# 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

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topBeneš, 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

top- 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
- 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)
- Cahn, J. W., Hilliard, J. E., 10.1063/1.1730447, J. Chem. Phys. (1959), 31 688-699. (1959) DOI10.1063/1.1730447
- Taylor, J. E., Cahn, J. W., 10.1007/BF02186838, J. Statist. Phys. (1994), 77 183-197. (1994) Zbl0844.35044MR1300532DOI10.1007/BF02186838
- 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
- Bronsard, L., Stoth, B., 10.1137/S0036141094279279, SIAM J. Math. Anal. (1997), 28 769-807. (1997) Zbl0874.35009MR1453306DOI10.1137/S0036141094279279
- 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
- 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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