Computing the stochastic Allen-Cahn problem
Seminar Room 2, Newton Institute Gatehouse
AbstractOur main goal is the numerical approximation of the Allen-Cahn problem with additive white noise in one-dimensional space and the statistical validation (benchmarking) of numerical results.
One of the main difficulties for a rigorous numerical discretization of this SPDE, which is an important model for more complicated phase separation descriptions, is the presence of the time-space white noise as a forcing term and its interaction with the nonlinear term.
The discretization is conducted in two stages: (1) regularize the white noise and study the regularized problem, (2) approximate the regularized problem and derive a finite element Monte Carlo simulation scheme.
The numerical results are checked against theoretical results from the stochastic analysis of scaling limits estabilished by Funaki (1995) and Brassesco, De Masi & Presutti (1995). This requires an ad-hoc benchmarking technique based on statistical postprocessing of numerical data.Time allowing I will review recent advances on the topic where we analyze the behavior of one and multiple interfaces when the interface thickness parameter does not approach the limit. Asymptotic analysis shows that the stochastic solution and its approximation remain "close" to a set of functions where the interface makes sense. This is particularly useful to relate numerical results to theory and it has the potential to define stochastic diffuse interfaces in dimensions higher than 1.