[Paper Review] An Energy Stable Finite-Difference Scheme for Functionalized Cahn-Hilliard Equation and its Convergence Analysis

Wenqiang Feng, Zhen Guan+3 more|arXiv (Cornell University)|Oct 8, 2016

Solidification and crystal growth phenomena6 references3 citations

TL;DR

This paper proposes an unconditionally energy-stable and convergent finite difference scheme for the functionalized Cahn-Hilliard (FCH) equation, a sixth-order nonlinear parabolic PDE modeling phase separation in amphiphilic systems. By introducing auxiliary terms to convexify a non-convex energy functional, the authors enable a convex-splitting approach that ensures unique solvability and unconditional energy stability, with convergence proven via global $H_{\rm per}^{2}$ stability and an efficient preconditioned steepest descent solver for the $H^{-1}$-gradient flow system.

ABSTRACT

We present and analyze an unconditionally energy stable and convergent finite difference scheme for the Functionalized Cahn-Hilliard equation. One key difficulty associated with the energy stability is based on the fact that one nonlinear energy functional term in the expansion appears as non-convex, non-concave. To overcome this subtle difficulty, we add two auxiliary terms to make the combined term convex, which in turns yields a convex-concave decomposition of the physical energy. As a result, an application of the convex splitting methodology assures both the unique solvability and the unconditional energy stability of the proposed numerical scheme. To deal with a 4-Laplacian solver in an $H^{-1}$ gradient flow at each time step, we apply an efficient preconditioned steepest descent algorithm to solve the corresponding nonlinear systems. In addition, a global in time $H_{ m per}^2$ stability of the numerical scheme is established at a theoretical level, which in turn ensures the full order convergence analysis of the scheme. A few numerical results are presented, which confirm the stability and accuracy of the proposed numerical scheme.

Motivation & Objective

• To develop a numerically stable and convergent scheme for the highly nonlinear, sixth-order functionalized Cahn-Hilliard (FCH) equation arising in amphiphilic phase separation.
• To overcome the challenge of non-convexity in the energy functional, which complicates energy stability analysis and numerical implementation.
• To ensure unconditional energy stability and unique solvability of the numerical scheme without time step restrictions.
• To design an efficient nonlinear solver for the $H^{-1}$-gradient flow system arising at each time step, enabling large-scale simulations.
• To establish a rigorous convergence analysis based on global $H_{\rm per}^{2}$ stability of the numerical scheme.

Proposed method

• Introduce two auxiliary terms to the energy functional to create a convex-concave decomposition, enabling application of the convex splitting methodology.
• Formulate a fully implicit finite difference scheme in time and central differences in space, ensuring unconditional energy stability.
• Apply a preconditioned steepest descent (PSD) algorithm to solve the nonlinear system arising from the $H^{-1}$-gradient flow at each time step.
• Use a variational formulation to derive the chemical potential $\mu$ from the energy functional, which includes a 4-Laplacian term and high-order nonlinearities.
• Establish global $H_{\rm per}^{2}$ stability of the numerical solution over time, which underpins the full-order convergence analysis.
• Implement periodic boundary conditions and use a semi-implicit time discretization to maintain mass conservation and energy dissipation.

Experimental results

Research questions

• RQ1 Can an unconditionally energy-stable finite difference scheme be constructed for the FCH equation despite the non-convexity of its energy functional?
• RQ2 How can the non-convex energy term be reformulated to allow for convex splitting and ensure unique solvability?
• RQ3 What is the convergence order of the proposed scheme, and can it be rigorously proven using stability estimates?
• RQ4 How efficiently can the high-order nonlinear system arising from the $H^{-1}$-gradient flow be solved in practice?
• RQ5 Does the numerical scheme preserve key physical properties such as energy dissipation and mass conservation over long-time simulations?

Key findings

• The proposed scheme achieves unconditional energy stability and unique solvability through a convex-concave decomposition of the energy functional via auxiliary terms.
• The scheme exhibits first-order convergence in time, with observed convergence rates of approximately 2.0–2.5 in space, confirming theoretical predictions.
• The preconditioned steepest descent solver demonstrates robust convergence, with residual reduction by a nearly constant factor per iteration.
• Numerical simulations confirm energy dissipation and mass conservation over long-time evolutions, consistent with physical expectations.
• The scheme successfully captures complex morphological structures such as micelle networks, bilayer-cylinder junctions, and Y-junctions, matching known experimental and simulation results.
• The global $H_{\rm per}^{2}$ stability of the scheme is established at the theoretical level, enabling a full-order convergence analysis.