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In this paper, we implement exponential integrators, specifically IntegratingFactor (IF) and Exponential Time Differencing (ETD) methods, usingpseudo-spectral techniques to solve phase-field equations within a Pythonframework. These exponential integrators have showcased robust performance andaccuracy when addressing stiff nonlinear partial differential equations. Wecompare these integrators to the well-known implicit-explicit (IMEX) Eulerintegrators used in phase-field modeling. The synergy between pseudo-spectraltechniques and exponential integrators yields significant benefits for modelingintricate systems governed by phase-field dynamics, such as solidificationprocesses and pattern formation. Our comprehensive Python implementationillustrates the effectiveness of this combined approach in solving phase-fieldmodel equations. The results obtained from this implementation highlight theaccuracy and computational advantages of the ETD method compared to othernumerical techniques.

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