# PIMS/CSC Distinguished Lecture Series: High-Order and Adaptive Time Stepping Methods for Energy Gradient Flows

The phase-field method is very useful for modeling interfacial phenomena. The key idea of the phase-field methodology is to replace sharp interfaces by thin transition regions where interfacial forces are smoothly distributed. One of the main reasons for the success of the phase-field methodology is that it is based on rigorous mathematics and thermodynamics. Most phase-field models satisfy a nonlinear stability relationship, usually expressed as a time-decreasing free-energy functional, which is called energy gradient stability. To obtain accurate numerical solutions of such problems, it is desirable to use high-order approximations in space and time. Yet because of the difficulties introduced by the combination of nonlinearity and stiffness, most computations have been limited to lower-order in time, and in most cases to constant time-stepping. On the other hand, numerical simulations for many physical problems require large time integration; as a result large time-stepping methods become necessary in some time regimes.