Noise regularization and computations for the 1-dimensional stochastic Allen–Cahn problem
Markos A. Katsoulakis, Georgios T. Kossioris, Omar Lakkis · Interfaces and Free Boundaries Mathematical Analysis Computation and Applications · 2007
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.