Article
Full entry |
PDF
(0.7 MB)
Feedback
PDF
(0.7 MB)
Feedback
Keywords:
Allen-Cahn equation; phase transitions; mean-curvature flow; finite-difference method
Allen-Cahn equation; phase transitions; mean-curvature flow; finite-difference method
Summary:
The paper presents the results of numerical solution of the Allen-Cahn equation with a non-local term. This equation originally mentioned by Rubinstein and Sternberg in 1992 is related to the mean-curvature flow with the constraint of constant volume enclosed by the evolving curve. We study this motion approximately by the mentioned PDE, generalize the problem by including anisotropy and discuss the computational results obtained.
References:
[1] Rubinstein, J., Sternberg, P.: Nonlocal reaction-diffusion equation and nucleation. IMA J. Appl. Math. (1992), 48 249-264. DOI 10.1093/imamat/48.3.249 | MR 1167735
[2] Allen, S., Cahn, J.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. (1979), 27 1084-1095.
[3] Cahn, J. W., Hilliard, J. E.: Free energy of a nonuniform system {III}. Nucleation of a two-component incompressible fluid. J. Chem. Phys. (1959), 31 688-699. DOI 10.1063/1.1730447
[4] Taylor, J. E., Cahn, J. W.: Linking anisotropic sharp and diffuse surface motion laws via gradient flows. J. Statist. Phys. (1994), 77 183-197. DOI 10.1007/BF02186838 | MR 1300532 | Zbl 0844.35044
[5] Bronsard, L., Kohn, R.: Motion by mean curvature as the singular limit of Ginzburg-{L}andau dynamics. J. Differential Equations (1991), 90 211-237. DOI 10.1016/0022-0396(91)90147-2 | MR 1101239 | Zbl 0735.35072
[6] Bronsard, L., Stoth, B.: Volume-preserving mean curvature flow as a limit of a nonlocal Ginzburg-{L}andau equation. SIAM J. Math. Anal. (1997), 28 769-807. DOI 10.1137/S0036141094279279 | MR 1453306 | Zbl 0874.35009
[7] Beneš, M.: Diffuse-interface treatment of the anisotropic mean-curvature flow. Appl. Math., Praha (2003), 48 437-453. DOI 10.1023/B:APOM.0000024485.24886.b9 | MR 2025297 | Zbl 1099.53044
[8] Beneš, M.: Mathematical and computational aspects of solidification of pure substances. Acta Math. Univ. Comenian. (2001), 70 123-152. MR 1865364 | Zbl 0990.80006
Search
Partner of
