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 bywhere 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.
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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.
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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).