Noise regularization and computations for the 1-dimensional stochastic Allen-Cahn problem
Abstract
We address the numerical discretization of the Allen-Cahn prob- lem with additive white noise in one-dimensional space. 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.
Cite
@article{arxiv.1111.6312,
title = {Noise regularization and computations for the 1-dimensional stochastic Allen-Cahn problem},
author = {Markos A. Katsoulakis and Georgios T. Kossioris and Omar Lakkis},
journal= {arXiv preprint arXiv:1111.6312},
year = {2013}
}
Comments
28 pages, 16 (4x4) figures, published in 2007; Interfaces and Free Boundaries 2007 vol. 9 (1)