Exponential Integrators for Phase-Field Equations using Pseudo-spectral Methods: A Python Implementation

Abstract: In this paper, we implement exponential integrators, specifically Integrating Factor (IF) and Exponential Time Differencing (ETD) methods, using pseudo-spectral techniques to solve phase-field equations within a Python framework. These exponential integrators have showcased robust performance and accuracy when addressing stiff nonlinear partial differential equations. We compare these integrators to the well-known implicit-explicit (IMEX) Euler integrators used in phase-field modeling. The synergy between pseudo-spectral techniques and exponential integrators yields significant benefits for modeling intricate systems governed by phase-field dynamics, such as solidification processes and pattern formation. Our comprehensive Python implementation illustrates the effectiveness of this combined approach in solving phase-field model equations. The results obtained from this implementation highlight the accuracy and computational advantages of the ETD method compared to other numerical techniques.