Strong convergence rate of finite difference approximations for stochastic cubic Schrodinger equations

doi:10.1016/j.jde.2017.05.002

	Strong convergence rate of finite difference approximations for stochastic cubic Schrodinger equations
	Cui, Jianbo; Hong, Jialin; Liu, Zhihui
	2017-10-05
Source Publication	JOURNAL OF DIFFERENTIAL EQUATIONS
ISSN	0022-0396
Volume	263 Issue:7 Pages:3687-3713
Abstract	In this paper, we derive a strong convergence rate of spatial finite difference approximations for both focusing and defocusing stochastic cubic Schrodinger equations driven by a multiplicative Q-Wiener process. Beyond the uniform boundedness of moments for high order derivatives of the exact solution, the key requirement of our approach is the exponential integrability of both the exact and numerical solutions. By constructing and analyzing a Lyapunov functional and its discrete correspondence, we derive the uniform boundedness of moments for high order derivatives of the exact solution and the first order derivative of the numerical solution, which immediately yields the well-posedness of both the continuous and discrete problems. The latter exponential integrability is obtained through a variant of a criterion given by Cox, Hutzenthaler and Jentzen [arXiv:1309.5595]. As a by-product of this exponential integrability, we prove that the exact and numerical solutions depend continuously on the initial data and obtain a large deviation type result on the dependence of the noise with first order strong convergence rate. (C) 2017 Elsevier Inc. All rights reserved.
Keyword	Stochastic cubic Schrodinger equation Strong convergence rate Central difference scheme Exponential integrability Continuous dependence
DOI	10.1016/j.jde.2017.05.002
Language	英语
Funding Project	National Natural Science Foundation of China[91630312] ; National Natural Science Foundation of China[91530118] ; National Natural Science Foundation of China[11290142]
WOS Research Area	Mathematics
WOS Subject	Mathematics
WOS ID	WOS:000406354200002
Publisher	ACADEMIC PRESS INC ELSEVIER SCIENCE
Citation statistics
Document Type	期刊论文
Identifier	http://ir.amss.ac.cn/handle/2S8OKBNM/26295
Collection	计算数学与科学工程计算研究所
Corresponding Author	Liu, Zhihui
Affiliation	Chinese Acad Sci, Acad Math & Syst Sci, Inst Computat Math & Sci Engn Comp, State Key Lab Sci & Engn Comp, Beijing 100190, Peoples R China
Recommended Citation GB/T 7714	Cui, Jianbo,Hong, Jialin,Liu, Zhihui. Strong convergence rate of finite difference approximations for stochastic cubic Schrodinger equations[J]. JOURNAL OF DIFFERENTIAL EQUATIONS,2017,263(7):3687-3713.
APA	Cui, Jianbo,Hong, Jialin,&Liu, Zhihui.(2017).Strong convergence rate of finite difference approximations for stochastic cubic Schrodinger equations.JOURNAL OF DIFFERENTIAL EQUATIONS,263(7),3687-3713.
MLA	Cui, Jianbo,et al."Strong convergence rate of finite difference approximations for stochastic cubic Schrodinger equations".JOURNAL OF DIFFERENTIAL EQUATIONS 263.7(2017):3687-3713.

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