• Title/Summary/Keyword: Cahn-Hilliard equation

Search Result 23, Processing Time 0.03 seconds

A DISCONTINUOUS GALERKIN METHOD FOR THE CAHN-HILLIARD EQUATION

  • CHOO S. M.;LEE Y. J.
    • Journal of applied mathematics & informatics
    • /
    • v.18 no.1_2
    • /
    • pp.113-126
    • /
    • 2005
  • The Cahn-Hilliard equation is modeled to describe the dynamics of phase separation in glass and polymer systems. A priori error estimates for the Cahn-Hilliard equation have been studied by the authors. In order to control accuracy of approximate solutions, a posteriori error estimation of the Cahn-Hilliard equation is obtained by discontinuous Galerkin method.

A CONSERVATIVE NONLINEAR DIFFERENCE SCHEME FOR THE VISCOUS CAHN-HILLIARD EQUATION

  • Choo, S.M.;Chung, S.K.
    • Journal of applied mathematics & informatics
    • /
    • v.16 no.1_2
    • /
    • pp.53-68
    • /
    • 2004
  • Numerical solutions for the viscous Cahn-Hilliard equation are considered using the Crank-Nicolson type finite difference method which conserves the mass. The corresponding stability and error analysis of the scheme are shown. The decay speeds of the solution in $H^1-norm$ are shown. We also compare the evolution of the viscous Cahn-Hilliard equation with that of the Cahn-Hilliard equation numerically and computationally, which has been given as an open question in Novick-Cohen[13].

ASYMTOTIC BEHAVIOUR OF THE VISCOUS CAHN-HILLIARD EQUATION

  • Choo, S.M.;Chung, S.K.
    • Journal of applied mathematics & informatics
    • /
    • v.11 no.1_2
    • /
    • pp.143-154
    • /
    • 2003
  • Analytical solutions for the viscous Cahn-Hilliard equation are considered. Existence and uniqueness of the solution are shown. The exponential decay of the solution in H$^2$-norm, which is an improvement of the result in Elliott and Zheng[5]. We also compare the early stages of evolution of the viscous Cahn-Hilliard equation with that of the Cahn-Hilliard equation, which has been given as an open question in Novick-Cohen[8].

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.

L^INFINITY ERROR ESTIMATES FOR FINITE DIFFERENCE SCHEMES FOR GENERALIZED CAHN-HILLIARD AND KURAMOTO-SIVASHINSKY EQUATIONS

  • Choo, S.M.
    • Journal of applied mathematics & informatics
    • /
    • v.23 no.1_2
    • /
    • pp.571-579
    • /
    • 2007
  • Finite difference schemes are considered for a generalization of the Cahn-Hilliard equation with Neumann boundary conditions and the Kuramoto-Sivashinsky equation with a periodic boundary condition, which is of the type $ut+\frac{{\partial}^2} {{\partial}x^2}\;g\;(u,\;u_x,\;u_{xx})=f(u,\;u_x,\;u_{xx})$. Stability and $L^{\infty}$ error estimates of approximate solutions for the corresponding schemes are obtained using the extended Lax-Richtmyer equivalence theorem.

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.

FINITE ELEMENT SCHEME FOR THE VISCOUS CAHN-HILLIARD EQUATION WITH A NONCONSTANT GRADIENT ENERGY COEFFICIENT

  • CHOO S. M.;KIM Y. H.
    • Journal of applied mathematics & informatics
    • /
    • v.19 no.1_2
    • /
    • pp.385-395
    • /
    • 2005
  • A finite element scheme is considered for the viscous Cahn-Hilliard equation with the nonconstant gradient energy coefficient. The scheme inherits energy decay property and mass conservation as for the classical solution. We obtain the corresponding error estimate using the extended Lax-Richtmyer equivalence theorem.

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.

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 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.