April  2022, 15(4): 687-711. doi: 10.3934/dcdss.2021071

Stable numerical methods for a stochastic nonlinear Schrödinger equation with linear multiplicative noise

1. 

Department of Mathematics, The University of Tennessee, Knoxville, TN 37996, USA

2. 

Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

*Corresponding author: Xiaobing Feng

Received  January 2021 Revised  April 2021 Published  April 2022 Early access  June 2021

Fund Project: The work of the first author was partially supported by the NSF grants DMS-2012414 and DMS-1620168

This paper is concerned with fully discrete finite element approximations of a stochastic nonlinear Schrödinger (sNLS) equation with linear multiplicative noise of the Stratonovich type. The goal of studying the sNLS equation is to understand the role played by the noises for a possible delay or prevention of the collapsing and/or blow-up of the solution to the sNLS equation. In the paper we first carry out a detailed analysis of the properties of the solution which lays down a theoretical foundation and guidance for numerical analysis, we then present a family of three-parameters fully discrete finite element methods which differ mainly in their time discretizations and contains many well-known schemes (such as the explicit and implicit Euler schemes and the Crank-Nicolson scheme) with different combinations of time discetization strategies. The prototypical $ \theta $-schemes are analyzed in detail and various stability properties are established for its numerical solution. An extensive numerical study and performance comparison are also presented for the proposed fully discrete finite element schemes.

Citation: Xiaobing Feng, Shu Ma. Stable numerical methods for a stochastic nonlinear Schrödinger equation with linear multiplicative noise. Discrete and Continuous Dynamical Systems - S, 2022, 15 (4) : 687-711. doi: 10.3934/dcdss.2021071
References:
[1]

V. BarbuM. Rockner and D. Zhang, Stochastic nonlinear Schrödinger equations, Nonlinear Anal., 136 (2016), 168-194.  doi: 10.1016/j.na.2016.02.010.

[2]

A. Bensoussan and R. Temam, Equations stochastiques du type Navier-Stokes, J. Funct. Anal., 13 (1973), 195-222.  doi: 10.1016/0022-1236(73)90045-1.

[3]

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

[4]

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.

[5]

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

[6]

J. Bourgain, Global Solutions of Nonlinear Schrödinger Equations, AMS, Providence, Rhode Island, 1999. doi: 10.1090/coll/046.

[7]

W. CaiJ. Li and Z. Chen, Unconditional convergence and optimal error estimates of the Euler semi-implicit scheme for a generalized nonlinear Schrödinger equation, Advances in Comp. Math., 42 (2016), 1311-1330.  doi: 10.1007/s10444-016-9463-2.

[8]

C. ChenJ. Hong and A. Prohl, Convergence of a $\theta$-scheme to solve the stochastic nonlinear Schrödinger equation with Stratonovich noise, Stoch PDE: Anal. Comp., 4 (2016), 274-318.  doi: 10.1007/s40072-015-0062-x.

[9]

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

[10]

A. Debussche and L. Di Menza, Numerical simulation of focusing stochastic nonlinear Schrödinger equations, Physica D, 162 (2002), 131-154.  doi: 10.1016/S0167-2789(01)00379-7.

[11]

X. Feng, B. Li and S. Ma, High-order mass- and energy-conserving SAV–Gauss collocation finite element methods for the nonlinear Schrödinger equation, SIAM J. Numer. Anal., (to appear).

[12]

X. FengH. Liu and S. Ma, Mass- and energy-conserved numerical schemes for nonlinear Schrödinger equations, Commun. Comput. Phys., 26 (2019), 1365-1396.  doi: 10.4208/cicp.2019.js60.05.

[13]

J. Liu, Order of convergence of splitting schemes for both deterministic and stochastic nonlinear Schrödinger equations, SIAM J. Numer. Anal., 51 (2013), 1911-1932.  doi: 10.1137/12088416X.

[14]

J. Liu, Mass-preserving splitting scheme for the stochastic Schrödinger equation with multiplicative noise, IMA. J. Numer. Anal., 33 (2013), 1469-1479.  doi: 10.1093/imanum/drs051.

