TY - JOUR

T1 - Strong convergence rates of semidiscrete splitting approximations for the stochastic Allen-Cahn equation

AU - Bréhier, Charles Edouard

AU - Cui, Jianbo

AU - Hong, Jialin

N1 - Funding Information:
National Natural Science Foundation of China (No. 91630312, No. 91530118, No.11021101, No. 11290142).
Publisher Copyright:
© 2018 The Author(s). Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

PY - 2019/10/16

Y1 - 2019/10/16

N2 - 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 $\mathcal{O}(\delta t)$.

AB - 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 $\mathcal{O}(\delta t)$.

KW - exponential integrability

KW - splitting scheme

KW - stochastic Allen-Cahn equation

KW - strong convergence rate

UR - http://www.scopus.com/inward/record.url?scp=85055977710&partnerID=8YFLogxK

U2 - 10.1093/imanum/dry052

DO - 10.1093/imanum/dry052

M3 - Journal article

AN - SCOPUS:85055977710

VL - 39

SP - 2096

EP - 2134

JO - IMA Journal of Numerical Analysis

JF - IMA Journal of Numerical Analysis

SN - 0272-4979

IS - 4

ER -