A structure-preserving explicit numerical scheme for the Allen–Cahn equation with a logarithmic potential

Abstract

This paper presents a stability analysis of a structure-preserving explicit finite difference method (FDM) for the Allen–Cahn (AC) equation with a logarithmic potential that has two arguments. Firstly, we compute the temporal step constraint that guarantees that if the initial condition is bounded by the two arguments of the minimum, then the numerical solutions are always bounded by them, i.e., the explicit numerical scheme satisfies the maximum principle. Secondly, we compute the temporal step constraint that guarantees that the discrete total energy of the system is non-increasing over time. To validate the preservation of the maximum principle and the decrease in discrete total energy, we perform numerical experiments.