An explicit numerical method for the conservative Allen-Cahn equation on a cubic surface

We introduced a fully explicit finite difference method (FDM) designed for numerically solving the conservative Allen-Cahn equation (CAC) on a cubic surface. In this context, the cubic surface refers to the combined areas of the six square faces that enclose the volume of a cube. The proposed numerical solution approach is structured into two sequential steps. First, the Allen-Cahn (AC) equation was solved by applying the fully explicit FDM, which is computationally efficient. Following this, the conservation term is resolved using the updated solution from the AC equation ensure consistency with the underlying conservation principles. To evaluate the effectiveness of proposed scheme, computational tests are performed to verify that the resulting numerical solution the CAC equation successfully conserves the discrete mass. Additionally, the solution is examined its ability to exhibit the property of constrained motion by mass conserving mean curvature, a critical characteristic of the CAC equation. These two properties are fundamental to the integrity and accuracy of the CAC equation.