errors and eigenvalues in the truncation error estimate.
• The energy stability is proven via a discrete energy dissipation law, and unique solvability is established through a fixed-point argument.
• Optimal convergence is derived in the ℓ∞(0,T;ℓ²) ∩ ℓ²(0,T;Hₕ²) norm using energy estimates and discrete Sobolev inequalities.

Experimental results

Research questions

• RQ1 Can a fourth-order finite difference scheme for the Cahn-Hilliard equation achieve energy stability with second-order temporal accuracy?
• RQ2 How can the truncation error of a long-stencil fourth-order finite difference be analyzed under reduced regularity assumptions?
• RQ3 What is the impact of using discrete Fourier analysis instead of Taylor expansion on the regularity requirement for the test function?
• RQ4 Can optimal convergence rates be established in the ℓ∞(0,T;ℓ²) ∩ ℓ²(0,T;Hₕ²) norm for such a scheme?
• RQ5 How does the proposed scheme compare in accuracy and robustness to existing second-order accurate schemes?

Key findings

• The proposed scheme achieves second-order accuracy in time and fourth-order accuracy in space, with optimal convergence in the ℓ∞(0,T;ℓ²) ∩ ℓ²(0,T;Hₕ²) norm.
• The truncation error is bounded in the discrete ℓ² norm with a bound proportional to h⁴, under an H⁶ regularity assumption, reducing the required regularity compared to classical C⁶ estimates.
• Energy stability is rigorously proven via a discrete energy dissipation law, ensuring the scheme preserves the physical energy decay property.
• Unique solvability of the discrete system is established through a fixed-point argument based on the scheme's structure.
• The discrete Fourier analysis approach successfully controls aliasing errors and eigenvalue behavior in 2D and 3D, enabling the ℓ² truncation error estimate.
• Numerical experiments confirm the scheme's robustness, accuracy, and optimal convergence rates in both 2D and 3D.