A practical and efficient numerical method for the Cahn–Hilliard equation in complex domains

Abstract

In this article, we present a practical and efficient numerical method for the Cahn–Hilliard (CH) equation in the two- and three-dimensional complex domains. We propose a simple mathematical model for the binary mixture in the complex domains. The model is based on the ternary CH system. An arbitrary domain is represented by the third phase, which is fixed during the temporal evolution of the other phases. By the local conservative property of the sum of the phases, the governing equation is simplified to a binary CH equation with a source term. For the numerical solution, we use a practically unconditionally gradient stable scheme. Various numerical experiments are performed on arbitrary domains. The numerical results show that the proposed algorithm can deal with the complex domains efficiently.