# HIGHER ORDER OPERATOR SPLITTING FOURIER SPECTRAL METHODS FOR THE ALLEN-CAHN EQUATION

• SHIN, JAEMIN (INSTITUTE OF MATHEMATICAL SCIENCES, EWHA WOMANS UNIVERSITY) ;
• LEE, HYUN GEUN (DEPARTMENT OF MATHEMATICS, KWANGWOON UNIVERSITY) ;
• LEE, JUNE-YUB (DEPARTMENT OF MATHEMATICS, EWHA WOMANS UNIVERSITY)
• Accepted : 2017.02.23
• Published : 2017.03.25

#### Abstract

The Allen-Cahn equation is solved numerically by operator splitting Fourier spectral methods. The basic idea of the operator splitting method is to decompose the original problem into sub-equations and compose the approximate solution of the original equation using the solutions of the subproblems. The purpose of this paper is to characterize higher order operator splitting schemes and propose several higher order methods. Unlike the first and the second order methods, each of the heat and the free-energy evolution operators has at least one backward evaluation in higher order methods. We investigate the effect of negative time steps on a general form of third order schemes and suggest three third order methods for better stability and accuracy. Two fourth order methods are also presented. The traveling wave solution and a spinodal decomposition problem are used to demonstrate numerical properties and the order of convergence of the proposed methods.

#### Acknowledgement

Supported by : National Research Foundation of Korea (NRF)

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