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August  2019, 24(8): 4169-4190. doi: 10.3934/dcdsb.2019077

Analysis of some splitting schemes for the stochastic Allen-Cahn equation

 1 Univ Lyon, CNRS, Université Claude Bernard Lyon 1, UMR5208, Institut Camille Jordan, 43 boulevard du 11 novembre 1918, F-69622 Villeurbanne, France 2 Université Paris-Saclay, CNRS - FR3487, Fédération de Mathématiques de CentraleSupélec, CentraleSupélec, 3 rue Joliot Curie, F-91190 Gif-sur-Yvette, France

* Corresponding author: Charles-Edouard Bréhier

Received  January 2018 Revised  October 2018 Published  April 2019

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.

Citation: Charles-Edouard Bréhier, Ludovic Goudenège. Analysis of some splitting schemes for the stochastic Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4169-4190. doi: 10.3934/dcdsb.2019077
References:

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References:
Mean square error order for $T = 1$, $\Delta x = 2.5~10^{-4}$ and $10^{5}$ independent realizations
Weak error order for $T = 1$, $\Delta x = 2.5~10^{-4}$ and $10^{5}$ independent realizations
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