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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

Special Issues: Nonlinear Evolution PDEs, Interfaces and Applications

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.
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Keywords 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. AIMS Mathematics, 2016, 1(3): 288-317. doi: 10.3934/Math.2016.3.288


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