August  2019, 24(8): 4295-4315. doi: 10.3934/dcdsb.2019120

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

Dedicated to Peter Kloeden's 70th Birthday

Received  March 2018 Revised  February 2019 Published  August 2019 Early access  June 2019

Fund Project: 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).

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.

Citation: Jialin Hong, Lijun Miao, Liying Zhang. Convergence analysis of a symplectic semi-discretization for stochastic nls equation with quadratic potential. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4295-4315. doi: 10.3934/dcdsb.2019120
References:
[1]

C. AntonJ. Deng and Y. Wong, Weak symplectic schemes for stochastic Hamiltonian equations, Electron. T. Numer. Ana., 43 (2014), 1-20. 

[2]

R. Anton and D. Cohen, Exponential Integrators for stochastic Schrödinger equations driven by Itô noise, J. Comput. Math., 2 (2018), 276-309.  doi: 10.4208/jcm.1701-m2016-0525.

[3]

W. Bao and Y. Cai, Mathematical theory and numerical methods for Bose-Einstein condensation, Kinet. Relat. Models, 6 (2013), 1-135.  doi: 10.3934/krm.2013.6.1.

[4]

R. BelaouarA. de Bouard and A. Debussche, Numerical analysis of the nonlinear Schrödinger equation with white noise dispersion, Stoch. Partial Differ. Equ. Anal. Comput., 3 (2015), 103-132.  doi: 10.1007/s40072-015-0044-z.

[5]

R. Carles, Critical nonlinear Schrödinger equations with and without harmonic potential, Math. Models Methods Appl. Sci., 12 (2002), 1513-1523.  doi: 10.1142/S0218202502002215.

[6]

C. Chen and J. Hong, Symplectic Runge-Kutta semidiscretization for stochastic Schrödinger equation, SIAM J. Numer. Anal., 54 (2016), 2569-2593.  doi: 10.1137/151005208.

[7]

C. ChenJ. Hong and L. Ji, Mean-Square convergence of a symplectic local discontinuous Galerkin method applied to stochastic linear Schrödinger equation, IMA J. Numer. Anal., 37 (2017), 1041-1065.  doi: 10.1093/imanum/drw023.

[8]

J. CuiJ. Hong and Z. Liu, Strong convergence rate of finite difference approximations for stochastic cubic Schrödinger equations, J. Differ. Equ., 263 (2017), 3687-3713.  doi: 10.1016/j.jde.2017.05.002.

[9]

A. de Bouard and A. Debussche, The stochastic nonlinear Schrödinger equation in $H^1$, Stochastic Anal. Appl., 21 (2003), 97-126.  doi: 10.1081/SAP-120017534.

[10]

A. de Bouard and A. Debussche, A semi-discrete scheme for the stochastic nonlinear Schrödinger equation, Numer. Math., 96 (2004), 733-770.  doi: 10.1007/s00211-003-0494-5.

[11]

A. de Bouard and A. Debussche, Weak and strong order of convergence of a semidiscrete scheme for the stochastic nonlinear Schrödinger equation, Appl. Math. Optim., 54 (2006), 369-399.  doi: 10.1007/s00245-006-0875-0.

[12]

D. FangL. Zhang and T. Zhang, On the well-posedness for stochastic Schrödinger equations with quadratic potential, Chin. Ann. Math. Ser. B, 32 (2011), 711-728.  doi: 10.1007/s11401-011-0670-3.

[13]

J. HongY. LiuH. Munthe-Kaas and A. Zanna, Globally conservative properties and error estimation of a multi-symplectic scheme for Schrödinger equations with variable coefficients, Appl. Numer. Math., 56 (2006), 814-843.  doi: 10.1016/j.apnum.2005.06.006.

[14]

S. JiangL. Wang and J. Hong, Stochastic multi-symplectic integrator for stochastic nonlinear Schrödinger equation, Commun. Comput. Phys., 14 (2013), 393-411.  doi: 10.4208/cicp.230212.240812a.

[15]

V. Konotop and L. Vázquez, Nonlinear Random Waves, World Scientific Publishing Co., Inc., River Edge, NJ, 1994. doi: 10.1142/9789814350259.

[16]

L. MengJ. Li and J. Tao, Blow-up for the stochastic nonlinear Schrödinger equations with quadratic potential and additive noise, Bound. Value Probl., 2015 (2015), 1-18.  doi: 10.1186/s13661-015-0394-5.

[17]

G. MilsteinY. Repin and M. Tretyakov, Numerical methods for stochastic systems preserving symplectic structure., SIAM J. Numer. Anal., 40 (2002), 1583-1604.  doi: 10.1137/S0036142901395588.

show all references

References:
[1]

C. AntonJ. Deng and Y. Wong, Weak symplectic schemes for stochastic Hamiltonian equations, Electron. T. Numer. Ana., 43 (2014), 1-20. 

