Abstract and Applied Analysis

Volume 2014 (2014), Article ID 427023, 7 pages

http://dx.doi.org/10.1155/2014/427023

Research Article

Symplectic Schemes for Linear Stochastic Schrödinger Equations with Variable Coefficients

Xiuling Yin1,2 and Yanqin Liu1,2

¹School of Mathematical Sciences, Dezhou University, Dezhou 253023, China

²The Center of Data Processing and Analyzing, Dezhou University, Dezhou 253023, China

Received 16 December 2013; Accepted 4 April 2014; Published 22 April 2014

Academic Editor: Adem Kılıçman

Copyright © 2014 Xiuling Yin and Yanqin Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper proposes a kind of symplectic schemes for linear Schrödinger equations with variable coefficients and a stochastic perturbation term by using compact schemes in space. The numerical stability property of the schemes is analyzed. The schemes preserve a discrete charge conservation law. They also follow a discrete energy transforming formula. The numerical experiments verify our analysis.

1. Introduction

Differential equations (DEs) are important models in sciences and engineering. By theoretical and numerical analysis of DEs, we can yield some mathematical explanation of many phenomena in applied sciences [1–5]. Time-dependent Schrödinger equations arise in quantum physics, optics, and many other fields [6, 7]. Some numerical methods for such equations, such as symplectic scheme and multisymplectic schemes, have been proposed in [8–14]. The schemes possess good numerical stability. Compact schemes are popular recently due to high accuracy, compactness, and economic resource in scientific computation [15–17]. In this paper, applying compact operators, we construct symplectic methods to the initial boundary problems of the linear Schrödinger equation with a variable coefficient and a stochastic perturbation term (denoted by LSES):

where , is a real differential function, is a differential function, is a small real number, and means Stratonovich product. is a real-valued white noise which is delta correlated in time and either smooth or delta correlated in space. For an integer , and mean the -order partial derivatives of with respect to and , respectively. The system (1) with is a deterministic system. When is small, we can think that (1) is perturbed by the stochastic term.

By multiplying (1) by or and then integrating it with respect to and , it is easy to verify the following result.

Proposition 1. Under the periodic boundary condition,

(a)the solution of (1) satisfies the charge conservation law

(b)the corresponding deterministic system ( ) of (1) possesses the energy conservation law

The paper is organized as follows. In Section 2, we give a symplectic structure of the LSES. In Section 3, we present the new symplectic methods to the LSES. First, we use a kind of compact schemes in discretization of spatial derivative. Then, in temporal discretization, we adopt the symplectic midpoint method. The new methods are denoted by LSC schemes. We also analyze the numerical stability of LSC schemes. We give two numerical examples to support our theory in Section 4. At last, we make some conclusions.

2. Symplectic Structure of the LSES

Let . The LSES (1) can be written in

Introducing the variable , (4) reads in stochastic symplectic context

where

The system satisfies the symplectic conservation law [7, 12, 18]:

Numerical methods which preserve the discrete symplectic conservation law are called symplectic methods. Symplectic methods have good numerical stability.

3. LSC Schemes

3.1. Compact Scheme

Introduce the following uniform mesh grids:

where and are spatial and temporal step sizes, respectively. Denote the numerical values of at the nodes by . The symbols and mean the numerical solution vectors at and with components , respectively. Furthermore, we denote

Introducing the following linear operators

we adopt formula [19]

to approximate , which means that

By Taylor expansion, we can derive a family of fourth-order schemes with

The leading term of the truncation error of the method is . If , we get a scheme with smaller stencil. A sixth-order scheme is obtained with

Denote two symmetric and cyclic matrices by

where and . Then the matrix form of (11) is

3.2. Discretization of the LSES

Applying the approximation (11) to linear system (4), we obtain the following semidiscretization stochastic Hamiltonian system:

where

In temporal discretization of (17), we apply the symplectic midpoint method

Its componentwise formulation is

According to the Fourier analysis, the LSC schemes (19) are unconditionally stable. In fact, we can derive

where

Then, with (19) and (21), we can obtain with

where . By direct computation, we can derive that the spectral radius of the matrix is 1 and . Therefore, the scheme (19) is unconditionally stable. Moreover, by symmetry, they are nondissipative.

