Numerical algorithms for the phase-field models using discrete cosine transform

We briefly review of numerical scheme based on the Fourier-spectral approach with discrete cosine transform (DCT) and its implementation. The DCT is a mathematical technique of expressing a set of discrete data as a sum of cosine functions that oscillate at different frequencies. In this study, we apply the DCT to numerically approach phase-field models equipped with homogeneous Neumann boundary conditions. The phase-field model is a powerful mathematical tool used to numerically simulate phase transformations in materials. This model describes many physical phenomena and is especially applicable to various phase transformation problems such as solidification, liquefaction, crystal growth, phase separations, and transitions. One of the most important concepts in the phase-field model is the order parameter. This is a variable that represents the state of the phase and usually has a value between 0 and 1. For example, in a system where solids and liquids coexist, the order parameter is set to 1 in the solid region and 0 in the liquid region. Additionally, the free energy functional calculates the free energy based on the spatial distribution of the order parameter, which is a key factor in determining the phase transformation process of the given system. For instance, phase-field models may include the following equations and properties. The Allen-Cahn equation describes the evolution of phase boundaries, representing the transition between different phases or states in a material system. The Cahn-Hilliard equation serves as a diffuse interface model for describing the spinodal decomposition in binary alloys. The nonlocal CH equation is utilized to simulate the microphase separation occurring within a diblock copolymer composed of distinct monomer types. The Swift-Hohenberg equation captures attention due to its intriguing perspective on pattern formation, owing to its possession of many qualitatively different stable equilibrium solutions. Furthermore, the phase-field crystal equation offers a simple dynamical density functional theory for crystalline solidification. The Fourier-spectral approach with DCT is characterized by both high accuracy and simplicity of implementation. We offer a detailed elucidation of this method along with its association with MATLAB usage, facilitating interested individuals to effortlessly employ the Fourierspectral approach with DCT in their research. To validate the effectiveness of the numerical methods, we perform various standard numerical experiments on phase-field models. Furthermore, the MATLAB code implementation can be found in the appendix.