Colloquium

Numerical Methods for Deterministic and Stochastic Phase Field Models

 Dr. Yukun Li

Department of Mathematics

The Ohio State University

 ABSTRACT: The phase field model is one of the most important approaches to formulate the moving interface problems, especially when topological changes happen at the interface. Unlike other models, the accuracy between different time discretization schemes needs special attention for the phase field models, and the efficiency has always been a key restriction of their usage. This talk analyzes the accuracy and efficiency of the Allen-Cahn model, which is a building block of the phase field models. Besides, the convergence of the numerical interfaces is proved to approximate the exact moving interface using a non-standard technique called spectrum estimate. The ideas above can also be applied to some other phase field models. Finally, motivated by the applications, we add the noise to the interface, for the first time, to consider the numerical approximations of the formulated stochastic partial differential equation (SPDE). This SPDE has a gradient-type multiplicative noise, and the analysis is much more involved compared with the SPDEs with additive noises. A few numerical examples are given to verify the theoretical results.