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arXiv · 1807.03171

Energy Stable Second Order Linear Schemes for the Allen-Cahn Phase-Field Equation

Lin Wang·Haijun Yu
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Abstract

Phase-field model is a powerful mathematical tool to study the dynamics of interface and morphology changes in fluid mechanics and material sciences. However, numerically solving a phase field model for a real problem is a challenge task due to the non-convexity of the bulk energy and the small interface thickness parameter in the equation. In this paper, we propose two stabilized second order semi-implicit linear schemes for the Allen-Cahn phase-field equation based on backward differentiation formula and Crank-Nicolson method, respectively. In both schemes, the nonlinear bulk force is treated explicitly with two second-order stabilization terms, which make the schemes unconditional energy stable and numerically efficient. By using a known result of the spectrum estimate of the linearized Allen-Cahn operator and some regularity estimate of the exact solution, we obtain an optimal second order convergence in time with a prefactor depending on the inverse of the characteristic interface thickness only in some lower polynomial order. Both 2-dimensional and 3-dimensional numerical results are presented to verify the accuracy and efficiency of proposed schemes.

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Lin Wang, Haijun Yu. 2018-07-06. Energy Stable Second Order Linear Schemes for the Allen-Cahn Phase-Field Equation. https://doi.org/10.4310/cms.2019.v17.n3.a2

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