Analysis of a splitting scheme for damped stochastic nonlinear Schrödinger equation with multiplicative noise

Jianbo Cui; Jialin Hong

doi:10.1137/17M1154904

Analysis of a splitting scheme for damped stochastic nonlinear Schrödinger equation with multiplicative noise

Jianbo Cui, Jialin Hong

Department of Applied Mathematics

Research output: Journal article publication › Journal article › Academic research › peer-review

7 Citations (Scopus)

Abstract

In this paper, we investigate the damped stochastic nonlinear Schrödinger (NLS) equation with multiplicative noise and its splitting-based approximation. When the damped effect is large enough, we prove that the solutions of both the damped stochastic NLS equation and the splitting scheme are exponentially stable and possess some exponential integrability. These properties show that the strong order of the scheme is ¹ ₂ and independent of time. Additionally, we analyze the regularity of the Kolmogorov equation with respect to the stochastic NLS equation. As a consequence, the weak order of the scheme is shown to be 1 and independent of time.

Original language	English
Pages (from-to)	2045-2069
Number of pages	25
Journal	SIAM Journal on Numerical Analysis
Volume	56
Issue number	4
DOIs	https://doi.org/10.1137/17M1154904
Publication status	E-pub ahead of print - 10 Jul 2018

Keywords

Damped stochastic nonlinear Schrödinger equation
Exponential integrability
Kolmogorov equation
Strong order
Weak order

ASJC Scopus subject areas

Numerical Analysis
Computational Mathematics
Applied Mathematics

More information

10.1137/17M1154904

Cite this

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title = "Analysis of a splitting scheme for damped stochastic nonlinear Schr{\"o}dinger equation with multiplicative noise",

abstract = "In this paper, we investigate the damped stochastic nonlinear Schr{\"o}dinger (NLS) equation with multiplicative noise and its splitting-based approximation. When the damped effect is large enough, we prove that the solutions of both the damped stochastic NLS equation and the splitting scheme are exponentially stable and possess some exponential integrability. These properties show that the strong order of the scheme is 1 2 and independent of time. Additionally, we analyze the regularity of the Kolmogorov equation with respect to the stochastic NLS equation. As a consequence, the weak order of the scheme is shown to be 1 and independent of time.",

keywords = "Damped stochastic nonlinear Schr{\"o}dinger equation, Exponential integrability, Kolmogorov equation, Strong order, Weak order",

author = "Jianbo Cui and Jialin Hong",

note = "Funding Information: ∗Received by the editors November 1, 2017; accepted for publication (in revised form) April 16, 2018; published electronically July 10, 2018. http://www.siam.org/journals/sinum/56-4/M115490.html Funding: This work was supported by National Natural Science Foundation of China (91630312, 91530118, and 11290142). †Corresponding author. LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China (jianbocui@lsec.cc.ac.cn). ‡School of Mathematical Science, University of Chinese Academy of Sciences, Beijing, 100049, China (hjl@lsec.cc.ac.cn). Publisher Copyright: {\textcopyright} 2018 Society for Industrial and Applied Mathematics.",

year = "2018",

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doi = "10.1137/17M1154904",

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TY - JOUR

T1 - Analysis of a splitting scheme for damped stochastic nonlinear Schrödinger equation with multiplicative noise

AU - Cui, Jianbo

AU - Hong, Jialin

N1 - Funding Information: ∗Received by the editors November 1, 2017; accepted for publication (in revised form) April 16, 2018; published electronically July 10, 2018. http://www.siam.org/journals/sinum/56-4/M115490.html Funding: This work was supported by National Natural Science Foundation of China (91630312, 91530118, and 11290142). †Corresponding author. LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China (jianbocui@lsec.cc.ac.cn). ‡School of Mathematical Science, University of Chinese Academy of Sciences, Beijing, 100049, China (hjl@lsec.cc.ac.cn). Publisher Copyright: © 2018 Society for Industrial and Applied Mathematics.

PY - 2018/7/10

Y1 - 2018/7/10

N2 - In this paper, we investigate the damped stochastic nonlinear Schrödinger (NLS) equation with multiplicative noise and its splitting-based approximation. When the damped effect is large enough, we prove that the solutions of both the damped stochastic NLS equation and the splitting scheme are exponentially stable and possess some exponential integrability. These properties show that the strong order of the scheme is 1 2 and independent of time. Additionally, we analyze the regularity of the Kolmogorov equation with respect to the stochastic NLS equation. As a consequence, the weak order of the scheme is shown to be 1 and independent of time.

AB - In this paper, we investigate the damped stochastic nonlinear Schrödinger (NLS) equation with multiplicative noise and its splitting-based approximation. When the damped effect is large enough, we prove that the solutions of both the damped stochastic NLS equation and the splitting scheme are exponentially stable and possess some exponential integrability. These properties show that the strong order of the scheme is 1 2 and independent of time. Additionally, we analyze the regularity of the Kolmogorov equation with respect to the stochastic NLS equation. As a consequence, the weak order of the scheme is shown to be 1 and independent of time.

KW - Damped stochastic nonlinear Schrödinger equation

KW - Exponential integrability

KW - Kolmogorov equation

KW - Strong order

KW - Weak order

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DO - 10.1137/17M1154904

M3 - Journal article

AN - SCOPUS:85053549006

VL - 56

SP - 2045

EP - 2069

JO - SIAM Journal on Numerical Analysis

JF - SIAM Journal on Numerical Analysis

SN - 0036-1429

IS - 4

ER -

Analysis of a splitting scheme for damped stochastic nonlinear Schrödinger equation with multiplicative noise

Abstract

Keywords

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