Efficient numerical methods for phase-field equations

Abstract

In this article, we overview recent developments of numerical methodsfor phase-field equations. The main difficulty fornumerically solving phase-field equations is about a severe restrictionon the time step due to nonlinearity and high order differential terms, while it usually requires a very long timesimulation to reach the steady state. It is knownthat phase-field models satisfy a nonlinear stability relationship, called energy stability,which means that the free energy functional decays in time.It has attracted more and more attention to design numerical schemes inheriting the energy stability so thatthe numerical simulation may use large time steps and keep the accuracy.For some popularly studied phase-field equations, this article will present several widely used highly efficient numerical schemes andshow an adaptive time-stepping strategy based on the changing rate in time of the energy functional, which could guarantee the accuracy and stability of the numerical solution and improves the computational efficiency significantly.

Topics & Concepts