Numerical approximation of the fractional Cahn-Hilliard equation by operator splitting method

Abstract

In this paper, we consider a fast explicit operator splitting method for a fractional Cahn-Hilliard equation with spatial derivative $(-{\varDelta })^{\frac {\alpha }{2}}$ (α ∈ (1,2]), where the choice α = 2 corresponds to the classical Cahn-Hilliard equation. The original problem is split into linear and nonlinear subproblems. For the linear part, the pseudo-spectral method is adopted, and thus an ordinary differential equation is obtained. For the nonlinear part, a second-order SSP-RK method together with the pseudo-spectral method is used. The stability and convergence of the proposed method in L2-norm are studied. We also carry out a comparative study of two classical definitions for fractional Laplacian $(-{\varDelta })^{\frac {\alpha }{2}}$ , and numerical results obtained using computational simulation of the fractional Cahn-Hilliard equation for a variety of choices of fractional order α are presented. It is observed that the fractional order α controls the sharpness of the interface, which is typically diffusive in integer-order phase-field models.