TY - JOUR
T1 - A semi-analytical Fourier spectral method for the Allen-Cahn equation
AU - Lee, Hyun Geun
AU - Lee, June Yub
N1 - 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 ).
PY - 2014/8
Y1 - 2014/8
N2 - 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.
AB - 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.
KW - First and second order convergence
KW - Phase-field method
KW - Stabilized semi-implicit Fourier spectral method
KW - Unconditional stability
UR - http://www.scopus.com/inward/record.url?scp=84904696558&partnerID=8YFLogxK
U2 - 10.1016/j.camwa.2014.05.015
DO - 10.1016/j.camwa.2014.05.015
M3 - Article
AN - SCOPUS:84904696558
SN - 0898-1221
VL - 68
SP - 174
EP - 184
JO - Computers and Mathematics with Applications
JF - Computers and Mathematics with Applications
IS - 3
ER -