ON EFFECTIVE NUMERICAL METHODS FOR PHASE-FIELD MODELS

In this article, we overview recent developments of modern computational methods for the approximate solution of phase-field problems. The main difficulty for developing a numerical method for phase field equations is a severe stability restriction on the time step due to nonlinearity and high order differential terms. It is known that the phase field models satisfy a nonlinear stability relationship called gradient stability, usually expressed as a time-decreasing free-energy functional. This property has been used recently to derive numerical schemes that inherit the gradient stability. The first part of the article will discuss implicit-explicit time discretizations which satisfy the energy stability. The second part is to discuss time-adaptive strategies for solving the phase-field problems, which is motivated by the observation that the energy functionals decay with time smoothly except at a few critical time levels. The classical operator-splitting method is a useful tool in time discrtization. In the final part, we will provide some preliminary results using operator-splitting approach.