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 L2 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(1)iψt+ψxx+aψ2ψ=0,ψ(0,t)=ψ(2π,t),where a is a real parameter. Now using ψ=p+iq, we can rewrite (1) as a pair of real-valued equations as follows:(2)pt+qxx+a(p2+q2)q=0,qt-pxx-a(p2+q2)p=0.Equations (2) can be expressed in the Hamiltonian form. Consider(3)dzdt=JδH(z)δz,where z=(p,q)T∈R2 and the Hamiltonian function, which is system energy, is(4)H(z)=∫02π12px2+qx2-a2p2+q22dx,J=01-10.The NLS equation (1) admits the energy conservation law. Consider(5)ε(t)=∫02πa4ψ4-12ψx2dx=ε0.
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 L2 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 D2 to approximate the second-order differential operator ∂xx. For the ordinary differential equation uxx=f, we set ux=v and vx=f. Applying the Fourier pseudospectral method to the two equations leads to D1u=v and D1v=f. Eliminating vector v gives D12u=f. In this work, we use D12u to approximate uxx instead of D2u and obtain the corresponding Fourier pseudospectral semidiscretization for the NLS equation (1) as follows:(6)iddtψj+D12Ψj+aψj2ψj=0,j=0,1,2,…,N-1,where Ψ=(ψ0,ψ1,ψ2,…,ψN-1)T. Equations (6) can be rewritten as(7)ddtpj+D12qj+apj2+qj2qj=0,ddtqj-D12pj-apj2+qj2pj=0,where p=(p0,p1,…,pN-1)T and q=(q0,q1,…,qN-1)T. Since D12 is symmetric, (7) is regarded as a Hamiltonian system with Hamiltonian. Consider (8)Hp,q=12pTD12p+qTD12q+a4∑j=0N-1pj2+qj22.
Now we discretize (6) with respect to time by the AVF method and obtain(9)iδtψjn+D12AtΨnj+aAtψjn2Atψjn16+16ψjn+1-ψjnψjn+1pjn-ψjnpjn+1=0,where δtψjn=(ψjn+1-ψjn)/τ, Atψjn=(ψjn+1+ψjn)/2, and pjn=Re(ψjn)=(ψjn+ψjn¯)/2. Obviously, scheme (9) can be reformed as a vector form. Consider(10)iδtΨn+D12AtΨn+aAtΨn2·AtΨn+16Ψn+1-Ψn16·(Ψn+1·Pn-Ψn·Pn+1)=0,where |Ψn|2=|Ψn|·|Ψn| and “·” denotes point multiplication between vectors; that is, (11)pn·qn=p0nq0n,p1nq1n,p2nq2n,…,pN-1nqN-1nT.
Equations (9) can also be rewritten as(12)pjn+1-pjnτ+D12qn+1/2j+a14(qjn+1+qjn)·pjn+12+pjn2+qjn+12+qjn2+16pjn+1-pjnqjn+1·pjn-qjn·pjn+1=0,qjn+1-qjnτ-D12pn+1/2j-a14(pjn+1+pjn)·pjn+12+pjn2+qjn+12+qjn2+16qjn+1-qjnpjn+1·qjn-pjn·qjn+1=0.
Next, we prove that scheme (10) conserves the discrete total energy. Let XN={u∣u=(u0,u1,…,uN-1)}⊆CN and define discrete inner product and discrete L2 norm over XN as (13)u,vN=h∑j=0N-1ujvj¯,uN=u,uN1/2.
Theorem 1.
With periodic boundary condition ψ(0,t)=ψ(2π,t), scheme (10) possesses the discrete global energy conservation law; namely,(14)εn+1=εn=⋯=ε1=ε0,where εn=(a/4)ΨnN,44-(1/2)D1ΨnN2 and ΨnN,44=(|Ψn|2,|Ψn|2)N.
Proof.
Taking the inner product of (10) with Ψn+1-Ψn yields(15)iδtΨn,Ψn+1-ΨnN+D12AtΨn,Ψn+1-ΨnN+aAtΨn2·AtΨn+16Ψn+1-Ψn16·Ψn+1·Pn-Ψn·Pn+1,Ψn+1-ΨnN=0.
