[Paper Review] Stability analysis for the Implicit-Explicit discretization of the Cahn-Hilliard equation

Dong Li, Tao Tang|arXiv (Cornell University)|Aug 9, 2020

Solidification and crystal growth phenomena32 references4 citations

TL;DR

This paper presents a rigorous stability analysis for the implicit-explicit (IMEX) time discretization of the Cahn-Hilliard equation without adding stabilization terms or modifying the problem structure. It establishes conditional energy stability under optimal time step constraints that scale naturally with energy and viscosity, resolving a long-standing open problem in phase field simulations by introducing a novel Trade-Energy-For-L∞ (TEFL) framework to handle the lack of maximum principle and stiffness from small mobility coefficients.

ABSTRACT

Implicit-Explicit methods have been widely used for the efficient numerical simulation of phase field problems such as the Cahn-Hilliard equation or thin film type equations. Due to the lack of maximum principle and stiffness caused by the effect of small dissipation coefficient, most existing theoretical analysis relies on adding additional stabilization terms, mollifying the nonlinearity or introducing auxiliary variables which implicitly either changes the structure of the problem or trades accuracy for stability in a subtle way. In this work we introduce a robust theoretical framework to analyze directly the stability of the standard implicit-explicit approach without stabilization or any other modification. We take the Cahn-Hilliard equation as a model case and prove energy stability under natural time step constraints which are optimal with respect to energy scaling. These settle several questions which have been open since the work of Chen and Shen \cite{CS98}.

Motivation & Objective

• To resolve the long-standing challenge of rigorously proving stability for standard IMEX schemes applied to the Cahn-Hilliard equation.
• To eliminate reliance on artificial stabilization, mollification, or auxiliary variables in theoretical analysis.
• To derive time step constraints that are optimal with respect to energy scaling and physically meaningful in the context of small mobility coefficients.
• To establish a robust theoretical framework—TEFL—that enables direct stability analysis under natural conditions.
• To confirm the robustness of IMEX methods for long-time, large-scale phase field simulations.

Proposed method

• The authors develop a novel Trade-Energy-For-L∞ (TEFL) framework to control the L∞ norm of the solution without modifying the equation or adding stabilization.
• They analyze the standard IMEX scheme proposed by Chen and Shen (2002), treating the nonlinear term explicitly and the linear diffusion implicitly.
• Energy stability is proven via a discrete energy dissipation argument, leveraging the gradient flow structure of the Cahn-Hilliard equation.
• Key estimates involve bounding the L∞ norm of the solution using L2 and H1 norms, with careful control of time step size via scaling analysis.
• The time step constraint is derived by balancing energy decay and L∞ growth, leading to constraints that are optimal in terms of energy scaling.
• The analysis is conducted in 2D and 3D periodic domains using Fourier-based techniques and induction on time steps.

Experimental results

Research questions

• RQ1 Can energy stability be rigorously proven for the standard IMEX scheme of the Cahn-Hilliard equation without additional stabilization?
• RQ2 What are the optimal time step constraints for energy stability in the presence of small mobility coefficients?
• RQ3 How can the lack of maximum principle and stiffness in small-viscosity regimes be handled in theoretical analysis?
• RQ4 Can a direct analysis be performed without introducing auxiliary variables or modifying the nonlinearity?
• RQ5 Is the TEFL framework generalizable to other phase field models with similar stiffness and nonlinearity?

Key findings

• The paper establishes conditional energy stability for the IMEX scheme of the Cahn-Hilliard equation under time step constraints that are optimal with respect to energy scaling.
• The time step constraint is derived as τ ≤ 0.0007 min{β₃⁻⁸, β₄⁻⁸ᐟ³, β₅⁻⁸}, where β₃, β₄, β₅ depend on initial energy and L² norms.
• The analysis avoids stabilization, mollification, or auxiliary variables, providing a direct and structure-preserving proof.
• The TEFL framework successfully controls the L∞ norm of the solution using energy and H¹ bounds, enabling stability under mild time step restrictions.
• The derived time step constraints are shown to be nearly optimal in terms of energy scaling, matching physical expectations from phase separation dynamics.
• The method is generalizable and paves the way for refined stability analysis of higher-order IMEX schemes and other phase field models.