Mustafa Almushairaand Fei Liu?

1School of Mathematics and Statistics,Huazhong University of Science and Technology,Wuhan 430074,China.

2Hubei Key Laboratory of Engineering Modeling and Scientific Computing,Huazhong University of Science and Technology,Wuhan 430074,China.

3Department of Mathematics,Faculty of Science,Sana’a University,Sana’a,Yemen.

Abstract.A new efficient compact difference scheme is proposed for solving a space fractional nonlinear Schr?dinger equation with wave operator.The scheme is proved to conserve the total mass and total energy in a discrete sense.Using the energy method,the proposed scheme is proved to be unconditionally stable and its convergence order is shown to be of O(h6+τ2)in the discrete L2norm with mesh size h and the time step τ.Moreover,a fast difference solver is developed to speed up the numerical computation of the scheme.Numerical experiments are given to support the theoretical analysis and to verify the efficiency,accuracy,and discrete conservation laws.

Key words:Space-fractional nonlinear Schr?dinger equations,fast difference solver,convergence,conservation laws.

1 Introduction

where F represents the Fourier transform acting on the spatial variable x,and F?1denotes its inverse.

When γ=2,the FNLSW(1.1)-(1.3)can be regarded as a generalization of the classical nonlinear Schr?dinger equation with wave operator(NLSW).NLSW is one of most important nonlinear Schr?dinger-type equations which is widely used to describe many physical phenomena,such as nonlinear optics[3,21],plasma physics[19]and bimolecular dynamics[26].

are the mass and energy,respectively.

As shown in[28]that the nonconservative schemes for Schr?dinger-type equations may lead to numerical blow-up,thus,the conservative and unconditionally stable schemes become very important for solving Schr?dinger and Schr?dinger-type equations.

In the past several years,various conservative and accurate numerical methods have been developed for the NLSW,including spectral methods[15,23],finite element methods[5,9],finite difference methods[2,6,16,17,25,29,30]and so on.For the FNLSW,to the authors’knowledge,the literature limited.For instance,in[20],a linearly implicit conservative scheme is constructed based on the finite difference method.The Galerkin finite element method is used to solve the FNLSW in[14].These limitations motivate us to develop an efficient and conservative scheme based on a high-order compact difference method and matrix transform technique(MTT)[10].By virtue of MTT,one can efficiently approximate the fractional operator(?Δ)γ/2by the matrix representation of the standard operator Δ.For more details,we refer to[1,8,10–12,27,32].

In this paper,a sixth-order compact finite difference is developed,which is a three-level linearly implicit one.It is shown to conserve the discrete mass and energy.We analyze the stability and convergence in the discrete L2norm as well.Besides,by FFT computations,a fast difference solver is designed to speed up the numerical computation of the scheme.

The remaining of this article is arranged as follows.Section 2 recalls some useful preliminaries.A new high-order conservative scheme is proposed in Section 3.In Section 4,the discrete two conservation laws of the difference scheme are discussed.In Section 5,the prior estimations for numerical solutions are obtained.Moreover,the convergence and stability of the new conservative scheme are proved.In Section 6,a fast iterative algorithm is presented and the numerical experiments are given to support the theoretical analysis.The last Section 7 concludes the article.

2 Preliminaries

In this section,the following definition and lemmas which will play an important role throughout this paper are recalled.

Definition 2.1.A matrix C=C(c0,c1,···,cn?1)in the following form

is called a circulant matrix where each row is a cyclic shift of the row above it.

Lemma 2.2([24]).If A and B are two circulant matrices,the AB is still a circulant matrix.

Lemma 2.3([24]).If a circulant matrix C is invertible,then its inverse matrix C?1is a circulant one as well.

Lemma 2.4([24]).For any a real circulant matrix C=C(c0,c1,···,cn?1),then all the eigenvalues can be constructed in the following form

Lemma 2.5([32]).Suppose that A is a symmetric positive definite matrix,then there exists an orthogonal matrix P such that

where Λ is a diagonal matrix which its entries being the eigenvalues of A.

3 Compact finite difference scheme

3.1 Notations

For simplicity,we write ‖·‖2as‖·‖.Throughout the paper,andrepresent the exact and numerical solution at the point(xm,tn),respectively,and C denotes a generic constant independent of mesh size h and time step τ,which may have different values in diverse occurrences.

