Analysis of Some Splitting Schemes for the Stochastic Allen-Cahn Equation
For researchers in stochastic PDEs, this provides a theoretically justified explicit scheme for a challenging nonlinear SPDE, though the convergence rate is low and the result is incremental.
The paper introduces and analyzes an explicit splitting scheme for the one-dimensional stochastic Allen-Cahn equation, proving strong convergence of order almost 1/4 in probability and L^2, supported by numerical experiments.
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 ($Ω$)-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.