We propose a novel second order in time numerical scheme for Cahn-Hilliard-NavierStokes phase field model with matched density. The scheme is based on second order convex-splitting for the Cahn-Hilliard equation and pressure-projection for the Navier-Stokes equation. We show that the scheme is mass-conservative, satisfies a modified energy law and is therefore unconditionally stable. Moreover, ...

In this paper we develop local discontinuous Galerkin (LDG) methods for the fourth-order nonlinear Cahn-Hilliard equation and system. The energy stability of the LDG methods is proved for the general nonlinear case. Numerical examples for the Cahn-Hilliard equation and the Cahn-Hilliard system in one and two dimensions are presented and the numerical results illustrate the accuracy and capabili...

In this article, we study the initial-boundary-value problem for a sixth-order Cahn-Hilliard type equation ut = D μ, μ = γDu− a(u)Du− a′(u) 2 |Du| + f(u) + kut, which describes the separation properties of oil-water mixtures, when a substance enforcing the mixing of the phases is added. The optimal control of the sixth order Cahn-Hilliard type equation under boundary condition is given and the ...

In this paper, we established traveling wave solutions of the nonlinear evolution equation. The sine-cosine method was used to construct travelling wave solutions of the Fitzhugh-Nagumo equation and Cahn-Allen equation. Graphical interpretation shows that obtained results include periodic and Soliton wave solutions. It is also shown that for α =−1, solutions of Fitzhugh-Nagumo equation coincide...

In this paper, we apply the spectral method to approximate the solutions of Cahn-Hilliard equation, which is a typical class of nonlinear fourth-order diffusion equations. Diffusion phenomena is widespread in the nature. Therefore, the study of the diffusion equation caught wide concern. Cahn-Hilliard equation was proposed by Cahn and Hilliard in 1958 as a mathematical model describing the diff...

In this paper, we present an efficient numerical method for multiphase image segmentation using a multiphase-field model. The method combines the vector-valued Allen– Cahn phase-field equation with initial data fitting terms containing prescribed interface width and fidelity constants. An efficient numerical solution is achieved using the recently developed hybrid operator splitting method for ...

We study the existence of weak solutions to a mixture model for tumour growth that consists of a Cahn–Hilliard–Darcy system coupled with an elliptic reactiondiffusion equation. The Darcy law gives rise to an elliptic equation for the pressure that is coupled to the convective Cahn–Hilliard equation through convective and source terms. Both Dirichlet and Robin boundary conditions are considered ...

Numerical method for Cahn-Hilliard equation has been well-studied, but few can be generalized to fractional Cahn-Hilliard equation. In this project to modified the numerical method proposed by Brian Wetton et al in the paper High accuracy solutions to energy gradient flows from material science models[1]. The method they described in the paper is a pseudo-spectral method suitable for considerin...

A formal asymptotic method is used to derive a differential-algebraic system of equations characterizing the metastable motion of a pattern of n (n ≥ 2) internal layers for the one-dimensional viscous Cahn-Hilliard modeling slow phase separation. Similar slow motion results are obtained for the Cahn-Hilliard equation and the constrained Allen-Cahn equation by introducing a homotopy parameter in...

A phase separation process model formed by the Cahn-Hilliard equation and the heat equation is considered. The mobility coefficient in the Cahn-Hilliard equation is assumed to be an increasing function of the temperature. This property is of importance at the latest stage of the cooling process. The existence and uniqueness of a weak solution is proved, and a uniform cooling problem is studied....