• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.26 no.4
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• pp.333-342
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• 2022

In this paper, we perform a direct comparison study of real experimental data for domain rearrangement and the Cahn-Hilliard (CH) equation on the dynamics of morphological evolution. To validate a mathematical model for physical phenomena, we take initial conditions from experimental images by using an image segmentation technique. The image segmentation algorithm is based on the Mumford-Shah functional and the Allen-Cahn (AC) equation. The segmented phase-field profile is similar to the solution of the CH equation, that is, it has hyperbolic tangent profile across interfacial transition region. We use unconditionally stable schemes to solve the governing equations. As a test problem, we take domain rearrangement of lipid bilayers. Numerical results demonstrate that comparison of the evolutions with experimental data is a good benchmark test for validating a mathematical model.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.27 no.4
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• pp.272-280
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• 2023

We study the numerical analysis for the Cahn-Hilliard (CH) equation using the decoupled projection (DP) method. The CH equation is a fourth order nonlinear partial differential equation that is hard to solve. Therefore, various of numerical schemes have been proposed to solve the CH equation. To verify the relation of each existing scheme for the CH equation, we consider the DP method for linear convex splitting schemes. We present the numerical experiments to demonstrate our analysis. Throughout this study, it is expected to construct a novel numerical scheme using the relation with existing numerical schemes.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.23 no.1
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• pp.1-18
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• 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.

• Journal of applied mathematics & informatics
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• v.29 no.5_6
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• pp.1129-1141
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• 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.

• Journal of the Korean Mathematical Society
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• v.42 no.6
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• pp.1121-1136
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• 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.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.17 no.3
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• pp.197-207
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• 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.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.19 no.1
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• pp.47-56
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• 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.

• Communications of the Korean Mathematical Society
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• v.22 no.3
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• pp.453-464
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• 2007

A computationally efficient numerical scheme is presented for the phase-field model of two-phase systems for anisotropic interfacial energy. The scheme is solved by using a nonlinear multigrid method. When the coefficient for the anisotropic interfacial energy is sufficiently high, the interface of the system shows corners or missing crystallographic orientations. Numerical simulations with high and low anisotropic coefficients show excellent agreement with exact equilibrium shapes. We also present spinodal decomposition, which shows the robustness of the pro-posed scheme.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.13 no.3
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• pp.225-236
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• 2009

Image inpainting is the process of reconstructing lost or deteriorated parts of images using information from surrounding areas. We propose a computationally efficient and fast phase-field method which uses automatic switching parameter, adaptive time step, and automatic stopping of calculation. The algorithm is based on an energy functional. We demonstrate the performance of our new method and compare it with a previous method.

• Bulletin of the Korean Mathematical Society
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• v.52 no.4
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• pp.1059-1068
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• 2015

The paper is devoted to studying a fourth-order degenerate parabolic equation, which arises in fluid, phase transformation and biology. Based on the existence and uniqueness of one semi-discrete problem, two types of approximate solutions are introduced. By establishing some necessary uniform estimates for those approximate solutions, the existence and uniqueness of the corresponding parabolic problem are obtained. Moreover, the long time asymptotic behavior is established by the entropy functional method.