An efficient boundary integral method for the Mullins-Sekerka problem
type: Item
instance of: scholarly article
title: An efficient boundary integral method for the Mullins-Sekerka problem
zbMATH Open document ID: 0879.65092
author: Jingyi Zhu
author: Xinfu Chen
author: Thomas Y. Hou
published in: Journal of Computational Physics
publication date: 1997-12-11T00:00:00Z
review text: The paper is concerned with the numerical treatment of a free boundary problem which is sometimes called Mullins-Sekerka problem in studying solidification and melting of materials with negligible specific heat. In a different context the considered problem can be regarded as an asymptotic limit of the Cahn-Hilliard equation. The problem is given by the Laplace equation in two space dimensions. On the free boundary one has the Gibbs-Thomson law (surface tension effects) and a condition describing the motion of the interface. This motion is an area preserving and curve shortening motion. After a careful introduction the authors develop an efficient method for the numerical solution which is essentially based on a reformulation of the problem by a boundary integral representation. The second step of the approach consists of the consideration of the evolution of the interface in a \(\theta\)-\(L\) formulation (\(\theta\)-tangent angle, \(L\)-total arclength). This allows to solve the evolution equation implicitly in the Fourier space. Finally, the discretized integral system is solved for the velocity, where the singular part is handled by discrete Fourier transforms. In Section 3 the authors discuss details of the numerical implementation of the proposed method. After deriving some special solutions to be used for the verification of the numerical solution, the authors report in Section 5 on their numerical results obtained for several examples. The paper is concluded by a discussion of possible extensions and applications of this approach to other problems.
Mathematics Subject Classification ID: 65Z05
Mathematics Subject Classification ID: 65M70
Mathematics Subject Classification ID: 35Q72
Mathematics Subject Classification ID: 35R35
Mathematics Subject Classification ID: 80A22
Mathematics Subject Classification ID: 35J05
zbMATH DE Number: 947699
zbMATH Keywords: free boundary problem
zbMATH Keywords: Mullins-Sekerka problem
zbMATH Keywords: boundary integral method
zbMATH Keywords: Cahn-Hilliard equation
zbMATH Keywords: Laplace equation
zbMATH Keywords: Gibbs-Thomson law
zbMATH Keywords: evolution equation
zbMATH Keywords: discrete Fourier transforms
zbMATH Keywords: numerical examples
MaRDI profile type: Publication
full work available at URL: jcph.1996.0173
OpenAlex ID: W2041785731
DOI: 10.1006/JCPH.1996.0173
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description: scientific article; zbMATH DE number 947699