Error estimate of the finite volume scheme for the Allen-Cahn equation
type: Item
instance of: scholarly article
title: Error estimate of the finite volume scheme for the Allen-Cahn equation
zbMATH Open document ID: 1397.65153
author: Pavel Strachota
author: Michal Beneš
published in: BIT
publication date: 2018-06-26T00:00:00Z
review text: The authors consider the numerical solution of the initial-boundary value problem for the Allen-Cahn equation \[ \xi \frac{\partial p}{\partial t} = \xi \Delta p + \frac{1}{\xi} f_0(p) + c_0W \text{ in } (0,T) \times \Omega, \] where \(\Omega \subset \mathbb{R}^d\) is a bounded polygonal domain with \(d \in \{2,3\}\), \(f_0(p):=p(1-p)(p-1/2)\), \(\xi>0\) is a constant parameter and \(c_0W\) is a constant forcing term. The authors study only the isotropic case. \vskip 0.8em For the numerical solution the method of lines is used with finite volumes as space discretization. Finite volumes turn out to be more suitable than finite differences because they produce edge-symmetric stencils. A first-order error estimate is proved for the semidiscrete finite volume problem. The proof is based on a maximum principle also provided by the authors. Numerical results are shown for a 2D and a 3D example, in which the finite volume scheme uses a uniform rectangular mesh and the time integration is carried out by the fourth-order Runge-Kutta-Merson solver.
reviewed by: Rolf Dieter Grigorieff
Mathematics Subject Classification ID: 65M08
Mathematics Subject Classification ID: 65M15
Mathematics Subject Classification ID: 65M20
Mathematics Subject Classification ID: 80A22
Mathematics Subject Classification ID: 74N05
Mathematics Subject Classification ID: 35Q74
zbMATH DE Number: 6894733
zbMATH Keywords: Allen-Cahn equation
zbMATH Keywords: phase transition
zbMATH Keywords: isotropic material
zbMATH Keywords: microstructure growth
zbMATH Keywords: solidification
zbMATH Keywords: semidiscrete scheme
zbMATH Keywords: finite volume method
zbMATH Keywords: error estimate
MaRDI profile type: Publication
full work available at URL: s10543-017-0687-4
OpenAlex ID: W2763139309
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DOI: 10.1007/S10543-017-0687-4
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description: scientific article; zbMATH DE number 6894733