The Allen–Cahn equation with a space-dependent mobility and a source term for general motion by mean curvature

We propose the Allen–Cahn (AC) equation with a space-dependent mobility and a source term for general motion by mean curvature. Using the space-dependent mobility, we can control the temporal evolution dynamics. Furthermore, by using the source term, we can control the growth and shrinkage of the interfaces. To efficiently solve the governing equation, we use an operator splitting method that splits the main equation into the modified AC equation and the source term equation. The modified AC model is numerically computed using a fully explicit Euler method, and the source term equation is solved analytically. The overall numerical schemes preserve the maximum principle if the time step size satisfies a certain condition. To show the performance of the proposed mathematical model and its corresponding numerical scheme, we conduct several computational experiments. The numerical results confirm the efficiency and robust performance of the proposed model and its numerical algorithm, rendering the proposed model as a versatile tool for a wide range of applications.