An unconditionally stable adaptive finite difference scheme for the Allen–Cahn equation

We propose an unconditionally stable adaptive finite difference scheme for the Allen–Cahn (AC) equation. The AC equation is a reaction-diffusion equation used to model phase separation in multi-component alloy systems. It describes the temporal evolution of the order parameter, which denotes different phases, and incorporates both diffusion and nonlinear reaction terms to capture the interfacial dynamics between phases. A fundamental aspect of the dynamics of the AC equation is motion by mean curvature, which implies that an initial interface shrinks as time progresses. Therefore, it is highly efficient to reduce the computational domain as the interface shrinks. We use an operator splitting technique with a finite difference method and a closed-form solution. We conduct computational tests to validate the effectiveness of the proposed approach. The computational tests demonstrate that the proposed algorithm is effective, reliable, and robust across various test cases.