Katsoulakis, MAKossioris, GTLakkis, O2024-04-262024-04-262007-01-01https://hdl.handle.net/20.500.14394/34689<p>The published version is located at <a href="http://www.ems-ph.org/journals/show_abstract.php?issn=1463-9963&vol=9&iss=1&rank=1">http://www.ems-ph.org/journals/show_abstract.php?issn=1463-9963&vol=9&iss=1&rank=1</a></p>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.Allen-CahnStochastic PDEFinite ElementsRegularityMean Curvature FlowPhysical Sciences and Mathematicsarticle