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

Xiaobing Feng; Shu Ma

doi:10.3934/dcdss.2021071

Article Contents

2022, Volume 15, Issue 4: 687-711. Doi: 10.3934/dcdss.2021071

This issue Previous Article Solving the linear transport equation by a deep neural network approach Next Article Stochastic quasi-subgradient method for stochastic quasi-convex feasibility problems

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

Xiaobing Feng^1, , and
Shu Ma^2,

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
^*Corresponding author: Xiaobing Feng

Received: December 31, 2020

Revised: March 31, 2021

Early access: June 2021

Published: April 2022

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

Abstract Full Text(HTML) Figure(9) / Table(1) Related Papers Cited by

Abstract

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.

Keywords:

Stochastic nonlinear Schrödinger equation,
multiplicative noises,
conserved quantities,
explicit and implicit schemes,
finite element methods,
stability analysis.

Mathematics Subject Classification: Primary: 65M12, 65M60.

Citation:

Full Text(HTML)

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 $.

Download: Full-size image PowerPoint slide

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 $

Download: Full-size image PowerPoint slide

Figure 3. Soliton propagation when $ t\in [0, 2] $: graph of the exact solution $ |u(\cdot,t)| $ with $ \sigma = 0 $.

Download: Full-size image PowerPoint slide

Figure 4. Soliton propagation when $ t\in [0, 2] $: numerical solutions with $ \sigma = 0 $, $ h = 0.2 $ and $ \Delta t = 0.025 $.

Download: Full-size image PowerPoint slide

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)

Download: Full-size image PowerPoint slide

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.

Download: Full-size image PowerPoint slide

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} $

Download: Full-size image PowerPoint slide

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} $

Download: Full-size image PowerPoint slide

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} $)

Download: Full-size image PowerPoint slide

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

| Show Table

DownLoad: CSV

Related Papers

Cited by

References

[1]	V. Barbu, M. 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. Cai, J. 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. Chen, J. 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. Cui, J. 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. Feng, H. 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. Lu, Y. 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. Taghizadeh, M. 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.