An efficient hybrid numerical method for the high-order Allen-Cahn equation

초록

This paper presents an efficient hybrid numerical algorithm to solve the high-order Allen-Cahn (AC) equation, which uses a polynomial free energy potential with a high-order exponent. The high-order AC equation preserves fine structures, shows traveling wave phenomena, suppresses noise for larger polynomial orders, and accurately captures interface motion driven by mean curvature. The proposed scheme combines an operator splitting method, a finite difference discretization, and an interpolation technique to overcome the challenge of solving nonlinear implicit equations that arise from the high-order polynomial potential. A theoretical condition on the time step is derived to guarantee monotonicity and ensure that the computational solution obtained by the proposed method satisfies the discrete maximum principle. Compared to fully explicit methods, the proposed approach allows a significantly larger time step size and thus results in improved numerical efficiency. Computational tests are performed to evaluate the accuracy, stability, and ability of the proposed algorithm to capture key physical behaviors of the phase-field model, such as curvature-driven interface motion and the suppression of high-frequency noise. The computational results demonstrate that higher polynomial orders lead to cleaner interface evolution by eliminating spurious oscillations and preserving non-random features. Furthermore, the method reliably captures the shrinking of interfaces even with relatively low numerical resolution. The proposed hybrid algorithm thus provides a robust and practical numerical method for simulating the complex dynamics of high-order AC equations in multi-phase systems and has potential applications in materials science, image processing, and biological modeling.

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