Analysis of Some Splitting Schemes for the Stochastic Allen-Cahn Equation

TL;DR

This paper introduces and analyzes a splitting scheme for the stochastic Allen-Cahn equation driven by space-time white noise, demonstrating strong and weak convergence properties through theoretical proofs and numerical experiments.

Contribution

It presents a novel explicit splitting scheme that uses the exact nonlinear solution, with rigorous convergence analysis for the stochastic Allen-Cahn equation.

Findings

Moment boundedness of the numerical solution

Strong convergence order almost 1/4 in L^2

Convergence in probability of order almost 1/4

Abstract

We introduce and analyze an explicit time discretization scheme for the one-dimensional stochastic Allen-Cahn, driven by space-time white noise. The scheme is based on a splitting strategy, and uses the exact solution for the nonlinear term contribution. We first prove boundedness of moments of the numerical solution. We then prove strong convergence results: first, L^2 ($\Omega$)-convergence of order almost 1/4, localized on an event of arbitrarily large probability, then convergence in probability of order almost 1/4. The theoretical analysis is supported by numerical experiments, concerning strong and weak orders of convergence.