An unconditionally gradient stable numerical method for solving the Allen–Cahn equation

We consider an unconditionally gradient stable scheme for solving the Allen–Cahn equation representing a model for anti-phase domain coarsening in a binary mixture. The continuous problem has a decreasing total energy. We show the same property for the corresponding discrete problem by using eigenvalues of the Hessian matrix of the energy functional. We also show the pointwise boundedness of the numerical solution for the Allen–Cahn equation. We describe various numerical experiments we performed to study properties of the Allen–Cahn equation.