Unconditionally maximum principle preserving scheme for the Allen–Cahn equation with a logarithmic free energy

Abstract

We present a novel computational scheme that unconditionally satisfies the maximum principle to solve the Allen–Cahn (AC) equation with a logarithmic Flory–Huggins potential. The proposed scheme uses an operator splitting approach integrated with a frozen coefficient technique without using any stabilization term. Due to the use of the frozen coefficient method, we can apply a closed-form solution to the highly nonlinear term. This combination allows the development of a scheme that rigorously maintains the maximum principle throughout the computation. We provide a detailed analytical proof of the discrete maximum principle for the proposed scheme to ensure its theoretical robustness. To validate the high performance, we conduct computational experiments, which confirm the unconditional stability and demonstrate the method's effectiveness in preserving the maximum principle. These numerical results highlight the proposed scheme's potential as a reliable tool for accurately solving the AC equation.