[Paper Review] The operator-splitting method for Cahn-Hilliard is stable

Dong Li, Chaoyu Quan|arXiv (Cornell University)|Jul 3, 2021

Solidification and crystal growth phenomena23 references4 citations

TL;DR

This paper establishes the first unconditional energy stability for the operator-splitting method applied to the Cahn-Hilliard equation on the 2D periodic torus. By combining semigroup estimates, Sobolev embedding, and induction, the authors prove uniform bounds on $ H^{k_0} $ norms and energy decay for small time steps $ \tau < \tau_* $, ensuring long-term stability and convergence of the numerical scheme under minimal assumptions on initial data.

ABSTRACT

We prove energy stability of a standard operator-splitting method for the Cahn-Hilliard equation. We establish uniform bound of Sobolev norms of the numerical solution and convergence of the splitting approximation. This is the first unconditional energy stability result for the operator-splitting method for the Cahn-Hilliard equation. Our analysis can be extended to many other models.

Motivation & Objective

• To establish the first unconditional energy stability result for the operator-splitting method applied to the Cahn-Hilliard equation.
• To prove uniform boundedness of the numerical solution in $ H^{k_0} $ norms for all time steps, under minimal regularity assumptions on initial data.
• To extend the analysis to other gradient-flow models by developing a general framework for stability and convergence.
• To resolve the open problem of energy stability for operator-splitting schemes in the context of the Cahn-Hilliard equation.

Proposed method

• The method employs a standard Strang splitting: first solving the linear diffusion part $ \partial_t u = -\nu \Delta^2 u $ via the semigroup $ S_L(\tau) = e^{-\tau \nu \Delta^2} $, then solving the nonlinear part $ \partial_t u = \Delta(u^3 - u) $ via $ S_N(\tau)w = w + \tau \Delta(w^3 - w) $.
• The numerical scheme is defined as $ u^{n+1} = S_L(\tau) S_N(\tau) u^n $, with time step $ \tau > 0 $, and analyzed on the 2D periodic torus $ \mathbb{T}^2 $.
• Key estimates rely on smoothing properties of the semigroup $ e^{-\tau \nu \Delta^2} $, including $ L^p $-boundedness and decay of high-frequency modes via spectral multipliers.
• The proof uses induction and bootstrap arguments to control $ L^\infty $, $ H^1 $, and higher-order Sobolev norms of the solution, leveraging embedding theorems and energy-type inequalities.
• A crucial step involves bounding the discrete energy $ E_1(u^{n+1}) $, defined as a combination of $ \||\nabla|^{-1}(e^{\tau \nu \Delta^2} - 1)^{1/2} u^{n+1}\|_{L^2}^2 $ and the potential energy $ \int ((u^{n+1})^2 - 1)^2 dx $.
• The analysis is extended to $ H^{k_0} $ regularity via recursive expansion of the scheme and application of smoothing estimates for the heat semigroup.

Experimental results

Research questions

• RQ1 Can the operator-splitting method for the Cahn-Hilliard equation be proven unconditionally energy stable, without restrictive assumptions on the numerical solution?
• RQ2 Does the numerical solution remain uniformly bounded in $ H^{k_0} $ norms for all time steps, given initial data in $ H^{k_0} $?
• RQ3 What is the minimal time step $ \tau_* $ that guarantees stability and convergence of the splitting scheme?
• RQ4 Can the framework be generalized to other gradient-flow models with similar structure?
• RQ5 How do the $ L^\infty $ and $ H^1 $ norms of the numerical solution behave under the splitting scheme?

Key findings

• The paper proves unconditional energy stability for the operator-splitting method applied to the Cahn-Hilliard equation, with $ E_1(u^{n+1}) \leq E_1(u^n) $ for all $ n \geq 1 $, under a time step restriction.
• A uniform bound $ \sup_{n \geq 0} \|u^n\|_{H^{k_0}} \leq B_1 < \infty $ is established for initial data in $ H^{k_0}(\mathbb{T}^2) $, $ k_0 \geq 2 $, with $ B_1 $ depending on $ \|u^0\|_{H^{k_0}} $, $ \nu $, and $ k_0 $.
• The time step restriction is $ \tau < \tau_* = c \cdot \min\{ \alpha^{-8}, \alpha^{-8/3} \} \nu^3 $, where $ \alpha $ depends on initial data and $ \nu $, with $ c > 0 $ a small absolute constant.
• An $ L^\infty $ bound is derived: $ \sup_n \|u^n\|_{\infty} \leq \alpha (\nu \tau)^{-1/8} + \alpha \tau (\nu \tau)^{-7/8} $, showing controlled growth under small $ \tau $.
• The scheme is shown to converge in $ H^1 $ and $ L^\infty $, with convergence rates implied by the uniform bounds and error estimates in the proof.
• The analysis is robust and extends to other models with similar gradient-flow structure, particularly those involving $ \Delta^2 $ and double-well potential terms.