TY - JOUR
T1 - Strong convergence rate of finite difference approximations for stochastic cubic Schrödinger equations
AU - Cui, Jianbo
AU - Hong, Jialin
AU - Liu, Zhihui
N1 - Funding Information:
This work was supported by National Natural Science Foundation of China (No. 91630312, No. 91530118 and No. 11290142).
Publisher Copyright:
© 2017 Elsevier Inc.
PY - 2017/10/5
Y1 - 2017/10/5
N2 - In this paper, we derive a strong convergence rate of spatial finite difference approximations for both focusing and defocusing stochastic cubic Schrödinger 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.
AB - In this paper, we derive a strong convergence rate of spatial finite difference approximations for both focusing and defocusing stochastic cubic Schrödinger 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.
KW - Central difference scheme
KW - Continuous dependence
KW - Exponential integrability
KW - Stochastic cubic Schrödinger equation
KW - Strong convergence rate
UR - http://www.scopus.com/inward/record.url?scp=85019349656&partnerID=8YFLogxK
U2 - 10.1016/j.jde.2017.05.002
DO - 10.1016/j.jde.2017.05.002
M3 - Journal article
AN - SCOPUS:85019349656
SN - 0022-0396
VL - 263
SP - 3687
EP - 3713
JO - Journal of Differential Equations
JF - Journal of Differential Equations
IS - 7
ER -