Numerical Approximation of the Stochastic Cahn-Hilliard Equation and the (Stochastic) Hele-Shaw Problem

Overview

Abstract​

The Cahn-Hilliard equation is a fourth order parabolic partial differential equation (PDE) that is widely used as a phenomenological model to describe the evolution of interfaces in many practical problems, such as, the microstructure formation in materials, fluid flow, etc. It has been observed in the engineering literature that the stochastic version of the Cahn-Hilliard equation provides a better description of the experimentally observed evolution of complex microstructure. The equation belongs to a class of so-called phase-field models where the interface is replaced by a diffuse layer with small thickness proportional to an interfacial thickness parameter. It can be shown that for vanishing interfacial thickness the deterministic as well as the stochastic Cahn-Hilliard equation (with proper scaling of the noise) both converge to a sharp-interface limit which is given by the deterministic Hele-Shaw problem. We propose a time implicit numerical approximation of the stochastic Cahn-Hilliard equation which is robust with respect to the interfacial thickness parameter. We show that, with suitable scaling of the noise, the sharp-interface limit of the proposed numerical approximation converges to the deterministic Hele-Shaw problem. In addition, we present numerical evidence that without the scaling of the noise the sharp-interface limit of the stochastic Cahn-Hilliard equation is a stochastic version of the Hele-Shaw problem. We propose a numerical approximation of the stochastic Hele-Shaw problem and present computational results which demonstrate the respective convergence of the stochastic Cahn-Hilliard equation to the deterministic or the stochastic version of the Hele-Shaw problem depending on scaling of the noise term. This is joint work with D. Antonopoulou, R. Nurnberg and A. Prohl.