• Title, Summary, Keyword: Cahn-Hilliard equation

Search Result 16, Processing Time 0.038 seconds

THE EXISTENCE OF GLOBAL ATTRACTOR FOR CONVECTIVE CAHN-HILLIARD EQUATION

  • Zhao, Xiaopeng;Liu, Bo
    • Journal of the Korean Mathematical Society
    • /
    • v.49 no.2
    • /
    • pp.357-378
    • /
    • 2012
  • In this paper, we consider the convective Cahn-Hilliard equation. Based on the regularity estimates for the semigroups, iteration technique and the classical existence theorem of global attractors, we prove that the convective Cahn-Hilliard equation possesses a global attractor in $H^k$($k\geq0$) space, which attracts any bounded subset of $H^k({\Omega})$ in the $H^k$-norm.

SPECTRAL APPROXIMATIONS OF ATTRACTORS FOR CONVECTIVE CAHN-HILLIARD EQUATION IN TWO DIMENSIONS

  • ZHAO, XIAOPENG
    • Bulletin of the Korean Mathematical Society
    • /
    • v.52 no.5
    • /
    • pp.1445-1465
    • /
    • 2015
  • In this paper, the long time behavior of the convective Cahn-Hilliard equation in two dimensions is considered, semidiscrete and completely discrete spectral approximations are constructed, error estimates of optimal order that hold uniformly on the unbounded time interval $0{\leq}t<{\infty}$ are obtained.

DYNAMICAL BIFURCATION OF THE ONE-DIMENSIONAL CONVECTIVE CAHN-HILLIARD EQUATION

  • Choi, Yuncherl
    • Korean Journal of Mathematics
    • /
    • v.22 no.4
    • /
    • pp.621-632
    • /
    • 2014
  • In this paper, we study the dynamical behavior of the one-dimensional convective Cahn-Hilliard equation(CCHE) on a periodic cell [$-{\pi},{\pi}$]. We prove that as the control parameter passes through the critical number, the CCHE bifurcates from the trivial solution to an attractor. We describe the bifurcated attractor in detail which gives the final patterns of solutions near the trivial solution.

A NONLINEAR CONVEX SPLITTING FOURIER SPECTRAL SCHEME FOR THE CAHN-HILLIARD EQUATION WITH A LOGARITHMIC FREE ENERGY

  • Kim, Junseok;Lee, Hyun Geun
    • Bulletin of the Korean Mathematical Society
    • /
    • v.56 no.1
    • /
    • pp.265-276
    • /
    • 2019
  • For a simple implementation, a linear convex splitting scheme was coupled with the Fourier spectral method for the Cahn-Hilliard equation with a logarithmic free energy. However, an inappropriate value of the splitting parameter of the linear scheme may lead to incorrect morphologies in the phase separation process. In order to overcome this problem, we present a nonlinear convex splitting Fourier spectral scheme for the Cahn-Hilliard equation with a logarithmic free energy, which is an appropriate extension of Eyre's idea of convex-concave decomposition of the energy functional. Using the nonlinear scheme, we derive a useful formula for the relation between the gradient energy coefficient and the thickness of the interfacial layer. And we present numerical simulations showing the different evolution of the solution using the linear and nonlinear schemes. The numerical results demonstrate that the nonlinear scheme is more accurate than the linear one.

THREE-DIMENSIONAL VOLUME RECONSTRUCTION BASED ON MODIFIED FRACTIONAL CAHN-HILLIARD EQUATION

  • CHOI, YONGHO;LEE, SEUNGGYU
    • Journal of the Korean Society for Industrial and Applied Mathematics
    • /
    • v.23 no.3
    • /
    • pp.203-210
    • /
    • 2019
  • We present the three-dimensional volume reconstruction model using the modified Cahn-Hilliard equation with a fractional Laplacian. From two-dimensional cross section images such as computed tomography, magnetic resonance imaging slice data, we suggest an algorithm to reconstruct three-dimensional volume surface. By using Laplacian operator with the fractional one, the dynamics is changed to the macroscopic limit of Levy process. We initialize between the two cross section with linear interpolation and then smooth and reconstruct the surface by solving modified Cahn-Hilliard equation. We perform various numerical experiments to compare with the previous research.

A CONSTRAINED CONVEX SPLITTING SCHEME FOR THE VECTOR-VALUED CAHN-HILLIARD EQUATION

  • LEE, HYUN GEUN;LEE, JUNE-YUB;SHIN, JAEMIN
    • Journal of the Korean Society for Industrial and Applied Mathematics
    • /
    • v.23 no.1
    • /
    • pp.1-18
    • /
    • 2019
  • In contrast to the well-developed convex splitting schemes for gradient flows of two-component system, there were few efforts on applying the convex splitting idea to gradient flows of multi-component system, such as the vector-valued Cahn-Hilliard (vCH) equation. In the case of the vCH equation, one need to consider not only the convex splitting idea but also a specific method to manage the partition of unity constraint to design an unconditionally energy stable scheme. In this paper, we propose a constrained Convex Splitting (cCS) scheme for the vCH equation, which is based on a convex splitting of the energy functional for the vCH equation under the constraint. We show analytically that the cCS scheme is mass conserving and unconditionally uniquely solvable. And it satisfies the constraint at the next time level for any time step thus is unconditionally energy stable. Numerical experiments are presented demonstrating the accuracy, energy stability, and efficiency of the proposed cCS scheme.

