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

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:263

Article

Cui, Jianbo¹  Hong, Jialin¹  Liu, Zhihui¹ 

[1] Chinese Acad Sci, Acad Math & Syst Sci, Inst Computat Math & Sci Engn Comp, State Key Lab Sci & Engn Comp, Beijing 100190, Peoples R China

关键词: Stochastic cubic Schrodinger equation;    Strong convergence rate;    Central difference scheme;    Exponential integrability;    Continuous dependence;   

DOI  :  10.1016/j.jde.2017.05.002

来源: Elsevier

【 摘 要 】

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.

【 授权许可】

Free