[15]

W. LuY. Huang and H. Liu, Mass preserving discontinuous Galerkin methods for Schrödinger equations, J. Comput. Phys., 282 (2015), 210-226.  doi: 10.1016/j.jcp.2014.11.014.

[16]

N. TaghizadehM. Mirzazadeh and F. Farahrooz, Exact solutions of the nonlinear Schrödinger equation by the first integral method, J. Math. Anal. Appl., 374 (2011), 549-553.  doi: 10.1016/j.jmaa.2010.08.050.

[17]

T. Tao, Nonlinear Dispersive Equations, AMS, Providence, Rhode Island, 2006. doi: 10.1090/cbms/106.

[18]

J. Wang, A new error analysis of Crank-Nicolson Galerkin FEMs for a generalized nonlinear Schrödinger equation, J. Scient. Comp., 60 (2014), 390-407.  doi: 10.1007/s10915-013-9799-4.

[19]

Y. Xu and C. W. Shu, Local discontinuous Galerkin methods for nonlinear Schrödinger equations, J. Comput. Phys., 205 (2005), 72-97.  doi: 10.1016/j.jcp.2004.11.001.

show all references

References:
[1]

V. BarbuM. Rockner and D. Zhang, Stochastic nonlinear Schrödinger equations, Nonlinear Anal., 136 (2016), 168-194.  doi: 10.1016/j.na.2016.02.010.

[2]

A. Bensoussan and R. Temam, Equations stochastiques du type Navier-Stokes, J. Funct. Anal., 13 (1973), 195-222.  doi: 10.1016/0022-1236(73)90045-1.

[3]

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

[4]

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.

[5]

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

[6]

J. Bourgain, Global Solutions of Nonlinear Schrödinger Equations, AMS, Providence, Rhode Island, 1999. doi: 10.1090/coll/046.

[7]

W. CaiJ. Li and Z. Chen, Unconditional convergence and optimal error estimates of the Euler semi-implicit scheme for a generalized nonlinear Schrödinger equation, Advances in Comp. Math., 42 (2016), 1311-1330.  doi: 10.1007/s10444-016-9463-2.

[8]

C. ChenJ. Hong and A. Prohl, Convergence of a $\theta$-scheme to solve the stochastic nonlinear Schrödinger equation with Stratonovich noise, Stoch PDE: Anal. Comp., 4 (2016), 274-318.  doi: 10.1007/s40072-015-0062-x.

[9]

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

[10]

A. Debussche and L. Di Menza, Numerical simulation of focusing stochastic nonlinear Schrödinger equations, Physica D, 162 (2002), 131-154.  doi: 10.1016/S0167-2789(01)00379-7.

[11]

X. Feng, B. Li and S. Ma, High-order mass- and energy-conserving SAV–Gauss collocation finite element methods for the nonlinear Schrödinger equation, SIAM J. Numer. Anal., (to appear).

[12]

X. FengH. Liu and S. Ma, Mass- and energy-conserved numerical schemes for nonlinear Schrödinger equations, Commun. Comput. Phys., 26 (2019), 1365-1396.  doi: 10.4208/cicp.2019.js60.05.

[13]

J. Liu, Order of convergence of splitting schemes for both deterministic and stochastic nonlinear Schrödinger equations, SIAM J. Numer. Anal., 51 (2013), 1911-1932.  doi: 10.1137/12088416X.

[14]

J. Liu, Mass-preserving splitting scheme for the stochastic Schrödinger equation with multiplicative noise, IMA. J. Numer. Anal., 33 (2013), 1469-1479.  doi: 10.1093/imanum/drs051.

[15]

W. LuY. Huang and H. Liu, Mass preserving discontinuous Galerkin methods for Schrödinger equations, J. Comput. Phys., 282 (2015), 210-226.  doi: 10.1016/j.jcp.2014.11.014.

[16]

N. TaghizadehM. Mirzazadeh and F. Farahrooz, Exact solutions of the nonlinear Schrödinger equation by the first integral method, J. Math. Anal. Appl., 374 (2011), 549-553.  doi: 10.1016/j.jmaa.2010.08.050.

