[Paper Review] Revisit of Semi-Implicit Schemes for Phase-Field Equations

Tao Tang|arXiv (Cornell University)|Jun 12, 2020

Solidification and crystal growth phenomena9 references4 citations

TL;DR

This paper revisits semi-implicit schemes for the Allen-Cahn equation with general potential functions, proving that the discrete maximum principle and energy stability hold under a time-step restriction. The scheme treats the Laplacian term implicitly and the nonlinear term explicitly, ensuring bounded solutions and monotonic energy decay without requiring convex splitting or stabilization terms.

ABSTRACT

It is a very common practice to use semi-implicit schemes in various computations, which treat selected linear terms implicitly and the nonlinear terms explicitly. For phase-field equations, the principal elliptic operator is treated implicitly to reduce the associated stability constraints while the nonlinear terms are still treated explicitly to avoid the expensive process of solving nonlinear equations at each time step. However, very few recent numerical analysis is relevant to semi-implicit schemes, while "stabilized" schemes have become very popular. In this work, we will consider semi-implicit schemes for the Allen-Cahn equation with {\em general potential} function. It will be demonstrated that the maximum principle is valid and the energy stability also holds for the numerical solutions. This paper extends the result of Tang \& Yang (J. Comput. Math., 34(5):471--481, 2016) which studies the semi-implicit scheme for the Allen-Cahn equation with {\em polynomial potentials}.

Motivation & Objective

• To establish rigorous stability guarantees for semi-implicit schemes in phase-field models, particularly for the Allen-Cahn equation with general nonlinear potentials.
• To demonstrate that the maximum principle and energy stability hold for the semi-implicit scheme without requiring convex splitting or stabilization terms.
• To extend prior results on polynomial potentials to general smooth potential functions, enhancing the generality of existing stability frameworks.
• To provide a theoretical foundation for the effectiveness of semi-implicit schemes in long-time simulations, where stability and boundedness are critical.

Proposed method

• Uses a semi-implicit time discretization where the Laplacian term is treated implicitly and the nonlinear term explicitly, solving the resulting linear system at each time step.
• Applies monotone scheme arguments and mathematical induction to prove the discrete maximum principle under a time-step condition involving the derivative of the potential function.
• Employs discrete integration by parts and Taylor expansion to analyze the energy evolution and establish energy stability.
• Derives a time-step restriction based on the maximum of the derivative of the potential function, ensuring energy decay.
• Considers periodic boundary conditions and uses a finite difference spatial discretization with central differences for the Laplacian.
• Relies on the boundedness of the solution, which is guaranteed by the maximum principle, to control the energy evolution.

Experimental results

Research questions

• RQ1 Does the semi-implicit scheme for the Allen-Cahn equation with a general potential function satisfy the discrete maximum principle?
• RQ2 Can energy stability be proven for the semi-implicit scheme without convex splitting or stabilization terms?
• RQ3 Is the time-step restriction based on the derivative of the potential function sufficient to ensure both maximum principle and energy stability?
• RQ4 Can the stability results for polynomial potentials be extended to general smooth potential functions?
• RQ5 What conditions on the potential and time step are necessary and sufficient for energy decay and solution boundedness in the semi-implicit scheme?

Key findings

• The discrete maximum principle holds for the semi-implicit scheme if the time step satisfies Δt ≤ 1 / max|f′(u)| over the solution range, ensuring γ₋ ≤ φⱼⁿ ≤ γ₊.
• Energy stability is proven: the discrete energy Eₕ(φⁿ⁺¹) ≤ Eₕ(φⁿ) for all n, under the same time-step condition.
• The proof relies on the boundedness of the solution (from the maximum principle) and the use of discrete integration by parts and Taylor expansion.
• The energy decay is established via a bound involving the second derivative of the potential, with the time-step restriction ensuring non-increasing energy.
• The result extends prior work on polynomial potentials to general smooth potentials, broadening the applicability of semi-implicit schemes.
• The framework suggests that similar energy stability may hold for the Cahn-Hilliard equation with logarithmic free energy, though deeper analysis is required.