Higher order operator splitting Fourier spectral methods for the Allen-Cahn equation
For researchers in computational PDEs, this work provides higher-order numerical methods for phase-field simulations, though it is an incremental extension of existing operator splitting techniques.
The paper develops higher-order operator splitting Fourier spectral methods for the Allen-Cahn equation, proposing three third-order and two fourth-order schemes that handle negative time steps for improved stability and accuracy. Numerical tests on traveling wave and spinodal decomposition problems verify the convergence orders.
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.