We address the numerical discretization of the Allen-Cahn problem with additive white noise in one-dimensional space. Our main focus is to understand the behavior of the discretized equation with respect to a small ``interface thickness'' parameter and the noise intensity. The discretization is conducted in two stages: (1) regularize the white noise and study the regularized problem, (2) approximate the regularized problem. We address (1) by introducing a piecewise constant random approximation of the white noise with respect to a space-time mesh. We analyze the regularized problem and study its relation to both the original problem and the deterministic Allen-Cahn problem. Step (2) is then performed leading to a practical Monte-Carlo method combined with a Finite Element-Implicit Euler scheme. The resulting numerical scheme is tested against theoretical benchmark results concerning the behavior of the solution as the interface thickness goes to zero.
Cite this article
Markos A. Katsoulakis, Georgios T. Kossioris, Omar Lakkis, Noise regularization and computations for the 1-dimensional stochastic Allen–Cahn problem. Interfaces Free Bound. 9 (2007), no. 1, pp. 1–30DOI 10.4171/IFB/154