Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 758954, 11 pages
http://dx.doi.org/10.1155/2015/758954

Research Article

Convergent Analysis of Energy Conservative Algorithm for the Nonlinear Schrödinger Equation

Lv Zhong-Quan,1,2,3 Gong Yue-Zheng,³ and Wang Yu-Shun³

¹College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
²College of Science, Nanjing Forestry University, Nanjing 210037, China
³Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210097, China

Received 26 September 2014; Accepted 27 February 2015

Academic Editor: Slim Choura

Copyright © 2015 Lv Zhong-Quan et al. 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.

Using average vector field method in time and Fourier pseudospectral method in space, we obtain an energy-preserving scheme for the nonlinear Schrödinger equation. We prove that the proposed method conserves the discrete global energy exactly. A deduction argument is used to prove that the numerical solution is convergent to the exact solution in discrete norm. Some numerical results are reported to illustrate the efficiency of the numerical scheme in preserving the energy conservation law.

1. Introduction

The nonlinear Schrödinger (NLS) equation describes a wide range of physical phenomena, such as hydrodynamics, plasma physics, nonlinear optics, self-focusing in laser pulses, propagation of heat pulses in crystals, and description of the dynamics of Bose-Einstein condensate at extremely low temperature [1, 2]. It plays an essential role in mathematical and physical context, and more and more focus is concentrated upon its numerical solvers in recent years [3, 4]. For the NLS equation, construction and theoretical analysis of numerical algorithms have achieved fruitful results [5–14].

The general form of the NLS equation with the initial value and the periodic boundary condition is

where is a real parameter. Now using , we can rewrite (1) as a pair of real-valued equations as follows:

Equations (2) can be expressed in the Hamiltonian form. Consider

where and the Hamiltonian function, which is system energy, is

The NLS equation (1) admits the energy conservation law. Consider

Quispel and McLaren [15] proposed the average vector field (AVF) method, which is a second-order energy-preserving method, and they also provided the corresponding high-order method which is of fourth-order accuracy. The second-order energy-preserving method has been applied to solve the partial differential equation [16]. However, to our knowledge, the current papers are most concentrated on construction of energy-preserving scheme, and very few papers discussed convergent analysis of the energy-preserving scheme. In this paper, we develop an energy conservative algorithm for the NLS equation by using AVF method in time and Fourier pseudospectral method in space and analyze the proposed method.

The paper is organized as follows. In Section 2, a new conservative scheme is proposed for the NLS equation. We prove that the method preserves the energy conservation law. In Section 3, a deduction argument is used to prove that the numerical solution is convergent to the exact solution in discrete norm. The solitary wave behaviors for the NLS equation are simulated by the new scheme in Section 4. In Section 5, it is devoted to the conclusions.

2. Construction of Conservative Algorithm for the NLS Equation

In this section, we apply the Fourier pseudospectral method in space and the AVF method in time to construct an energy-preserving algorithm for the NLS equation.

One usually uses second-order Fourier spectral differentiation matrix to approximate the second-order differential operator . For the ordinary differential equation , we set and . Applying the Fourier pseudospectral method to the two equations leads to and . Eliminating vector gives . In this work, we use to approximate instead of and obtain the corresponding Fourier pseudospectral semidiscretization for the NLS equation (1) as follows:

where . Equations (6) can be rewritten as

where and . Since is symmetric, (7) is regarded as a Hamiltonian system with Hamiltonian. Consider

Now we discretize (6) with respect to time by the AVF method and obtain

where , , and . Obviously, scheme (9) can be reformed as a vector form. Consider

where and “” denotes point multiplication between vectors; that is,

Equations (9) can also be rewritten as

Next, we prove that scheme (10) conserves the discrete total energy. Let and define discrete inner product and discrete norm over as

Theorem 1. With periodic boundary condition , scheme (10) possesses the discrete global energy conservation law; namely,

where and .

Proof. Taking the inner product of (10) with yields

The first term becomes

which is purely imaginary. The second term can be reduced to

Noticing that , we have

Therefore, the real part of (15) is

So (19) gives the energy conservation law (14).

3. Convergence Analysis

Let , with the inner product and the norm . For any positive integer , the seminorm and the norm of are denoted by and , respectively. Let be the set of infinitely differentiable functions with period , defined on . is the closure of in . In this section, let be a generic positive constant which may be dependent on the regularity of exact solution and the initial data but independent of the time step and the grid size .

For even , set

where the summation is defined by

It is obviously that . Denote the orthogonal projection operator and the interpolation operator . Note that and satisfy the following properties:

(1), ;(2), ;(3), ; , .

Lemma 2. For , .

