Numerical Studies of Discrete Approximations to the Allen--Cahn Equation in the Sharp Interface Limit

The numerical approximations to the Allen–Cahn type diffuse interface models are studied, with a particular focus on their performance in the sharp interface limit and the effectiveness of high order discretization schemes. Different spatial discretizations of an energy functional in the diffuse interface framework are compared first. Discretizations of the time-dependent equation using various different time-stepping schemes and an adaptive finite element spatial approximations are then analyzed. Assessments of the numerical accuracy in different parameter regimes are carried out. The analysis and computational findings provide insight on the effectiveness of the discretization schemes and offer guidance to parameter choices when numerical simulations are conducted within the diffuse interface framework.