Regularized Dirac delta functions for phase field models

The phase field model is a highly successful computational technique for capturing the evolution and topological change of complex interfaces. The main computational advantage of phase field models is that an explicit tracking of the interface is unnecessary. The regularized Dirac delta function is an important ingredient in many interfacial problems that phase field models have been applied. The delta function can be used to postprocess the phase field solution and represent the surface tension force. In this paper, we present and compare various types of delta functions for phase field models. In particular, we analytically show which type of delta function works relatively well regardless of whether an interfacial phase transition is compressed or stretched. Numerical experiments are presented to show the performance of each delta function. Numerical results indicate that (1) all of the considered delta functions have good performances when the phase field is locally equilibrated; and (2) a delta function, which is the absolute value of the gradient of the phase field, is the best in most of the numerical experiments.