  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.
