On the maximum principle preserving schemes for the generalized Allen-Cahn equation
type: Item
instance of: scholarly article
title: On the maximum principle preserving schemes for the generalized Allen-Cahn equation
zbMATH Open document ID: 1361.65059
author: Jie Shen
author: Tao Tang
author: Jiang Yang
published in: Communications in Mathematical Sciences
publication date: 2016-11-22T00:00:00Z
reviewed by: Gisbert Stoyan
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 65M20
Mathematics Subject Classification ID: 65M15
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 35B50
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 65M50
zbMATH DE Number: 6655257
zbMATH Keywords: Allen-Cahn equation
zbMATH Keywords: finite differences
zbMATH Keywords: maximum principle
zbMATH Keywords: maximum norm stability
zbMATH Keywords: error estimate
zbMATH Keywords: semidiscretization
zbMATH Keywords: nonlinear convection-diffusion-reaction equation
zbMATH Keywords: numerical experiment
zbMATH Keywords: time step control
MaRDI profile type: Publication
full work available at URL: cms.2016.v14.n6.a3
OpenAlex ID: W2507484451
DOI: 10.4310/CMS.2016.V14.N6.A3
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review text: The authors investigate the 1D Allen-Cahn equation (a nonlinear convection-diffusion-reaction equation) along with homogeneous Neumann or Dirichlet conditions, or with periodic boundary conditions. Continuing their former work (see [\textit{T. Tang} and \textit{J. Yang}, J. Comput. Math. 34, No. 5, 451--461 (2016; \url{doi:10.4208/jcm.1603-m2014-0017})]) and considering a polynomial and a logarithmic potential, they give a semi-discrete scheme and a fully discretized semi-implicit difference scheme for the numerical solution of the problem. Assuming suitable (sufficient) restrictions on the time step, they prove (based on properties of diagonally dominant matrices) maximum norm estimates for both schemes. The results can be generalized to the multidimensional case and for rectangular domains. In special cases of the potentials and mobilities considered they are able to explicitly give (sufficient) time step bounds.NEWLINENEWLINEAssuming the solution of the original problem to be sufficiently smooth (which is not specified concretely), they then show an error estimate for the fully discrete scheme in the maximum norm guaranteeing first order in time and space -- provided one starts from the values of the initial condition.NEWLINENEWLINEIn their numerical experiments verifying their theoretical results they consider 1D and 2D cases and add an adaptive time step control. Let us remark that two figures (6.3 and 6.4) probably were in colors but are not distinguishable here.
description: scientific article; zbMATH DE number 6655257