Fourier Spectral Methods for Phase Field and Interface Dynamics: Coarsening and Pattern Formation in Energy-Based Models

Fourier Spectral Methods for Phase Field and Interface Dynamics: Coarsening and Pattern Formation in Energy-Based Models | Authorea try { document.documentElement.classList.add('js'); } catch (e) { } var _gaq = _gaq || []; _gaq.push(['_setAccount', 'G-8VDV14Y67G']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })(); Skip to main content Preprints Collections Wiley Open Research IET Open Research Ecological Society of Japan All Collections About About Authorea FAQs Contact Us Quick Search anywhere Search for preprint articles, keywords, etc. Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed. Data may be preliminary. 19 August 2025 V1 Latest version Share on Fourier Spectral Methods for Phase Field and Interface Dynamics: Coarsening and Pattern Formation in Energy-Based Models Authors : Kolade M. Owolabi 0000-0001-9290-3458 , Edson Pindza , and Eben Mare 0000-0001-5966-3296 [email protected] Authors Info & Affiliations https://doi.org/10.22541/au.175561927.73365905/v1 234 views 174 downloads Contents Abstract Supplementary Material Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract Phase-field models have become essential tools for simulating interface dynamics, microstructure evolution, and pattern formation in materials science, fluid mixtures, and biological systems. This paper presents a comprehensive study of Fourier spectral methods applied to three prototypical phase-field models: the Allen–Cahn equation, the Cahn-Hilliard equation, and the Phase-Field Crystal (PFC) model. We review the mathematical formulation of each model, deriving them from variational free-energy functionals and highlighting their roles in capturing phenomena such as domain coarsening, spinodal decomposition, and periodic microstructure formation. A detailed account of numerical solution strategies is provided, with emphasis on Fourier spectral discretization in space, time-integration schemes (including semi-implicit and exponential time-differencing methods), stability and convergence considerations, and techniques for aliasing control in nonlinear term evaluations. To investigate the spatiotemporal dynamics of each model, we conduct extensive numerical simulations in both two and three spatial dimensions. These simulations illustrate a range of pattern formation and coarsening dynamics, including evolving morphologies, lattice structures, and interface instabilities. We present visualizations of microstructure evolution, perform spectral analyses of the evolving fields, and compare dynamics across dimensions. Selected applications are discussed in materials science (e.g., alloy microstructure and grain coarsening), fluid dynamics (e.g., spinodal decomposition and multiphase interactions), and biological systems (e.g., tissue modeling and morphogenesis). Finally, we critically assess the advantages and limitations of Fourier spectral methods for phase-field simulations and outline possible extensions, including multi-component systems, stochastic fluctuations, and anisotropic effects. The aim is to demonstrate that Fourier spectral methods, when applied in both 2D and 3D, provide an accurate and efficient framework for simulating energy-based interface dynamics, while also indicating key challenges and future research directions. Supplementary Material File (main2.pdf) Download 5.00 MB Information & Authors Information Version history V1 Version 1 19 August 2025 Copyright This work is licensed under a Non Exclusive No Reuse License. Keywords fourier spectral methods interface dynamics microstructure evolution pattern formation phase-field models Authors Affiliations Kolade M. Owolabi 0000-0001-9290-3458 University of Pretoria Department of Mathematics and Applied Mathematics View all articles by this author Edson Pindza University of South Africa Department of Decision Sciences View all articles by this author Eben Mare 0000-0001-5966-3296 [email protected] University of Pretoria Department of Mathematics and Applied Mathematics View all articles by this author Metrics & Citations Metrics Article Usage 234 views 174 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Kolade M. Owolabi, Edson Pindza, Eben Mare. Fourier Spectral Methods for Phase Field and Interface Dynamics: Coarsening and Pattern Formation in Energy-Based Models. Authorea . 19 August 2025. DOI: https://doi.org/10.22541/au.175561927.73365905/v1 If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download. For more information or tips please see 'Downloading to a citation manager' in the Help menu . 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