Computational methods for the three-dimensional phase-field equations using discrete cosine transform

초록

This paper introduces the Fourier spectral method based on the discrete cosine transform (DCT). This approach is highly accurate and easy to code. We use it to solve several three-dimensional (3D) phase-field equations. These equations include the Allen–Cahn (AC) equation, Cahn–Hilliard (CH) equation, Swift–Hohenberg (SH) equation, and phase-field crystal (PFC) equation. Such parabolic partial differential equations (PDEs) are important for various scientific studies. Each model has a distinct role and is designed to resolve specific physical phenomena or mechanisms. It captures different aspects of the underlying processes and is used according to the requirements of the problem under consideration. The AC equation is a reaction–diffusion model that describes domain coarsening. The CH equation captures the phase separation of binary mixtures. The SH equation is famous for pattern formation in dissipative systems. The PFC model describes crystal growth at the atomic scale. We provide a clear guide to this numerical method and explain its MATLAB implementation. This allows readers to use the technique easily in their own research. We also perform standard numerical simulations to demonstrate the efficiency of the method. The full MATLAB scripts are provided in the Appendix. © 2026 Elsevier B.V.

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