Unconditionally stable method for the high-order Allen–Cahn equation

We propose an unconditionally stable algorithm for the Allen–Cahn (AC) equation that incorporates a high-order free energy. The high-order AC equation improves the preservation of interfacial dynamics and suppresses noise. The proposed method guarantees unconditional stability, which is essential for precise phase transition modeling and preserving detailed characteristics. To effectively solve the governing equation, it is divided into two subproblems, each of which is solved separately. The nonlinear operator is handled using a frozen coefficient method, followed by a closed-form solution. The linear operator is solved by applying the discrete cosine transform. To verify the effectiveness of the proposed algorithm, we carried out various computational simulations in two- and three-dimensional space. The proposed method ensures unconditional stability, and therefore allows stable solutions even with relatively large time steps. Moreover, we investigate the notable characteristics of the high-order AC equation, particularly its enhanced capability to effectively handle phase separation phenomena in the presence of significant noise and complex phase interfaces.