A semi-implicit spectral method for stochastic nonlocal phase-field models

The classical phase-field model has been introduced as a model for non-isothermal phase separation processes in materials. In order to overcome some of the model's shortcomings, a variety of extensions have recently been proposed which include nonlocal interactions as well as stochastic noise terms. In this paper, we study these extensions from a functional-analytic and numerical point of view. More precisely, we present a functional-analytic framework for establishing existence, uniqueness, and qualitative dynamical results, and then propose a spectral method for solving stochastic nonlocal phase-field models. In particular, we establish convergence properties of our method and study the effects of noise regularity and of the nonlocal interaction term on these convergence properties. Finally, numerical studies relating to the associated phase separation process are presented.