Stability of the Semi-Implicit Method for the Cahn-Hilliard Equation with Logarithmic Potentials

The Cahn-hilliard Equation

1. Steady states. There are many two component systems in which phase separation can be induced by rapidly cooling the system. Thus, if a two component system, which is spatially uniform at temperature T1, is rapidly cooled to a second sufficiently lower temperature T2, then the cooled system will separate into regions of higher and lower concentration. A phenomenological description of the beh...

The Cahn–Hilliard Equation

The Fourier spectral method for the Cahn-Hilliard equation

In this paper, a Fourier spectral method for numerically solving Cahn-Hilliard equation with periodic boundary conditions is developed. We establish their semi-discrete and fully discrete schemes that inherit the energy dissipation property and mass conservation property from the associated continuous problem. we prove existence and uniqueness of the numerical solution and derive the optimal er...

Unconditional Energy Stability Analysis of a Second Order Implicit-Explicit Local Discontinuous Galerkin Method for the Cahn-Hilliard Equation

Abstract In this article, we present a second-order in time implicit-explicit (IMEX) local discontinuous Galerkin (LDG) method for computing the Cahn-Hilliard equation, which describes the phase separation phenomenon. It is well-known that the Cahn-Hilliard equation has a nonlinear stability property, i.e., the free-energy functional decreases with respect to time. The discretized Cahn-Hilliard...

A discontinuous Galerkin method for the Cahn-Hilliard equation

A discontinuous Galerkin finite element method has been developed to treat the high-order spatial derivatives appearing in the Cahn–Hilliard equation. The Cahn–Hilliard equation is a fourth-order nonlinear parabolic partial differential equation, originally proposed to model phase segregation of binary alloys. The developed discontinuous Galerkin approach avoids the need for mixed finite elemen...