@article{Kim2025, 
author = {Hyundong Kim and Zhengang Li and Xinpei Wu and Hyunho Shin and Yunjae Nam and Junseok Kim},
title = {An unconditionally stable hybrid numerical method for the gradient flow for the high-order Modica–Mortola functional},
year = {2025},
journal = {Electronic Research Archive},
volume = {33},
number = {10},
pages = {6375-6390},
keywords = {Modica–Mortola model, high-order multi-well free energy, multi-phase system, Fourier spectral approach, stable scheme},
url = {https://www.sciopen.com/article/10.3934/era.2025281},
doi = {10.3934/era.2025281},
abstract = {This paper presents a numerically stable, time-accurate algorithm for simulating the gradient flow associated with the Modica–Mortola functional with a uniformly spaced multi-well potential. The scheme uses operator splitting; the nonlinear component is updated analytically, while the linear part is advanced by a Fourier spectral discretization. The method is unconditionally stable, preserves pointwise boundedness independently of the time step size, and attains spectral accuracy in space and first-order accuracy in time. We provide a theoretical analysis establishing unconditional stability and boundedness, and present comprehensive numerical experiments that demonstrate the accuracy and robustness of the proposed approach.}
}