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arXiv · 2109.01253

New efficient time-stepping schemes for the anisotropic phase-field dendritic crystal growth model

Minghui Li·Mejdi Azaiez·Chuanju Xu
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Abstract

In this paper, we propose and analyze a first-order and a second-order time-stepping schemes for the anisotropic phase-field dendritic crystal growth model. The proposed schemes are based on an auxiliary variable approach for the Allen-Cahn equation and delicate treatment of the terms coupling the Allen-Cahn equation and temperature equation. The idea of the former is to introduce suitable auxiliary variables to facilitate construction of high order stable schemes for a large class of gradient flows. We propose a new technique to treat the coupling terms involved in the crystal growth model and introduce suitable stabilization terms to result in totally decoupled schemes, which satisfy a discrete energy law without affecting the convergence order. A delicate implementation demonstrates that the proposed schemes can be realized in a very efficient way. That is, it only requires solving four linear elliptic equations and a simple algebraic equation at each time step. A detailed comparison with existing schemes is given, and the advantage of the new schemes are emphasized. As far as we know this is the first second-order scheme that is totally decoupled, linear, unconditionally stable for the dendritic crystal growth model with variable mobility parameter.

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Minghui Li, Mejdi Azaiez, Chuanju Xu. 2021-09-03. New efficient time-stepping schemes for the anisotropic phase-field dendritic crystal growth model. https://arxiv.org/abs/2109.01253

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