ANALYSIS OF SOME SPLITTING SCHEMES FOR THE STOCHASTIC ALLEN-CAHN EQUATION

Charles-Edouard Bréhier (1, 2) , Ludovic Goudenège (3)

1 ICJ - Institut Camille Jordan
2 PSPM - Probabilités, statistique, physique mathématique
3 FR3487 - Fédération de Mathématiques de l'Ecole Centrale Paris

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.

Domaines: Analyse numérique [math.NA]; Probabilités [math.PR]; Equations aux dérivées partielles [math.AP]

Charles-Edouard Bréhier, Ludovic Goudenège. ANALYSIS OF SOME SPLITTING SCHEMES FOR THE STOCHASTIC ALLEN-CAHN EQUATION. Discrete and Continuous Dynamical Systems - Series B, 2019, ⟨10.3934/dcdsb.2019077⟩. ⟨hal-01688333⟩