[17]

T. Tao, Nonlinear Dispersive Equations, AMS, Providence, Rhode Island, 2006. doi: 10.1090/cbms/106.

[18]

J. Wang, A new error analysis of Crank-Nicolson Galerkin FEMs for a generalized nonlinear Schrödinger equation, J. Scient. Comp., 60 (2014), 390-407.  doi: 10.1007/s10915-013-9799-4.

[19]

Y. Xu and C. W. Shu, Local discontinuous Galerkin methods for nonlinear Schrödinger equations, J. Comput. Phys., 205 (2005), 72-97.  doi: 10.1016/j.jcp.2004.11.001.

Figure 1.  Evolution of mass $ \mathscr{M}_h(t)- \mathscr{M}_h(0) $ and energy $ \mathscr{H}_h(t)- \mathscr{H}_h(0) $, with $ \sigma = 0 $ and $ \tau = 0.05, h = 0.2 $.
Figure 2.  Evolution of mass $ \mathbb{E}[ \mathscr{M}_h(t)]- \mathbb{E}[ \mathscr{M}_h(0)] $ and energy $ \mathbb{E}[ \mathscr{H}_h(t)]- \mathbb{E}[ \mathscr{H}_h(0)] $, with $ \sigma = 0.05 $, $ \tau = 0.05, h = 0.2 $ and $ M = 500 $
Figure 3.  Soliton propagation when $ t\in [0, 2] $: graph of the exact solution $ |u(\cdot,t)| $ with $ \sigma = 0 $.
Figure 4.  Soliton propagation when $ t\in [0, 2] $: numerical solutions with $ \sigma = 0 $, $ h = 0.2 $ and $ \Delta t = 0.025 $.
Figure 5.  Plots in $ (x,t) $ plane of $ |U| $ for one trajectory: (a) $ \sigma $ = 0.001, (b) $ \sigma $ = 0.1, (c) $ \sigma $ = 0.5, (d) contour plot of $ |U| $ for $ \sigma $ = 0.5 (multiplicative noise)
Figure 6.  Rates of convergence with τ ∈ {2-i; 1 ≤ i ≤ 5}. left: σ = 0, T = 0.1 , right: σ = 0.05, T = 0.5.
Figure 7.  The sensitivity of $ E[ \mathscr{M}^n_h] $ in different subdomains using different time step sizes. (a) Crank-Nicolson scheme : $ \theta_i = \frac{1}{2}, i = 1,2,3 $; (b) Implicit Euler scheme : $ \theta _i = 1, i = 1,2,3 $ ; (c) Hybrid scheme 1: $ \theta_1 = \frac{1}{2} ,\theta_2 = 1, \theta_3 = \frac{1}{2} $; (d) Hybrid scheme 2: $ \theta_1 = 1, \theta_2 = \frac{1}{2}, \theta_3 = \frac{1}{2} $
Figure 8.  The sensitivity of $ E[ \mathscr{H}^n_h] $ in different subdomains using different time step sizes. (a) Crank-Nicolson scheme : $ \theta_i = \frac{1}{2}, i = 1,2,3 $; (b) Implicit Euler scheme : $ \theta _i = 1, i = 1,2,3 $ ; (c) Hybrid scheme 1: $ \theta_1 = \frac{1}{2} ,\theta_2 = 1, \theta_3 = \frac{1}{2} $; (d) Hybrid scheme 2: $ \theta_1 = 1, \theta_2 = \frac{1}{2}, \theta_3 = \frac{1}{2} $
Figure 9.  The different increasing speeds between the Euler Explicit scheme $ (\theta_i = 0,i = 1,2,3) $ and the Hybrid scheme 1 with ($ \theta_1 = \frac{1}{2}, \theta_2 = 1, \theta_3 = \frac{1}{2} $)
Table 1.  The comparison between the $ {\theta} $-scheme and other commonly used numerical schemes $ (i = 1,2,3). $
1 $\theta_i= 0$ Explicit Euler scheme 3 ${\theta_i}=1 $ Implicit Euler scheme
2 $\theta_i=\frac{1}{2}$ Crank-Nicolson scheme 4 Others Some hybrid schemes
1 $\theta_i= 0$ Explicit Euler scheme 3 ${\theta_i}=1 $ Implicit Euler scheme
2 $\theta_i=\frac{1}{2}$ Crank-Nicolson scheme 4 Others Some hybrid schemes
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