3.2 Compact difference scheme

We will present a new compact finite difference for solving the system(1.1)-(1.3)based on MTT[10].The main advantage of the MTT is the fact that if A is the approximate matrix representation of the standard Laplace operator,then Aγ/2is the approximate matrix representation of the fractional Laplacian of order γ such that the matrix A is diagonalizable.

4 Discrete conservation laws

5 Convergence and stability of the difference scheme

6 Implementation and numerical experiment

6.1 Fast iterative algorithm

6.2 Numerical experiment

In this subsection,we give some numerical examples to support our theoretical analysis.At first,we define the error in the discrete L2norm as

where UNdenotes the exact or reference solution when the analytical solution is unavailable and uNdenotes the computational solution corresponding to time increment τ and space increment h at the final time T.The corresponding spatial and temporal convergence order is calculated by the following formula

respectively.

Example 6.1.To test the accuracy and efficiency of the scheme(3.11)-(3.12),we consider the following periodic-boundary and initial value problem of FNLSW with the source term

where

In this example,the analytical solution isu(x,t)=exp(?it)sin(πx).

This example is considered as a benchmark problem in order to investigate the performance in terms of accuracy and efficiency of the proposed scheme.In Table 1,we list the error in the discreteL2-norm and their cosponsoring temporal and spatial convergence orders and CPU times.From Table 1,one clearly observes that the convergence in time and space directions agree with expected orders of the scheme which are second-and sixth-order accurate,respectively.It is worth mentioning that since the source termf(x,t)is not equivalent to zero,the discrete conservation laws are no longer valid,thus the verification is omitted here.

Table 1:Errors of the numerical solution of Example 6.1 and their corresponding spatial and temporal convergence rates at T=1.

Example 6.2.Consider the following periodic-boundary and initial value problem of

FNLSW

In this experiment,we principally do the following works.Firstly,we investigate the convergence order of the proposed scheme.Since the analytical solution is unavailable,we takeT=1,L=10 and choose the numerical solution withh=1/160,τ=1/40960 as the reference solution.Then we get the following spatial and temporal convergence rates figure forh=1/10,1/20,1/40,1/80 andτ=1/10,1/80,1/640,1/5120,respectively.From Figure 1,it is obvious that the proposed scheme is convergence inL2norm,and the convergence order isO(h6+τ2).

Figure 1:The convergence order for Example 6.2 at T=1.

Next,we investigate the two discrete conservation laws given in Lemma 4.4.We takeT=10,000,L=30,andh=τ=0.05.In Tables 2 and 3,the values of discrete massQnand energyEnare reported at different timestfor various values ofγ=2,1.8,1.6,1.4,1.2.Moreover,the evolution of the discrete massQnand energyEnfor varyingγare displayed in Figure 2.

Table 2:The discrete mass Qnat different times with h=τ=0.05 for Example 6.2 with varying values of γ.

Table 3:The discrete mass Enat different times with h=τ=0.05 for Example 6.2 with varying values of γ.

From Tables 2,3 and Figure 2,they indicate that the proposed scheme preserves the discrete mass and energy very well and it is suitable for long-time simulation.

At last,the evolution of the numerical solution for different valuesγ=2,1.7,1.5,1.3 is depicted in Figure 3.It shows that the shape of solitons is affected by varying values ofγ.In other words,whenγbecomes smaller,the shape of solitons will change faster.

Figure 2:The evolution of discrete mass and energy for Example 6.2 with varying values of γ.

Figure 3:The wave propagation for Example 6.2 with different γ.

7 Conclusion

In this work,we construct a sixth-order compact difference scheme for solving a periodic initial value problem of the space fractional nonlinear Schr?dinger equation with wave operator base on MTT.We prove that the scheme conserves the mass and energy in the discrete sense.Using energy method,the scheme is proved to be unconditionally stable and to be convergent with order O(h6+τ2)in the discrete L2norm.Furthermore,a fast difference solver is designed to speed up the calculation of the scheme.The discrete conservation laws,accuracy and effectiveness for long-time simulation are confirmed by numerical examples.

Acknowledgement

The authors would like to thank the editor and referees for their valuable comments and suggestions which improved the paper.