scholarly journals The Spectral Method for the Cahn-Hilliard Equation with Concentration-Dependent Mobility

2012 ◽  
Vol 2012 ◽  
pp. 1-35 ◽  
Author(s):  
Shimin Chai ◽  
Yongkui Zou
Keyword(s):  
Theoretical Analysis ◽  
Convergence Analysis ◽  
Spectral Method ◽  
Numerical Experiments ◽  
Numerical Solutions ◽  
Type Equation ◽  
Euler Method ◽  
Implicit Euler Method ◽  
Hilliard Equation ◽  
Cahn Hilliard Equation

This paper is concerned with the numerical approximations of the Cahn-Hilliard-type equation with concentration-dependent mobility. Convergence analysis and error estimates are presented for the numerical solutions based on the spectral method for the space and the implicit Euler method for the time. Numerical experiments are carried out to illustrate the theoretical analysis.

scholarly journals A Stabilized Fourier Spectral Method for the Fractional Cahn-Hilliard Equation

2018 ◽  
Vol 1 (3) ◽  
Author(s):  
Liquan Mei
Keyword(s):  
Spectral Method ◽  
Numerical Experiments ◽  
Second Order ◽  
Fourier Spectral Method ◽  
Numerical Schemes ◽  
Hilliard Equation ◽  
Discretization Schemes ◽  
Energy Stable ◽  
Cahn Hilliard Equation ◽  
Crank Nicolson

In this paper, the second order accurate (in time) energy stable numerical schemes are presented for the Fractional Cahn-Hilliard (CH) equation. Combining the stabilized technique, we apply the implicit Crank-Nicolson formula (CN) to derive second order temporal accuracy, and we use the Fourier spectral method for space discrete to obtain the fully discretization schemes. It is shown that the schemes are unconditionally energy stable. A few numerical experiments are presented to conclude the article.

scholarly journals On the sharp interface limit of a model for phase separation on biological membranes

Analysis ◽  
2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Helmut Abels ◽  
Johannes Kampmann
Keyword(s):  
Lipid Raft ◽  
Cell Membranes ◽  
System Modeling ◽  
Bulk Diffusion ◽  
Type Equation ◽  
Sharp Interface ◽  
Interface Problem ◽  
Hilliard Equation ◽  
System A ◽  
Cahn Hilliard Equation

AbstractWe rigorously prove the convergence of weak solutions to a model for lipid raft formation in cell membranes which was recently proposed in [H. Garcke, J. Kampmann, A. Rätz and M. Röger, A coupled surface-Cahn–Hilliard bulk-diffusion system modeling lipid raft formation in cell membranes, Math. Models Methods Appl. Sci. 26 2016, 6, 1149–1189] to weak (varifold) solutions of the corresponding sharp-interface problem for a suitable subsequence. In the system a Cahn–Hilliard type equation on the boundary of a domain is coupled to a diffusion equation inside the domain. The proof builds on techniques developed in [X. Chen, Global asymptotic limit of solutions of the Cahn–Hilliard equation, J. Differential Geom. 44 1996, 2, 262–311] for the corresponding result for the Cahn–Hilliard equation.

A Uniquely Solvable, Energy Stable Numerical Scheme for the Functionalized Cahn–Hilliard Equation and Its Convergence Analysis

2018 ◽  
Vol 76 (3) ◽  
pp. 1938-1967 ◽  
Author(s):  
Wenqiang Feng ◽  
Zhen Guan ◽  
John Lowengrub ◽  
Cheng Wang ◽  
Steven M. Wise ◽  
...  
Keyword(s):  
Convergence Analysis ◽  
Numerical Scheme ◽  
Hilliard Equation ◽  
Energy Stable ◽  
Cahn Hilliard Equation

scholarly journals The Cahn–Hilliard Equation with Generalized Mobilities in Complex Geometries

2019 ◽  
Vol 2019 ◽  
pp. 1-10
Author(s):  
Jaemin Shin ◽  
Yongho Choi ◽  
Junseok Kim
Keyword(s):  
Growth Rate ◽  
Finite Difference ◽  
Difference Scheme ◽  
Pore Size ◽  
Finite Difference Scheme ◽  
Numerical Experiments ◽  
Complex Geometries ◽  
Hilliard Equation ◽  
Controlled Porosity ◽  
Cahn Hilliard Equation

In this study, we apply a finite difference scheme to solve the Cahn–Hilliard equation with generalized mobilities in complex geometries. This method is conservative and unconditionally gradient stable for all positive variable mobility functions and complex geometries. Herein, we present some numerical experiments to demonstrate the performance of this method. In particular, using the fact that variable mobility changes the growth rate of the phases, we employ space-dependent mobility to design a cylindrical biomedical scaffold with controlled porosity and pore size.

scholarly journals A Note on the Solutions for a Higher-Order Convective Cahn–Hilliard-Type Equation

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1835
Author(s):  
Giuseppe Maria Coclite ◽  
Lorenzo di Ruvo
Keyword(s):  
Type Equation ◽  
Higher Order ◽  
Crystal Surfaces ◽  
Hilliard Equation ◽  
The Cauchy Problem ◽  
The Mean ◽  
Oil Water ◽  
Cahn Hilliard Equation ◽  
Well Posed ◽  
Dynamics Of Phase Transitions

The higher-order convective Cahn-Hilliard equation describes the evolution of crystal surfaces faceting through surface electromigration, the growing surface faceting, and the evolution of dynamics of phase transitions in ternary oil-water-surfactant systems. In this paper, we study the H3 solutions of the Cauchy problem and prove, under different assumptions on the constants appearing in the equation and on the mean of the initial datum, that they are well-posed.

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