Noise regularization and computations for the 1-dimensional stochastic Allen-Cahn problem
Publication Date
2007
Journal or Book Title
INTERFACES AND FREE BOUNDARIES
Abstract
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.
Pages
1-30
Volume
9
Issue
1
Recommended Citation
Katsoulakis, MA; Kossioris, GT; and Lakkis, O, "Noise regularization and computations for the 1-dimensional stochastic Allen-Cahn problem" (2007). INTERFACES AND FREE BOUNDARIES. 443.
Retrieved from https://scholarworks.umass.edu/math_faculty_pubs/443
Comments
The published version is located at http://www.ems-ph.org/journals/show_abstract.php?issn=1463-9963&vol=9&iss=1&rank=1