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

  • *Corresponding author: Xiaobing Feng

    *Corresponding author: Xiaobing Feng 

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

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  • 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.

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

    Citation:

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  • 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
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