Symplectic Schemes for Linear Stochastic Schrödinger Equations with Variable Coefficients

This paper proposes a kind of symplectic schemes for linear Schr¨odinger 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.


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][2][3][4][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][9][10][11][12][13][14]. The schemes possess good numerical stability. Compact schemes are popular recently due to high accuracy, compactness, and economic resource in scientific computation [15][16][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 2 = −1, ( ) is a real differential function, 0 ( ) 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 = 0 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 ( = 0) of (1) possesses the energy conservation law 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.

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.

LSC Schemes
3.1. Compact Scheme. Introduce the following uniform mesh grids: where ℎ = / 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 4 , 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 ((7− 26 )/240)( 8 ) ℎ 4 . If = 0, we get a scheme with smaller stencil. A sixth-order scheme is obtained with Denote two symmetric and cyclic matrices by where 0 = 16 + 36 and 1 = −9 − 24 . Then the matrix form of (11) is

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 Then, with (19) and (21), we can obtain +1 = with where = + ( ) − ∘̇+ (1/2) . By direct computation, we can derive that the spectral radius of the matrix is 1 and ‖ ‖ 2 = 1. Therefore, the scheme (19) is unconditionally stable. Moreover, by symmetry, they are nondissipative.
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 ( +1 − ) and summing over , we obtain  Since A and B 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 Abstract and Applied Analysis  According to the Green formula, we obtain that Then, Therefore, from the above analysis, we give the following result.
The exact solution of its deterministic system is ( , ) = exp[ ( + ( /3))] cos 2 . 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.
The exact solution of its deterministic system is ( , , ) = sin 2 .
Abstract and Applied Analysis 7 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.

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.