The first term becomes(16)iτΨn+1-Ψn,Ψn+1-ΨnN=iτΨn+1-ΨnN2,which is purely imaginary. The second term can be reduced to(17)12D12Ψn+1+Ψn,Ψn+1-ΨnN=-12(D1Ψn+1N2-D1ΨnN2)+iImD1Ψn+1,D1ΨnN.Noticing that Pn=(Ψn+Ψn¯)/2, we have(18)aAt(Ψn2)·AtΨn+16Ψn+1-Ψn16·Ψn+1·Pn-Ψn·Pn+1,Ψn+1-ΨnN=a14Ψn+1N,44-ΨnN,44-i2Ψn+12+Ψn2,ImΨn+1·Ψn¯N+i6ImΨn+1·Ψn¯,Ψn+1-Ψn2N.Therefore, the real part of (15) is(19)-12D1Ψn+1N2-D1ΨnN2+a4Ψn+1N,44-ΨnN,44=0.So (19) gives the energy conservation law (14).
3. Convergence Analysis
Let I=[0,2π], L2(I) with the inner product (·,·) and the norm ·. For any positive integer r, the seminorm and the norm of Hr(I) are denoted by |·|r and ·r, respectively. Let C(p)∞(I) be the set of infinitely differentiable functions with period 2π, defined on R. H(p)r(I) is the closure of C(p)∞(I) in Hr(I). In this section, let C 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 τ and the grid size h.
For even N, set (20)VN=u∣u(x)=∑k≤N/2u^keikx,u^k¯=u^-k,k≤N2,VN′′=u∣ux=∑k≤N/2′′u^keikx,u^k¯=u^-k,lll∑k≤N/2′′k≤N2,u^N/2=u^-N/2,where the summation ∑′′ is defined by (21)∑|k|≤N/2′′ψk=12ψ-N/2+∑|k|<N/2ψk+12ψN/2.It is obviously that VN′′⊆VN. Denote the orthogonal projection operator PN:L2(I)→VN and the interpolation operator IN:L2(I)→VN′′. Note that PN and IN satisfy the following properties:
PN∂xu=∂xPNu, IN∂xu≠∂xINu;
(PN-2u,v)N=(PN-2u,v), ∀v∈VN;
PNu=u, ∀u∈VN; INu=u, ∀u∈VN′′.
Lemma 2.
For u∈VN′′, u≤uN≤2u.
Lemma 3 (see [<xref ref-type="bibr" rid="B1">17</xref>]).
If 0≤l≤r and u∈H(p)r(I), then (22)PNu-ul≤CNl-rur,PNul≤Cul,and if r>1/2, then (23)INu-ul≤CNl-rur,INul≤Cul.
Lemma 4.
Suppose u*=PN-2u, u∈H(p)r(I), and r>1/2; then u*-uN≤CN-r|u|r.
Proof.
According to Lemmas 2 and 3, we have (24)u*-uN=INu*-uN=u*-INuN≤2u*-INu≤2u*-u+u-INu≤CN-r|u|r.
Suppose that the discrete function {wn∣n=0,1,2,…,M;Mτ=T} satisfies the following inequality: (25)wn-wn-1≤Aτwn+Bτwn-1+Cnτ,where A, B, and Cn(n=0,1,2,…,M) are nonnegative constants. Then (26)max1≤n≤Mwn≤w0+∑l=1MCle2(A+B)T,where τ is sufficiently small, such that (A+B)τ≤(M-1)/2M,(M>1).
An equivalent form of full-discrete Fourier pseudospectral scheme (12) is to find (pcn,qcn)T∈(VN′′)2, so that, for any Φn=(Φ1n,Φ2n)T∈(VN′′)2, then(27)pcn+1-pcnτ,Φ1nN-∂xqcn+1/2,∂xΦ1nN+fpcn,pcn+1,qcn,qcn+1,Φ1nN=0,(28)qcn+1-qcnτ,Φ2nN+∂xpcn+1/2,∂xΦ2nN-fqcn,qcn+1,pcn,pcn+1,Φ2nN=0,where (29)fx,y,u,v=a14u+v·x2+y2+u2+v2+16(y-x)(v·x-u·y).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,(30)max1≤n≤Npcn∞≤C,max1≤n≤Nqcn∞≤C.
