Convergence analysis of a symplectic semi-discretization for stochastic nls equation with quadratic potential

Jialin Hong; Lijun Miao; Liying Zhang

doi:10.3934/dcdsb.2019120

Article Contents

2019, Volume 24, Issue 8: 4295-4315. Doi: 10.3934/dcdsb.2019120

This issue Previous Article Malliavin regularity and weak approximation of semilinear SPDEs with Lévy noise Next Article Approximation of the interface condition for stochastic Stefan-type problems

Convergence analysis of a symplectic semi-discretization for stochastic nls equation with quadratic potential

1.
LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China
2.
School of Mathematical Science, University of Chinese Academy of Sciences, Beijing, 100049, China
3.
School of Mathematics, Liaoning Normal University, Dalian, 116029, China
4.
Department of Mathematics, College of Sciences, China University of Mining and Technology, Beijing, 100083, China

^* Corresponding author: Lijun Miao
^* Corresponding author: Lijun Miao

Dedicated to Peter Kloeden's 70th Birthday

Received: February 28, 2018

Revised: January 31, 2019

Early access: June 2019

Published: July 2019

The first author is supported by the NNSFC (NO. 91530118, NO. 11290142 and NO. 91630312). The third author is supported by the NNSFC (NO.11601514, NO.11801556 and NO.11771444)

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Abstract

In this paper, we investigate the convergence in probability of a stochastic symplectic scheme for stochastic nonlinear Schrödinger equation with quadratic potential and an additive noise. Theoretical analysis shows that our symplectic semi-discretization is of order one in probability under appropriate regularity conditions for the initial value and noise. Numerical experiments are given to simulate the long time behavior of the discrete averaged charge and energy as well as the influence of the external potential and noise, and to test the convergence order.

Keywords:

Stochastic nonlinear Schrödinger equation,
quadratic potential,
additive noise,
stochastic symplectic scheme,
convergence analysis.

Mathematics Subject Classification: Primary: 60H15, 60H35, 65P10.

Citation:

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Figure 1. Rates of convergence for $ \epsilon = 0 $ (left) and $ \epsilon = \sqrt{2} $ (right)

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Figure 2. The profile of numerical solution $ |u(x, t)| $ for one trajectory with different noise when $ \theta = -1 $. The left figure is the case of $ \epsilon = 0.05 $, The right figure is the case of $ \epsilon = 0.5. $

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Figure 3. The profile of $ |u(t, x)| $ when $ \theta = -1 $ (left), $ 0 $ (middle), $ 1 $ (right), respectively

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Figure 4. The evolution of the averaged discrete charge as $ \theta = -1, 0, 1, \lambda = 1, \sigma = 1 $ (left); $ \epsilon = 0, 0.05, 0.5, \theta = -1, \lambda = 1, \sigma = 1 $ (right)

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Figure 5. The evolution of the averaged discrete energy as $ \theta = -1, 0, 1, \lambda = 1, \sigma = 1 $ (left); $ \epsilon = 0, 0.05, 0.5, \theta = -1, \lambda = 1, \sigma = 1 $ (right)

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Figure 6. The global errors of charge conservation law for our proposed scheme (left) and Crank-Nicolson scheme (right) as $ \epsilon = 0 $

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Figure 7. The evolution of the averaged charge for our proposed scheme (left) and Crank-Nicolson scheme (right) at $ T = 100 $

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Figure 8. The evolution of the averaged charge for our proposed scheme (left) and Crank-Nicolson scheme (right) at $ T=1000 $

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