Noise regularization and computations for the 1-dimensional stochastic Allen-Cahn problem

Details Noise regularization and computations for the 1-dimensional stochastic Allen-Cahn problem Noise regularization and computations for the 1-dimensional stochastic Allen-Cahn problem

2011-11-27

arxiv.org/abs/1111.6312v1

Mathematics

Numerical Analysis

28 pages, 16 (4x4) figures, published in 2007; Interfaces and Free Boundaries 2007 vol. 9 (1)

Scientific paper

10.4171/IFB/154

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.

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