Energy methods 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

Parallel domain decomposition methods for the 3D Cahn-Hilliard equation

Domain decomposition methods are studied in a scalable parallel solver for the Cahn-Hilliard equation in 3D. The discretization is based on a stabilized implicit cell-centered finite difference scheme together with an adaptive time-stepping strategy. A Newton-KrylovSchwarz algorithm is applied to solve the nonlinear system of equations arising at each time step. In the Schwarz preconditioner, w...

On large time-stepping methods for the Cahn–Hilliard equation

In this work, we will analyze a class of large time-stepping methods for the Cahn–Hilliard equation. The equation is discretized by Fourier spectral method in space and semi-implicit schemes in time. For first-order semi-implicit scheme, the stability and convergence properties are investigated based on an energy approach. Here stability means that the decay of energy is preserved. The numerica...

The convective Cahn-Hilliard equation

We consider the convective Cahn–Hilliard equation with periodic boundary conditions as an infinite dimensional dynamical system and establish the existence of a compact attractor and a finite dimensional inertial manifold that contains it. Moreover, Gevrey regularity of solutions on the attractor is established and used to prove that four nodes are determining for each solution on the attractor...