The Allen–Cahn Equation

  • Sören BartelsEmail author
Part of the Springer Series in Computational Mathematics book series (SSCM, volume 47)


The mathematical description of phase separation and melting processes is often based on phase field models. These descriptions define a phase field variable that specifies particular phases as the solution of a semilinear parabolic partial differential equation. A small parameter that defines the width of the interfaces between different phases enters classical stability estimates in a critical way, and refined arguments are required to improve this dependence. Applying those estimates to analyzing approximation schemes for the simplest case of the Allen–Cahn equation in terms of a priori and a posteriori error estimates is carried out. The stability of various implicit and semi-implicit time-stepping schemes is discussed.


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© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Abteilung für Angewandte MathematikAlbert-Ludwigs-Universität FreiburgFreiburgGermany

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