Structure-preserving finite difference schemes for the Cahn-Hilliard equation with dynamic boundary conditions in the one-dimensional case

@article{Fukao2017StructurepreservingFD,
  title={Structure-preserving finite difference schemes for the Cahn-Hilliard equation with dynamic boundary conditions in the one-dimensional case},
  author={Takeshi Fukao and Shuji Yoshikawa and Saori Wada},
  journal={Communications on Pure and Applied Analysis},
  year={2017},
  volume={16},
  pages={1915-1938}
}
The structure-preserving finite difference schemes for the one dimensional Cahn-Hilliard equation with dynamic boundary conditions are studied. A dynamic boundary condition is a sort of transmission condition that includes the time derivative, namely, it is itself a time evolution equation. The Cahn-Hilliard equation with dynamic boundary conditions is well-treated from various viewpoints. The standard type consists of a dynamic boundary condition for the order parameter, and the Neumann… 

Figures from this paper

A structure-preserving scheme for the Allen–Cahn equation with a dynamic boundary condition
We propose a structure-preserving finite difference scheme for the Allen–Cahn equation with a dynamic boundary condition using the discrete variational derivative method [ 9 ]. In this method, how to
A second-order accurate structure-preserving scheme for the Cahn-Hilliard equation with a dynamic boundary condition
TLDR
The proposed structure-preserving finite difference scheme for the Cahn–Hilliard equation with a dynamic boundary condition using the discrete variational derivative method (DVDM) is proposed and it is shown that the proposed scheme is second-order accurate in space, although the previous structure- Preserving scheme proposed by Fukao–Yoshikawa–Wada is first-order inaccurate in space.
The Cahn-Hilliard Equation with Forward-Backward Dynamic Boundary Condition via Vanishing Viscosity
An asymptotic analysis for a system with equation and dynamic boundary condition of Cahn–Hilliard type is carried out as the coefficient of the surface diffusion acting on the phase variable tends to
On a transmission problem for equation and dynamic boundary condition of Cahn–Hilliard type with nonsmooth potentials
This paper is concerned with well‐posedness of the Cahn–Hilliard equation subject to a class of new dynamic boundary conditions. The system was recently derived in Liu–Wu (Arch. Ration. Mech. Anal.
Convergence of a Robin boundary approximation for a Cahn–Hilliard system with dynamic boundary conditions
We prove the existence of unique weak solutions to an extension of a Cahn–Hilliard model proposed recently by C Liu and H Wu (2019 Arch. Ration. Mech. Anal. 233 167–247), in which the new dynamic
Numerical Approximations and Error Analysis of the Cahn-Hilliard Equation with Dynamic Boundary Conditions
TLDR
A first-order in time, linear and energy stable numerical scheme, which is based on the stabilized linearly implicit approach, is proposed, which has been proved and the semi-discrete-in-time error estimates are carried out.
Numerical approximations and error analysis of the Cahn-Hilliard equation with reaction rate dependent dynamic boundary conditions
TLDR
A first-order in time, linear and energy stable scheme for solving the Cahn-Hilliard equation with reaction rate dependent dynamic boundary conditions is proposed and the corresponding semi-discretized-in-time error estimates are derived.
Operator splitting for abstract Cauchy problems with dynamical boundary conditions
In this work we study operator splitting methods for a certain class of coupled abstract Cauchy problems, where the coupling is such that one of the problems prescribes a "boundary type" extra
The Cahn–Hilliard equation and some of its variants
Our aim in this article is to review and discuss the Cahn–Hilliard equation, as well as someof its variants. Such variants have applications in, e.g., biology and image inpainting.
...
...

References

SHOWING 1-10 OF 25 REFERENCES
A stable and conservative finite difference scheme for the Cahn-Hilliard equation
TLDR
A stable and conservative finite difference scheme to solve numerically the Cahn-Hilliard equation which describes a phase separation phenomenon and inherits characteristic properties, the conservation of mass and the decrease of the total energy, from the equation.
Convergence of Cahn-Hilliard systems to the Stefan problem with dynamic boundary conditions
TLDR
It is clear that the state of the mushy region of the Stefan problem is characterized by an asymptotic limit of the fourth-order system, which has a double-well structure, which raises the possibility of the numerical application of the Cahn-Hilliard system to the degenerate parabolic equation.
A NUMERICAL ANALYSIS OF THE CAHN-HILLIARD EQUATION WITH DYNAMIC BOUNDARY CONDITIONS
We consider a finite element space semi-discretization of the Cahn- Hilliard equation with dynamic boundary conditions. We prove optimal error estimates in energy norms and weaker norms, assuming
A numerical analysis of the Cahn–Hilliard equation with non-permeable walls
TLDR
The numerical analysis of the Cahn–Hilliard equation in a bounded domain with non-permeable walls, endowed with dynamic-type boundary conditions is considered and the stability of a fully discrete scheme based on the backward Euler scheme for the time discretization is proved.
Numerical analysis of parabolic problems with dynamic boundary conditions
Space and time discretizations of parabolic differential equations with dynamic boundary conditions are studied in a weak formulation that fits into the standard abstract formulation of parabolic
A Variational Approach to a Cahn–Hilliard Model in a Domain with Nonpermeable Walls
We deal with the well-posedness and the long time behavior of a Cahn–Hilliard model with a singular bulk potential and suitable dynamic boundary conditions. We assume that the system is confined in a
Asymptotic behavior of a Cahn-Hilliard equation with Wentzell boundary conditions and mass conservation
In this article, we consider a Cahn-Hilliard model with boundary conditions of Wentzell type and mass conservation. We show that each solution of this problem converges to a steady state as time
...
...