Higher order operator splitting Fourier spectral methods for the Allen-Cahn equation

TL;DR

This paper develops higher order operator splitting Fourier spectral methods for the Allen-Cahn equation, improving stability and accuracy through novel third and fourth order schemes tested on wave and spinodal decomposition problems.

Contribution

It introduces new third and fourth order operator splitting Fourier spectral methods with enhanced stability and accuracy for solving the Allen-Cahn equation.

Findings

Third order methods effectively handle negative time steps.

Fourth order methods demonstrate higher convergence accuracy.

Numerical tests confirm improved stability and convergence rates.

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. 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.