Strong convergence rates of semidiscrete splitting approximations for the stochastic Allen-Cahn equation

Charles Edouard Bréhier, Jianbo Cui, Jialin Hong

Research output: Journal article publicationJournal articleAcademic researchpeer-review

26 Citations (Scopus)

Abstract

This article analyses an explicit temporal splitting numerical scheme for the stochastic Allen-Cahn equation driven by additive noise in a bounded spatial domain with smooth boundary in dimension $d\leqslant 3$. The splitting strategy is combined with an exponential Euler scheme of an auxiliary problem. When $d=1$ and the driving noise is a space-time white noise we first show some a priori estimates of this splitting scheme. Using the monotonicity of the drift nonlinearity we then prove that under very mild assumptions on the initial data this scheme achieves the optimal strong convergence rate $\mathcal{O}(\delta t^{\frac 14})$. When $d\leqslant 3$ and the driving noise possesses some regularity in space we study exponential integrability properties of the exact and numerical solutions. Finally, in dimension $d=1$, these properties are used to prove that the splitting scheme has a strong convergence rate $\mathcal{O}(\delta t)$.

Original languageEnglish
Pages (from-to)2096-2134
Number of pages39
JournalIMA Journal of Numerical Analysis
Volume39
Issue number4
DOIs
Publication statusPublished - 16 Oct 2019
Externally publishedYes

Keywords

  • exponential integrability
  • splitting scheme
  • stochastic Allen-Cahn equation
  • strong convergence rate

ASJC Scopus subject areas

  • Mathematics(all)
  • Computational Mathematics
  • Applied Mathematics

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