Numerical solutions of the Allen–Cahn equation with the [formula omitted]-Laplacian

Abstract

We investigate the behavior of the numerical solutions of the p-Laplacian Allen–Cahn equation. Because of the p-Laplacian’s challenging numerical properties, many different methods have been proposed for the discretized p-Laplacian. In this paper, we provide and analyze a numerical scheme for the boundedness of solutions and energy decay properties. For a comprehensive understanding of the effect of p-Laplacian and its relationship in the context of phase-field modeling, we compare the temporal evolution and compute the eigenpairs of the classical, fractional, and p-Laplacian in the Allen–Cahn equations. As for the p-Laplacian Allen–Cahn equation, we characterize different morphological changes of numerical solutions under various numerical tests such as phase separation, equilibrium profile, boundedness of solution, energy decay, traveling wave solution, geometric motions, and comparison of the Allen–Cahn equations with the three different Laplacians. Our results imply that the interface profile along the two-phase boundary lines changes more steeply than classical one as the p order decreases, therefore, the p-Laplacian Allen–Cahn equation can be applied for the description of phase interface where it is important to maintain sharply.