Article Contents
Article Contents

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

• * Corresponding author: Lijun Miao

Dedicated to Peter Kloeden's 70th Birthday

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.

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

 Citation:

• 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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