New approximate method for the Allen–Cahn equation with double-obstacle constraint and stability criteria for numerical simulations

Tomoyuki Suzuki; Keisuke Takasao; Noriaki Yamazaki; Tomoyuki Suzuki; Keisuke Takasao; Noriaki Yamazaki

doi:10.3934/Math.2016.3.288

AIMS Mathematics

2016, Volume 1, Issue 3: 288-317. doi: 10.3934/Math.2016.3.288

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New approximate method for the Allen–Cahn equation with double-obstacle constraint and stability criteria for numerical simulations

1.
Department of Mathematics, Faculty of Engineering, Kanagawa University, 3-27-1 Rokkakubashi, Kanagawa-ku, Yokohama, 221-8686, Japan
2.
Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba Meguro-ku, Tokyo, 153-8914, Japan

Received: 26 September 2016 Accepted: 17 October 2016 Published: 28 October 2016

In a numerical study, we consider the Allen–Cahn equation with a double-obstacle constraint. The constraint is a multivalued function that is provided by the subdi erential of the indicator function on a closed interval. Therefore, performing a numerical simulation of our problem poses diffculties. We propose a new approximate method for the constraint and demonstrate its validity. Moreover, we present stability criteria for the standard forward Euler method guaranteeing stable numerical experiments when using the approximating equation.
- Allen-Cahn equation,
- constraint,
- double obstacle,
- stability,
- subdi erential,
- numerical simulations
Citation: Tomoyuki Suzuki, Keisuke Takasao, Noriaki Yamazaki. New approximate method for the Allen–Cahn equation with double-obstacle constraint and stability criteria for numerical simulations[J]. AIMS Mathematics, 2016, 1(3): 288-317. doi: 10.3934/Math.2016.3.288

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Abstract

In a numerical study, we consider the Allen–Cahn equation with a double-obstacle constraint. The constraint is a multivalued function that is provided by the subdi erential of the indicator function on a closed interval. Therefore, performing a numerical simulation of our problem poses diffculties. We propose a new approximate method for the constraint and demonstrate its validity. Moreover, we present stability criteria for the standard forward Euler method guaranteeing stable numerical experiments when using the approximating equation.

References

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