[Paper Review] Benchmark Computation of Morphological Complexity in the Functionalized Cahn-Hilliard Gradient Flow

Andrew Christlieb, Keith Promislow+4 more|arXiv (Cornell University)|Jun 8, 2020

Solidification and crystal growth phenomena4 citations

TL;DR

This paper introduces benchmark computations for the functionalized Cahn-Hilliard (FCH) gradient flow to study morphological complexity in amphiphilic systems, using second-order time discretization with Fourier pseudo-spectral spatial methods. It compares BDF2-based schemes—IMEX, SAV, ETD, and fully implicit PSD—finding that SAV and IMEX offer the best computational efficiency, while PSD achieves the highest accuracy at the cost of greater computational effort.

ABSTRACT

Reductions of the self-consistent mean field theory model of amphiphilic molecules in solvent can lead to a singular family of functionalized Cahn-Hilliard energies. We modify these energies, mollifying the singularities to stabilize the computation of the gradient flows and develop a series of benchmark problems that emulate the "morphological complexity" observed in experiments. These benchmarks investigate the delicate balance between the rate of absorption of amphiphilic material onto an interface and a least energy mechanism to disperse the arriving mass. The result is a trichotomy of responses in which two-dimensional interfaces either lengthen by a regularized motion against curvature, undergo pearling bifurcations, or split directly into networks of interfaces. We evaluate a number of schemes that use second order BDF2-type time stepping coupled with Fourier pseudo-spectral spatial discretization. The BDF2-type schemes are either based on a fully implicit time discretization with a PSD nonlinear solver, or upon IMEX, SAV, ETD approaches. All schemes use a fixed local truncation error target with adaptive time-stepping to achieve the error target. Each scheme requires proper "preconditioning" to achieve robust performance that can enhance efficiency by several orders of magnitude.

Motivation & Objective

• To develop physically motivated benchmark problems that replicate the morphological complexity observed in amphiphilic systems during phase separation.
• To investigate the delicate balance between the rate of amphiphilic material arrival at an interface and the energy-minimizing redistribution mechanism.
• To evaluate and compare the performance of multiple second-order time integration schemes for stiff FCH gradient flows under adaptive time-stepping with fixed local truncation error.
• To establish a fair computational efficiency comparison by measuring cost in terms of Fast Fourier Transform (FFT) calls required to achieve a target global error.
• To demonstrate that energy decay must be coupled with accuracy, as energy stability alone can mask incorrect results in highly nonlinear systems.

Proposed method

• Formulate a mollified, nonsingular version of the functionalized Cahn-Hilliard (FCH) energy to stabilize numerical computation of gradient flows.
• Use second-order backward differentiation formula (BDF2) time discretization combined with Fourier pseudo-spectral spatial discretization for high accuracy and spectral convergence.
• Implement four numerical schemes: (1) fully implicit with preconditioned steepest descent (PSD), (2) linearly implicit IMEX, (3) scalar auxiliary variable (SAV), and (4) exponential time differencing (ETD) for comparison.
• Apply adaptive time-stepping with a fixed local truncation error target to ensure consistent accuracy across all schemes.
• Employ preconditioning in all schemes to enhance robustness and efficiency, especially for stiff nonlinear systems.
• Use the SAV approach to ensure unconditional modified energy stability, with provable energy decay properties under the scheme.

Experimental results

Research questions

• RQ1 How do different second-order time integration schemes perform in resolving the morphological complexity of FCH gradient flows under adaptive time-stepping?
• RQ2 What is the trade-off between accuracy and computational cost (measured in FFT calls) when solving stiff FCH systems with different numerical schemes?
• RQ3 Can energy stability alone ensure accurate simulation outcomes, or is accuracy a necessary complement to energy decay in complex phase-field systems?
• RQ4 How does the rate of amphiphilic material delivery influence the selection of interfacial morphologies—curvature-driven motion, pearling, or network formation?
• RQ5 To what extent does preconditioning enhance the performance of nonlinear and linearly implicit schemes in FCH simulations?

Key findings

• The fully implicit BDF2-PSD scheme achieves the smallest global discretization error at a fixed local truncation error, indicating the highest accuracy among the tested schemes.
• The IMEX and SAV schemes are the most computationally efficient, requiring significantly fewer Fast Fourier Transform (FFT) calls than PSD to achieve the same global error level.
• The SAV scheme’s performance closely mirrors that of IMEX, with a factor of 1.4 increase in FFT calls due to the auxiliary variable system.
• The ETD scheme is included for comparison but is outperformed by the IMEX and SAV methods in terms of computational efficiency.
• Preconditioning is essential for robust performance, enabling several orders of magnitude improvement in efficiency across all schemes.
• The benchmark problems successfully reproduce the trichotomy of responses—curvature-driven motion, pearling bifurcations, and curve-splitting into networks—observed in experimental morphological complexity.