Theorem 2. Let . Then, is the discrete charge invariant of the LSC schemes (19), which implies the discrete charge conservation law of (2).

Scheme (19) can be rewritten in compact form

Multiplying (24) by and summing over , we obtain

Since and are symmetric, the first three summation terms in the above equality are purely imaginary, while the last four summation terms are real. Denote

Now, taking the imaginary parts of (25), we can get that

Denote

According to the Green formula, we obtain that

Then,

Therefore, from the above analysis, we give the following result.

Theorem 3. Under the periodic boundary condition, the LSC schemes (19) satisfy the discrete energy transforming law (27).

4. Numerical Examples

We use the LSC scheme to solve the LSESs and investigate its numerical behavior. According to the precise mathematical definition of the white noise [13, 14], we can simulate the noise as , where , is a sequence of independent random variables with normal law at each time increment. Denote

The numerical residuals of and are measured by and , respectively. For numerical computation, we take , , and .

Example 1. LSE with constant coefficients and periodic boundary condition.

Consider

The exact solution of its deterministic system is . The right side in the above system can be seen as a stochastic perturbation term.

Figure 1 plots the amplitude for one trajectory. Figure 2 shows the evolution of for one trajectory and the average norm over 50 trajectories. We see that the white noise produces stochastic perturbation on the solitary wave and the size of perturbation depends on the size of noise. Figure 2 plots the residuals of discrete charge of one trajectory, which verifies the conservation of discrete charge of the compact schemes. Figure 3 plots the residuals of discrete energy for one trajectory and the average norm over 50 trajectories. The figure tells us that the stochastic noise makes residuals of discrete energy increase linearly over time.

(a)

(b)

Figure 1: for one trajectory.

(a)

(b)

Figure 2: for one trajectory and the average norm over 50 trajectories (a). Residuals of discrete charge for one trajectory (b).

(a)

(b)

Figure 3: Residuals of discrete energy for one trajectory and the average energy over 50 trajectories.

Example 2. LSE with a variable coefficient and periodic boundary condition.

Consider

The exact solution of its deterministic system is .

Figure 4 plots the amplitude for one trajectory. Figure 5 shows the evolution of for one trajectory and the average norm over 50 trajectories. We see that the white noise produces stochastic perturbation on the solitary wave and the size of perturbation depends on the size of noise. Figure 5 plots the residuals of discrete charge of one trajectory, which verifies the conservation of discrete charge of the compact schemes. Figure 6 plots the residuals of discrete energy for one trajectory and the average norm over 50 trajectories. The figure tells us that the stochastic noise makes residuals of discrete energy increase linearly over time.

(a)

(b)

Figure 4: for one trajectory.

(a)

(b)

Figure 5: for one trajectory and the average norm over 50 trajectories (a). Residuals of discrete charge for one trajectory (b).

(a)

(b)

Figure 6: Residuals of discrete energy for one trajectory and the average energy over 50 trajectories.

5. Conclusion

In this paper, we apply a symplectic scheme in time and a kind of compact difference schemes in space to solve the LSES. The methods are unconditionally stable. Under periodic boundary conditions, they preserve a discrete charge invariant and satisfy a discrete energy transforming law.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

Xiuling Yin is supported by the Director Innovation Foundation of ICMSEC and AMSS, the Foundation of CAS, NNSFC (nos. 91130003 and 11021101). Yanqin Liu is supported by the Natural Science Foundation of Shandong Province (nos. ZR2013AQ005 and BS2013HZ026).

References

A. Ashyralyev, A. Hanalyev, and P. E. Sobolevskii, “Coercive solvability of the nonlocal boundary value problem for parabolic differential equations,” Abstract and Applied Analysis, vol. 6, no. 1, pp. 53–61, 2001.

