Convergent Analysis of Energy Conservative Algorithm for the Nonlinear Schrödinger Equation

Using average vector fieldmethod 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 L 2 norm. Some numerical results are reported to illustrate the efficiency of the numerical scheme in preserving the energy conservation law.


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][6][7][8][9][10][11][12][13][14].
The general form of the NLS equation with the initial value and the periodic boundary condition is   +   +           2  = 0,  (0, ) =  (2, ) , where  is a real parameter.Now using  =  + , we can rewrite (1) as a pair of real-valued equations as follows: +   +  ( 2 +  2 )  = 0,   −   −  ( 2 +  2 )  = 0. ( Equations ( 2) can be expressed in the Hamiltonian form.Consider where  = (, )  ∈ R 2 and the Hamiltonian function, which is system energy, is The NLS equation (1) Quispel and McLaren [15] proposed the average vector field (AVF) method, which is a second-order energypreserving 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 2 Mathematical Problems in Engineering 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 energypreserving 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  2 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.

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 energypreserving algorithm for the NLS equation.

Convergence Analysis
Let  = [0, 2],  2 () 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 2, defined on R.   () () 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 ℎ.

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  = 2.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 [− 15,15] with the initial condition  (, 0) = sech () exp (2) . (57) We take  = 200 and the time step  = 10 −3 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   = ( 0 ,  1 , . . .,   )  , then the discrete conservation law  is We define the errors of the discrete conservation law on the th time level as We also test our new scheme on the following initial condition (, 0) = 0.5 + 0.025 cos() with the periodic boundary condition (0, ) = (4 √ 2, ).We take  = 4 √ 2,  = 2/.The initial condition is in the vicinity of the homoclinic orbit in [19].
In this case, we also take  = 200 and the time step  = 10 −3 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.

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 ‖ ⋅ ‖ 2 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.
conserves the energy exactly, which can be regarded as the energy stable algorithm.So we assume that the numerical solution is bounded; that is, 1≤≤          ∞ ≤ .

Table 1 :
15,15]ccuracy of new scheme(10)with initial condition (57) ( = 100).isthenumericalsolutiononthethtimeleveland  0 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 Err  with time are shown in Figure1.The discrete  2 norm of complex-valued function  is defined asWe consider that the problem is solved in [−15,15]till time  = 1 for accuracy test.Note that in Table1the spatial error ( = 100) is negligible and the error is dominated by the time discretization error.It shows that accuracy of space is very large.Table1clearly indicates that new scheme (10) is of second order in time.