A Positivity-Preserving Numerical Method for the Ternary Allen-Cahn Equation

초록

We propose a positivity-preserving computational method for the ternary Allen-Cahn (tAC) equation. To solve the tAC equation, we apply the operator splitting method. The tAC equation is decomposed into nonlinear and linear components. The nonlinear term is discretized using a finite difference method (FDM) combined with an explicit-implicit scheme. The resulting cubic polynomial admits a unique real solution under appropriate time step conditions. The unique solution is obtained by applying the analytical formula for solving cubic polynomials. The linear term, representing the diffusion equation, is discretized using a fully implicit FDM. The resulting tridiagonal matrix is solved by using the Thomas algorithm, which is a direct method for such systems. We analytically prove the positivity-preserving property of the proposed numerical method. To validate the performance of the proposed algorithm, several standard computational tests are conducted, and the results confirm the accuracy and effectiveness of the proposed algorithm.

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