A linear convex splitting scheme for the Cahn–Hilliard equation with a high-order polynomial free energy

In this article, we present an unconditionally energy stable linear scheme for the Cahn–Hilliard equation with a high-order polynomial free energy. The classical Cahn–Hilliard equation does not satisfy the maximum principle; hence the order parameter can be shifted out of the minimum values of the double-well potential. We adopt a high-order polynomial potential to diminish this effect and employ the efficient linear convex splitting scheme. Since the stabilizing factor gradually increases as the degree of potential becomes greater, we modify a non-physical part of potential as a fourth-order polynomial to reduce the stabilizing factor. Numerical results as well as theoretical results demonstrate the accuracy and energy stability of our method. Furthermore, we verify that some limitations arising from applications of the classical Cahn–Hilliard model can be resolved by adopting a high-order free energy.