# 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…

## 13 Citations

A structure-preserving scheme for the Allen–Cahn equation with a dynamic boundary condition

- MathematicsDiscrete & Continuous Dynamical Systems - A
- 2020

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

- MathematicsCommunications on Pure & Applied Analysis
- 2021

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

- MathematicsSIAM J. Math. Anal.
- 2022

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

- MathematicsMathematische Nachrichten
- 2020

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

- MathematicsNonlinearity
- 2020

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…

Structure-preserving finite difference schemes for nonlinear wave equations with dynamic boundary conditions

- MathematicsArXiv
- 2021

Numerical Approximations and Error Analysis of the Cahn-Hilliard Equation with Dynamic Boundary Conditions

- MathematicsArXiv
- 2020

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

- MathematicsJ. Sci. Comput.
- 2021

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

- MathematicsOperators and Matrices
- 2021

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

- Mathematics
- 2017

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.

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