COMPARISON OF DIFFERENT NUMERICAL SCHEMES FOR THE CAHN–HILLIARD EQUATION

Abstract

The Cahn–Hilliard equation was proposed as a phenomenological model for describing the process of phase separation of a binary alloy. The equation has been applied to many physical applications such as a morphological instability caused by elastic non-equilibrium,image inpainting, two- and three-phase fluid flow, phase separation, flow visualization and the formation of the quantum dots. To solve the Cahn–Hillard equation, many numerical methods have been proposed such as the explicit Euler’s, the implicit Euler’s, the Crank–Nicolson, the semi-implicit Euler’s, the linearly stabilized splitting and the non-linearly stabilized splitting schemes. In this paper, we investigate each scheme in finite-difference schemes by comparing their performances, especially stability and efficiency. Except the explicit Euler’s method, we use the fast solver which is called a multigrid method. Our numerical investigation shows that the linearly stabilized stabilized splitting scheme is not unconditionally gradient stable in time unlike the known result. And the Crank–Nicolson scheme is accurate but unstable in time,whereas the non-linearly stabilized splitting scheme has advantage over other schemes on the time step restriction.