2D/3D Pattern Formation Comparison Using Spectral Methods to Solve Nonlinear Partial Differential Equations of Condensed and Soft Matter

It is well known that nonlinear partial differential equations (NLPDEs) can only be solved numerically and that fourth-order NLPDEs in their derivatives require unconventional methods. This paper explains spectral numerical methods for obtaining a numerical solution by Fast Fourier Transform (FFT), implemented under Python in tis version 3.1 and their libraries (NumPy, Tkinter). Examples of NLPDEs typical of Condensed Matter Physics to be solved numerically are the conserved Cahn–Hilliard, Swift–Hohenberg and conserved Swift–Hohenberg equations. The last two equations are solved by the first- and second-order exponential integrator method, while the first of these equations is solved by the conventional FFT method. The Cahn–Hilliard equation, a phase-field model with an extended Ginzburg–Landau-like functional, is solved in two-dimensional (2D) to reproduce the evolution of the microstructure of an amorphous alloy Ce75Al25 − xGax, which is compared with the experimental micrography of the literature. Finally, three-dimensional (3D) simulations were performed using numerical solutions by FFT. The second-order exponential integrator method algorithm for the Swift–Hohenberg equation implementation is successfully obtained under Python by FFT to simulate different 3D patterns that cannot be obtained with the conventional FFT method. All these 2D/3D simulations have applications in Materials Science and Engineering.