An explicit conservative Saul’yev scheme for the Cahn–Hilliard equation

Abstract

We present an explicit conservative Saul’yev finite difference scheme for the Cahn–Hilliard (CH) equation, which models a phase separation phenomenon in binary alloys. The CH equation has been successfully used in various scientific and practical applications. A variety of numerical algorithms were developed to efficiently calculate the CH equation. Because of the highly nonlinear term and the biharmonic operator, numerical methods were mostly implicit schemes. Although a fully explicit scheme is very simple, the time-step restriction is very stringent and the stable time step size is not practicable in high-dimensional spaces. To overcome this severe time-step restriction and retain the simplicity of the explicit method for the CH model, we develop an explicit conservative numerical method based on the Saul’yev method. The proposed scheme has four main merits: (i) the phase-field variable can be directly updated without iterative algorithms; (ii) the numerical solution remains stable even if relatively larger time steps are used; (iii) the mass conservation of the CH equation can be satisfied; and (iv) the simulations in complex domains are easy to implement. The computational experiments confirm the superior performance of the proposed algorithm.