Time discretizations of anisotropic Allen-Cahn equations

We consider anisotropic Allen-Cahn equations with interfacial energy induced by an anisotropic surface energy density gamma . Assuming that gamma is positive, positively homogeneous of degree 1, strictly convex in tangential directions to the unit sphere and sufficiently smooth, we show the stabilit...

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Bibliographic Details
Published in:IMA journal of numerical analysis 2013-10, Vol.33 (4), p.1226-1244
Main Authors: Graser, C., Kornhuber, R., Sack, U.
Format: Article
Language:English
Subjects:
Anisotropy
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Summary:We consider anisotropic Allen-Cahn equations with interfacial energy induced by an anisotropic surface energy density gamma . Assuming that gamma is positive, positively homogeneous of degree 1, strictly convex in tangential directions to the unit sphere and sufficiently smooth, we show the stability of various time discretizations. In particular, we consider a fully implicit and a linearized time discretization of the interfacial energy combined with implicit and semiimplicit time discretizations of the double-well potential. In the semiimplicit variant, concave terms are taken explicitly. The arising discrete spatial problems are solved by globally convergent truncated nonsmooth Newton multigrid methods. Numerical experiments show the accuracy of the different discretizations. We also illustrate that pinch-off under anisotropic mean curvature flow is no longer invariant under rotation of the initial configuration for a fixed orientation of the anisotropy.
ISSN:0272-4979
1464-3642
DOI:10.1093/imanum/drs043