Theorem 6.
Suppose that the exact solutions p,q∈H1(0,T;Hpr(I))∩H3(0,T;L2(I)), r>1/2, and τ are small enough; then the solution of full-discrete Fourier pseudospectral scheme (12) converges to the solution of problems (3) with order O(N-r+τ2) in discrete L2 norm.
Proof.
Let p*=PN-2p and q*=PN-2q; we have from (2)(31)pt*+qxx*+PN-2ap2+q2q=0,qt*-pxx*-PN-2ap2+q2p=0,and then(32)p*n+1-p*nτ+∂xxq*n+1/2+PN-2ap2+q2qn+1/2=p*n+1-p*nτ-pt*n+1/2≜ξ1n,q*n+1-q*nτ-∂xxp*n+1/2-PN-2ap2+q2pn+1/2=q*n+1-q*nτ-qt*n+1/2≜ξ2n,where (p*)n+1/2=(p*n+p*n+1)/2 and so forth. Using Taylor’s expansion, we obtain (33)ξ1nN≤Cτ2,ξ2nN≤Cτ2.
For any Φn=(Φ1n,Φ2n)T∈(VN′′)2, (32) are equivalent to the following equations:(34)p*n+1-p*nτ,Φ1nN+∂xxq*n+1/2,Φ1nN+PN-2ap2+q2qn+1/2,Φ1nN=ξ1n,Φ1nN,q*n+1-q*nτ,Φ2nN-∂xxp*n+1/2,Φ2nN-PN-2ap2+q2pn+1/2,Φ2nN=ξ2n,Φ2nN.According to (PN-2u,v)N=(PN-2u,v), ∀v∈VN and PN∂xu=∂xPNu, we can deduce(35)p*n+1-p*nτ,Φ1nN-∂xq*n+1/2,∂xΦ1nN+PN-2ap2+q2qn+1/2,Φ1nN=ξ1n,Φ1nN,(36)q*n+1-q*nτ,Φ2nN+∂xp*n+1/2,∂xΦ2nN-PN-2ap2+q2pn+1/2,Φ2nN=ξ2n,Φ2nN.
Let εn=p*n-pcn and ηn=q*n-qcn. Subtracting (27)-(28) from (35)-(36), respectively, we obtain the error equations:(37)εn+1-εnτ,Φ1nN-∂xηn+1/2,∂xΦ1nN+PN-2ap2+q2qn+1/2ap2+q2qn+1/2-fpcn,pcn+1,qcn,qcn+1,Φ1nN=ξ1n,Φ1nN,ηn+1-ηnτ,Φ2nN+∂xεn+1/2,∂xΦ2nN-PN-2ap2+q2pn+1/2ap2+q2pn+1/2-fqcn,qcn+1,pcn,pcn+1,Φ2nN=ξ2n,Φ2nN.We take Φ1n=εn+1/2 and Φ2n=ηn+1/2, and then(38)εn+1-εnτ,εn+1+εn2N-∂xηn+1/2,∂xεn+1/2N+G1,εn+1/2N=ξ1n,εn+1/2N,(39)ηn+1-ηnτ,ηn+1+ηn2N+∂xεn+1/2,∂xηn+1/2N-G2,ηn+1/2N=ξ2n,ηn+1/2N,where (40)G1=PN-2ap2+q2qn+1/2-fpcn,pcn+1,qcn,qcn+1,G2=PN-2ap2+q2pn+1/2-fqcn,qcn+1,pcn,pcn+1.Adding (38) and (39), we obtain(41)12τεn+1N2+ηn+1N2-εnN2-ηnN2=ξ1n,εn+1/2N+ξ2n,ηn+1/2N-G1,εn+1/2N+G2,ηn+1/2N.Using Cauchy-Schwarz inequality, we have (42)ξ1n,εn+1/2N≤ξ1nN·εn+1/2N≤12ξ1nN2+18εn+εn+1N2≤Cτ4+14(εnN2+εn+1N2).Similarly, we have (43)ξ2n,ηn+1/2N≤Cτ4+14ηnN2+ηn+1N2,G1,εn+1/2N≤12G1N2+14(εnN2+εn+1N2),|G2,ηn+1/2N|≤12G2N2+14ηnN2+ηn+1N2.Using the triangle inequality, we obtain (44)G1N=PN-2ap2+q2qn+1/2-fpcn,pcn+1,qcn,qcn+1N≤PN-2ap2+q2qn+1/2-ap2+q2qn+1/2N+ap2+q2qn+1/2-fpn,pn+1,qn,qn+1N+fpn,pn+1,qn,qn+1-fp*n,p*n+1,q*n,q*n+1N+fp*n,p*n+1,q*n,q*n+1-fpcn,pcn+1,qcn,qcn+1N≜I+II+III+IV.According to Lemma 4, I≤CN-r. Using Taylor’s expansion, II≤Cτ2. Using the inequality(45)u·v-u~·v~≤u-u~·v+u~·v-v~and Lemma 4, III≤CN-r. According to inequality (45) and the boundedness of numerical solution (30), IV≤C(εnN+εn+1N+ηnN+ηn+1N).
