Conservative nonlinear difference scheme for the Cahn-Hilliard equation

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

Convergence analysis of a fully discrete finite difference scheme for the Cahn-Hilliard-Hele-Shaw equation

We present an error analysis for an unconditionally energy stable, fully discrete finite difference scheme for the Cahn-Hilliard-Hele-Shaw equation, a modified Cahn-Hilliard equation coupled with the Darcy flow law. The scheme, proposed in [47], is based on the idea of convex splitting. In this paper, we rigorously prove first order convergence in time and second order convergence in space. Ins...

A compact fourth-order finite difference scheme for the three-dimensional Cahn-Hilliard equation

This work extends the previous two-dimensional compact scheme for the Cahn–Hilliard equation (Lee et al., 2014) to three-dimensional space. The proposed scheme, derived by combining a compact formula and a linearly stabilized splitting scheme, has second-order accuracy in time and fourth-order accuracy in space. The discrete system is conservative and practically stable. We also implement the c...

A semidiscrete scheme for a one-dimensional Cahn-Hilliard equation

We analyze a semidiscrete scheme for the Cahn-Hilliard equation in one space dimension, when the interface length parameter is equal to zero. We prove convergence of the scheme for a suitable class of initial data, and we identify the limit equation. We also characterize the long-time behavior of the limit solutions. keywords: Nonconvex functionals, forward-backward parabolic equations, finite ...