LARGE TIME-STEPPING METHOD BASED ON THE FINITE ELEMENT DISCRETIZATION FOR THE CAHN-HILLIARD EQUATION

  • Yang, Yanfang;Feng, Xinlong;He, Yinnian
    • Journal of applied mathematics & informatics
    • /
    • v.29 no.5_6
    • /
    • pp.1129-1141
    • /
    • 2011
  • In this paper, a class of large time-stepping method based on the finite element discretization for the Cahn-Hilliard equation with the Neumann boundary conditions is developed. The equation is discretized by finite element method in space and semi-implicit schemes in time. For the first order fully discrete scheme, convergence property is investigated by using finite element analysis. Numerical experiment is presented, which demonstrates the effectiveness of the large time-stepping approaches.

ERROR ESTIMATES OF NONSTANDARD FINITE DIFFERENCE SCHEMES FOR GENERALIZED CAHN-HILLIARD AND KURAMOTO-SIVASHINSKY EQUATIONS

  • Choo, Sang-Mok;Chung, Sang-Kwon;Lee, Yoon-Ju
    • Journal of the Korean Mathematical Society
    • /
    • v.42 no.6
    • /
    • pp.1121-1136
    • /
    • 2005
  • Nonstandard finite difference schemes are considered for a generalization of the Cahn-Hilliard equation with Neumann boundary conditions and the Kuramoto-Sivashinsky equation with periodic boundary conditions, which are of the type $$U_t\;+\;\frac{{\partial}^2}{{\partial}x^2} g(u,\;U_x,\;U_{xx})\;=\;\frac{{\partial}^{\alpha}}{{\partial}x^{\alpha}}f(u,\;u_x),\;{\alpha}\;=\;0,\;1,\;2$$. Stability and error estimate of approximate solutions for the corresponding schemes are obtained using the extended Lax-Richtmyer equivalence theorem. Three examples are provided to apply the nonstandard finite difference schemes.

COMPARISON OF DIFFERENT NUMERICAL SCHEMES FOR THE CAHN-HILLIARD EQUATION

  • Lee, Seunggyu;Lee, Chaeyoung;Lee, Hyun Geun;Kim, Junseok
    • Journal of the Korean Society for Industrial and Applied Mathematics
    • /
    • v.17 no.3
    • /
    • pp.197-207
    • /
    • 2013
  • The Cahn-Hilliard equation was proposed as a phenomenological model for describing the process of phase separation of a binary alloy. The equation has been applied to many physical applications such as amorphological instability caused by elastic non-equilibrium, image inpainting, two- and three-phase fluid flow, phase separation, flow visualization and the formation of the quantum dots. To solve the Cahn-Hillard equation, many numerical methods have been proposed such as the explicit Euler's, the implicit Euler's, the Crank-Nicolson, the semi-implicit Euler's, the linearly stabilized splitting and the non-linearly stabilized splitting schemes. In this paper, we investigate each scheme in finite-difference schemes by comparing their performances, especially stability and efficiency. Except the explicit Euler's method, we use the fast solver which is called a multigrid method. Our numerical investigation shows that the linearly stabilized stabilized splitting scheme is not unconditionally gradient stable in time unlike the known result. And the Crank-Nicolson scheme is accurate but unstable in time, whereas the non-linearly stabilized splitting scheme has advantage over other schemes on the time step restriction.

A MODIFIED CAHN-HILLIARD EQUATION FOR 3D VOLUME RECONSTRUCTION FROM TWO PLANAR CROSS SECTIONS

  • Lee, Seunggyu;Choi, Yongho;Lee, Doyoon;Jo, Hong-Kwon;Lee, Seunghyun;Myung, Sunghyun;Kim, Junseok
    • Journal of the Korean Society for Industrial and Applied Mathematics
    • /
    • v.19 no.1
    • /
    • pp.47-56
    • /
    • 2015
  • In this paper, we present an implicit method for reconstructing a 3D solid model from two 2D cross section images. The proposed method is based on the Cahn-Hilliard model for the image inpainting. Image inpainting is the process of reconstructing lost parts of images based on information from neighboring areas. We treat the empty region between the two cross sections as inpainting region and use two cross sections as neighboring information. We initialize the empty region by the linear interpolation. We perform numerical experiments demonstrating that our proposed method can generate a smooth 3D solid model from two cross section data.