M. Pašić, “Fite-Wintner-Leighton-type oscillation criteria for second-order differential equations with nonlinear damping,” Abstract and Applied Analysis, vol. 2013, Article ID 852180, 10 pages, 2013.

W. M. Abd-Elhameed, E. H. Doha, and Y. H. Youssri, “New wavelets collocation method for solving second-order multipoint boundary value problems using Chebyshev polynomials of third and fourth kinds,” Abstract and Applied Analysis, vol. 2013, Article ID 542839, 9 pages, 2013.

K. Ivaz, A. Khastan, and J. J. Nieto, “A numerical method for fuzzy differential equations and hybrid fuzzy differential equations,” Abstract and Applied Analysis, vol. 2013, Article ID 735128, 10 pages, 2013.

C. Ashyralyyev, A. Dural, and Y. Sozen, “Finite difference method for the reverse parabolic problem,” Abstract and Applied Analysis, vol. 2012, Article ID 294154, 17 pages, 2012.

A. Phillips, Introduction to Quantum Mechanics, John Wiley & Sons, Chichester, UK, 2003.

G. Milstein and M. Tretyakov, Stochastic Numerics for Mathematical Physics, Kluwer Academic Publishers, Norwell, Mass, USA, 1995.

J. Hong and M.-Z. Qin, “Multisymplecticity of the centred box discretization for Hamiltonian PDEs with space dimensions,” Applied Mathematics Letters, vol. 15, no. 8, pp. 1005–1011, 2002.

L. Kong, J. Hong, L. Wang, and F. Fu, “Symplectic integrator for nonlinear high order Schrödinger equation with a trapped term,” Journal of Computational and Applied Mathematics, vol. 231, no. 2, pp. 664–679, 2009.

J. Hong, Y. Liu, H. Munthe-Kaas, and A. Zanna, “Globally conservative properties and error estimation of a multi-symplectic scheme for Schrödinger equations with variable coefficients,” Applied Numerical Mathematics, vol. 56, no. 6, pp. 814–843, 2006.

J. Hong, X. Liu, and C. Li, “Multi-symplectic Runge-Kutta-Nyström methods for nonlinear Schrödinger equations with variable coefficients,” Journal of Computational Physics, vol. 226, no. 2, pp. 1968–1984, 2007.

Y.-F. Tang, L. Vázquez, F. Zhang, and V. M. Pérez-García, “Symplectic methods for the nonlinear Schrödinger equation,” Computers & Mathematics with Applications, vol. 32, no. 5, pp. 73–83, 1996.

S. Jiang, L. Wang, and J. Hong, “Stochastic multi-symplectic integrator for stochastic nonlinear Schrödinger equation,” Communications in Computational Physics, vol. 14, no. 2, pp. 393–411, 2013.

A. Debussche and L. di Menza, “Numerical simulation of focusing stochastic nonlinear Schrödinger equations,” Physica D, vol. 162, no. 3-4, pp. 131–154, 2002.

Y. Ma, L. Kong, J. Hong, and Y. Cao, “High-order compact splitting multisymplectic method for the coupled nonlinear Schrödinger equations,” Computers & Mathematics with Applications, vol. 61, no. 2, pp. 319–333, 2011.

T. V. S. Sekhar and B. H. S. Raju, “An efficient higher order compact scheme to capture heat transfer solutions in spherical geometry,” Computer Physics Communications, vol. 183, no. 11, pp. 2337–2345, 2012.

A. Mohebbi and Z. Asgari, “Efficient numerical algorithms for the solution of “good” Boussinesq equation in water wave propagation,” Computer Physics Communications, vol. 182, no. 12, pp. 2464–2470, 2011.

E. Hairer, C. Lubich, and G. Wanner, Geometric Numerical Integration, vol. 31 of Springer Series in Computational Mathematics, Springer, Berlin, Germany, 2nd edition, 2006.

S. K. Lele, “Compact finite difference schemes with spectral-like resolution,” Journal of Computational Physics, vol. 103, no. 1, pp. 16–42, 1992.