[2]

R. Anton and D. Cohen, Exponential Integrators for stochastic Schrödinger equations driven by Itô noise, J. Comput. Math., 2 (2018), 276-309.  doi: 10.4208/jcm.1701-m2016-0525.

[3]

W. Bao and Y. Cai, Mathematical theory and numerical methods for Bose-Einstein condensation, Kinet. Relat. Models, 6 (2013), 1-135.  doi: 10.3934/krm.2013.6.1.

[4]

R. BelaouarA. de Bouard and A. Debussche, Numerical analysis of the nonlinear Schrödinger equation with white noise dispersion, Stoch. Partial Differ. Equ. Anal. Comput., 3 (2015), 103-132.  doi: 10.1007/s40072-015-0044-z.

[5]

R. Carles, Critical nonlinear Schrödinger equations with and without harmonic potential, Math. Models Methods Appl. Sci., 12 (2002), 1513-1523.  doi: 10.1142/S0218202502002215.

[6]

C. Chen and J. Hong, Symplectic Runge-Kutta semidiscretization for stochastic Schrödinger equation, SIAM J. Numer. Anal., 54 (2016), 2569-2593.  doi: 10.1137/151005208.

[7]

C. ChenJ. Hong and L. Ji, Mean-Square convergence of a symplectic local discontinuous Galerkin method applied to stochastic linear Schrödinger equation, IMA J. Numer. Anal., 37 (2017), 1041-1065.  doi: 10.1093/imanum/drw023.

[8]

J. CuiJ. Hong and Z. Liu, Strong convergence rate of finite difference approximations for stochastic cubic Schrödinger equations, J. Differ. Equ., 263 (2017), 3687-3713.  doi: 10.1016/j.jde.2017.05.002.

[9]

A. de Bouard and A. Debussche, The stochastic nonlinear Schrödinger equation in $H^1$, Stochastic Anal. Appl., 21 (2003), 97-126.  doi: 10.1081/SAP-120017534.

[10]

A. de Bouard and A. Debussche, A semi-discrete scheme for the stochastic nonlinear Schrödinger equation, Numer. Math., 96 (2004), 733-770.  doi: 10.1007/s00211-003-0494-5.

[11]

A. de Bouard and A. Debussche, Weak and strong order of convergence of a semidiscrete scheme for the stochastic nonlinear Schrödinger equation, Appl. Math. Optim., 54 (2006), 369-399.  doi: 10.1007/s00245-006-0875-0.

[12]

D. FangL. Zhang and T. Zhang, On the well-posedness for stochastic Schrödinger equations with quadratic potential, Chin. Ann. Math. Ser. B, 32 (2011), 711-728.  doi: 10.1007/s11401-011-0670-3.

[13]

J. HongY. LiuH. Munthe-Kaas and A. Zanna, Globally conservative properties and error estimation of a multi-symplectic scheme for Schrödinger equations with variable coefficients, Appl. Numer. Math., 56 (2006), 814-843.  doi: 10.1016/j.apnum.2005.06.006.

[14]

S. JiangL. Wang and J. Hong, Stochastic multi-symplectic integrator for stochastic nonlinear Schrödinger equation, Commun. Comput. Phys., 14 (2013), 393-411.  doi: 10.4208/cicp.230212.240812a.

[15]

V. Konotop and L. Vázquez, Nonlinear Random Waves, World Scientific Publishing Co., Inc., River Edge, NJ, 1994. doi: 10.1142/9789814350259.

[16]

L. MengJ. Li and J. Tao, Blow-up for the stochastic nonlinear Schrödinger equations with quadratic potential and additive noise, Bound. Value Probl., 2015 (2015), 1-18.  doi: 10.1186/s13661-015-0394-5.

[17]

G. MilsteinY. Repin and M. Tretyakov, Numerical methods for stochastic systems preserving symplectic structure., SIAM J. Numer. Anal., 40 (2002), 1583-1604.  doi: 10.1137/S0036142901395588.

Figure 1.  Rates of convergence for $ \epsilon = 0 $ (left) and $ \epsilon = \sqrt{2} $ (right)
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. $
Figure 3.  The profile of $ |u(t, x)| $ when $ \theta = -1 $ (left), $ 0 $ (middle), $ 1 $ (right), respectively
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)
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)
Figure 6.  The global errors of charge conservation law for our proposed scheme (left) and Crank-Nicolson scheme (right) as $ \epsilon = 0 $
Figure 7.  The evolution of the averaged charge for our proposed scheme (left) and Crank-Nicolson scheme (right) at $ T = 100 $
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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