A semi-analytical Fourier spectral method for the Allen-Cahn equation

Hyun Geun Lee, June Yub Lee

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In recent years, Fourier spectral methods have been widely used as a powerful tool for solving phase-field equations. To improve its effectiveness, many researchers have employed stabilized semi-implicit Fourier spectral (SIFS) methods which allow a much larger time step than a usual explicit scheme. Our mathematical analysis and numerical experiments, however, suggest that an effective time step is smaller than a time step specified in the SIFS schemes. In consequence, the SIFS scheme is inaccurate for a considerably large time step and may lead to incorrect morphologies in phase separation processes. In order to remove the time step constraint and guarantee the accuracy in time for a sufficiently large time step, we present a first and a second order semi-analytical Fourier spectral (SAFS) methods for solving the Allen-Cahn equation. The core idea of the methods is to decompose the original equation into linear and nonlinear subequations, which have closed-form solutions in the Fourier and physical spaces, respectively. Both the first and the second order methods are unconditionally stable and numerical experiments demonstrate that our proposed methods are more accurate than the stabilized semi-implicit Fourier spectral method.

Original languageEnglish
Pages (from-to)174-184
Number of pages11
JournalComputers and Mathematics with Applications
Issue number3
StatePublished - Aug 2014

Bibliographical note

Funding Information:
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education ( 2009-0093827 , 2012-002298 ).


  • First and second order convergence
  • Phase-field method
  • Stabilized semi-implicit Fourier spectral method
  • Unconditional stability


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