Maximum Principle-Preserving Explicit Scheme for the Ternary Allen-Cahn Equation

We investigate the stability of the ternary Allen-Cahn (tAC) equation solved by the fully explicit Euler method. The tAC equation models interface dynamics and phase separation in three-component systems, and is formulated as an $L^2$-gradient flow of the Ginzburg-Landau energy. We derive a sufficient condition on the time step size that satisfies the discrete maximum principle and guarantees the numerical stability of the fully explicit Euler method. Through detailed analysis, we confirm that the derived condition provides the largest possible time step size among those that preserve stability. Numerical experiments are conducted to verify the theoretical results, and the numerical solution violates the maximum principle and becomes unstable when the time step size exceeds the derived bound. Conversely, when the time step condition is satisfied, the computational solution preserves the maximum principle and maintains the energy dissipation property of the tAC equation. These results provide a rigorous theoretical foundation for determining the stability limit of the explicit Euler method applied to the tAC equation.