Lemma 3 (see [17]). If and , then

and if , then

Lemma 4. Suppose , , and ; then .

Proof. According to Lemmas 2 and 3, we have

Lemma 5 (Gronwall’s inequality [18]). Suppose that the discrete function satisfies the following inequality:

where , , and are nonnegative constants. Then

where is sufficiently small, such that .

An equivalent form of full-discrete Fourier pseudospectral scheme (12) is to find , so that, for any , then

where

Proposed scheme (10) conserves the energy exactly, which can be regarded as the energy stable algorithm. So we assume that the numerical solution is bounded; that is,

Theorem 6. Suppose that the exact solutions , , and are small enough; then the solution of full-discrete Fourier pseudospectral scheme (12) converges to the solution of problems (3) with order in discrete norm.

Proof. Let and ; we have from (2)

and then

where and so forth. Using Taylor’s expansion, we obtain

For any , (32) are equivalent to the following equations:

According to , and , we can deduce

Let and . Subtracting (27)-(28) from (35)-(36), respectively, we obtain the error equations:

We take and , and then

where

Adding (38) and (39), we obtain

Using Cauchy-Schwarz inequality, we have

Similarly, we have

Using the triangle inequality, we obtain

According to Lemma 4, . Using Taylor’s expansion, . Using the inequality

and Lemma 4, . According to inequality (45) and the boundedness of numerical solution (30), .
Therefore, we can deduce

Similarly, we have

Thus, we obtain

Let , and (48) can be rewritten as

According to Lemma 5, we have

According to Lemma 4 and noticing and , we have

Therefore, we get

Moreover, we have

Using the triangle inequality and Lemma 4, we obtain

This completes the proof.

4. Numerical Experiments

In this section, we conduct some tentative numerical experiments for this new scheme (10) to verify the theoretical conclusions, including the accuracy, the ability to preserve the first integrals of the nonlinear Schrödinger equation for long-time integration.

First we take the parameter . Then, we get the following:

We consider nonlinear Schrödinger equation (55) with the one-soliton solution as follows:

In order to analyze new scheme (10), the problem is solved in with the initial condition

We take and the time step for the new scheme (10). We check the ability of this new scheme preserving the first integral which is one of the important criteria to judge numerical schemes. The nonlinear Schrödinger equation with periodic boundary condition has the energy conservation law:

If the approximate solution of is , then the discrete conservation law is

We define the errors of the discrete conservation law on the th time level as

where is the numerical solution on the th time level and is the discrete initial value. Numerical solutions and exact solutions at different time levels and the changes of the errors between the exact solutions and the numerical solutions and with time are shown in Figure 1.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

Figure 1: The numerical solutions and the exact solutions at and the changes of the errors between the exact solutions and the numerical solutions and with time.

The discrete norm of complex-valued function is defined as

We consider that the problem is solved in till time for accuracy test. Note that in Table 1 the spatial error is negligible and the error is dominated by the time discretization error. It shows that accuracy of space is very large. Table 1 clearly indicates that new scheme (10) is of second order in time.

Table 1: Time accuracy of new scheme (10) with initial condition (57) .

error
Order

0.004

—

0.002

2.0020

0.001

2.0068

0.0005

2.0266

0.00025

2.0972

0.000125

2.0991

We also test our new scheme on the following initial condition with the periodic boundary condition . We take , . The initial condition is in the vicinity of the homoclinic orbit in [19].

In this case, we also take and the time step for new scheme (10). The corresponding waveforms at different time levels and the changes of errors of discrete conservation law with time are showed in Figure 2. We find that the numerical results we presented in the paper show that the new scheme is very robust and stable. Thus, our new scheme provides a new choice for solving the nonlinear Schrödinger equation.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 2: The numerical solutions at and the changes of with time.

5. Conclusions

In this paper, we derive a new method for the nonlinear Schrödinger system. We prove the proposed method preserves the energy conservation law exactly. A deduction argument is used to prove that the numerical solution is second-order convergent to the exact solutions in norm. Some numerical results are reported to illustrate the efficiency of the numerical scheme in preserving the energy conservation laws. Therefore, it will be a good choice for solving the nonlinear Schrödinger equation computation.

Conflict of Interests

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

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grants nos. 11271195, 41231173, and 11201169), the Postdoctoral Foundation of Jiangsu Province of China under Grant no. 1301030B, Open Fund Project of Jiangsu Key Laboratory for NSLSCS under Grant no. 201301, Fund Project for Highly Educated Talents of Nanjing Forestry University under Grant no. GXL201320, and the Project of Graduate Education Innovation of Jiangsu Province (Grant no. KYLX_0691).

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