Computational studies of non-local anisotropic Allen-Cahn equation

Michal Beneš; Shigetoshi Yazaki; Masato Kimura

Mathematica Bohemica (2011)

  • Volume: 136, Issue: 4, page 429-437
  • ISSN: 0862-7959

Abstract

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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.

How to cite

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Beneš, Michal, Yazaki, Shigetoshi, and Kimura, Masato. "Computational studies of non-local anisotropic Allen-Cahn equation." Mathematica Bohemica 136.4 (2011): 429-437. <http://eudml.org/doc/197198>.

@article{Beneš2011,
abstract = {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.},
author = {Beneš, Michal, Yazaki, Shigetoshi, Kimura, Masato},
journal = {Mathematica Bohemica},
keywords = {Allen-Cahn equation; phase transitions; mean-curvature flow; finite-difference method; Allen-Cahn equation; phase transitions; mean-curvature flow; finite-difference method},
language = {eng},
number = {4},
pages = {429-437},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Computational studies of non-local anisotropic Allen-Cahn equation},
url = {http://eudml.org/doc/197198},
volume = {136},
year = {2011},
}

TY - JOUR
AU - Beneš, Michal
AU - Yazaki, Shigetoshi
AU - Kimura, Masato
TI - Computational studies of non-local anisotropic Allen-Cahn equation
JO - Mathematica Bohemica
PY - 2011
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 136
IS - 4
SP - 429
EP - 437
AB - 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.
LA - eng
KW - Allen-Cahn equation; phase transitions; mean-curvature flow; finite-difference method; Allen-Cahn equation; phase transitions; mean-curvature flow; finite-difference method
UR - http://eudml.org/doc/197198
ER -

References

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  1. Rubinstein, J., Sternberg, P., 10.1093/imamat/48.3.249, IMA J. Appl. Math. (1992), 48 249-264. (1992) MR1167735DOI10.1093/imamat/48.3.249
  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. (1979) 
  3. Cahn, J. W., Hilliard, J. E., 10.1063/1.1730447, J. Chem. Phys. (1959), 31 688-699. (1959) DOI10.1063/1.1730447
  4. Taylor, J. E., Cahn, J. W., 10.1007/BF02186838, J. Statist. Phys. (1994), 77 183-197. (1994) Zbl0844.35044MR1300532DOI10.1007/BF02186838
  5. Bronsard, L., Kohn, R., 10.1016/0022-0396(91)90147-2, J. Differential Equations (1991), 90 211-237. (1991) Zbl0735.35072MR1101239DOI10.1016/0022-0396(91)90147-2
  6. Bronsard, L., Stoth, B., 10.1137/S0036141094279279, SIAM J. Math. Anal. (1997), 28 769-807. (1997) Zbl0874.35009MR1453306DOI10.1137/S0036141094279279
  7. Beneš, M., 10.1023/B:APOM.0000024485.24886.b9, Appl. Math., Praha (2003), 48 437-453. (2003) Zbl1099.53044MR2025297DOI10.1023/B:APOM.0000024485.24886.b9
  8. Beneš, M., Mathematical and computational aspects of solidification of pure substances, Acta Math. Univ. Comenian. (2001), 70 123-152. (2001) Zbl0990.80006MR1865364

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