Analysis of some splitting schemes for the stochastic Allen-Cahn equation. (English) Zbl 07099047
Summary: 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.
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.
MSC:
| 65C30 | Numerical solutions to stochastic differential and integral equations |
| 60H35 | Computational methods for stochastic equations (aspects of stochastic analysis) |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