Therefore, we can deduce (46)G1N2≤CN-2r+τ4+CεnN2+εn+1N2+ηnN2+ηn+1N2.Similarly, we have (47)G2N2≤CN-2r+τ4+C(εnN2+εn+1N2+ηnN2+ηn+1N2).
Thus, we obtain(48)12τεn+1N2+ηn+1N2-εnN2-ηnN2≤CN-2r+τ4+CεnN2+εn+1N2+ηnN2+ηn+1N2.
Let ωn=εnN2+ηnN2, and (48) can be rewritten as(49)ωn+1-ωn≤Cτ(ωn+1+ωn)+Cτ(N-2r+τ4).According to Lemma 5, we have(50)ωn≤ω0+τ∑l=1MC(N-2r+τ4)e4CT.According to Lemma 4 and noticing pc0=p0 and qc0=q0, we have (51)ω0=ε0N2+η0N2=p*0-p0N2+q*0-q0N2≤CN-2r.Therefore, we get(52)ωn≤C(N-2r+τ4).Moreover, we have (53)p*n-pcnN≤CN-r+τ2,q*n-qcnN≤CN-r+τ2.Using the triangle inequality and Lemma 4, we obtain (54)pn-pcnN≤pn-p*nN+p*n-pcnN≤CN-r+τ2,qn-qcnN≤qn-q*nN+q*n-qcnN≤CN-r+τ2.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 a=2. Then, we get the following:(55)iψt+ψxx+2ψ2ψ=0.
We consider nonlinear Schrödinger equation (55) with the one-soliton solution as follows:(56)ψx,t=sechx-4texp2ix-32t.
In order to analyze new scheme (10), the problem is solved in [-15,15] with the initial condition (57)u(x,0)=sech(x)exp(2xi).
We take N=200 and the time step τ=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: (58)F(ψ)=∫0La4ψ4-12ψx2dx.
If the approximate solution of ψ(x,t=jτ) is ψj=(ψ0,ψ1,…,ψN)T, then the discrete conservation law F is (59)Fh(ψ)=∑j=1N12ψj4-D1Ψj2h.
We define the errors of the discrete conservation law on the jth time level as(60)ErrF(jτ)=Fh(ψj)-Fh(ψ0),where ψj is the numerical solution on the jth time level and ψ0 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 ErrF with time are shown in Figure 1.
The numerical solutions and the exact solutions at t=1,10,14,50 and the changes of the errors between the exact solutions and the numerical solutions and ErrF with time.
t=1
0≤t≤1
t=10
0≤t≤10
t=14
0≤t≤14
t=50
0≤t≤50
0≤t≤1
0≤t≤10
0≤t≤14
0≤t≤50
The discrete L2 norm of complex-valued function ψ is defined as (61)ψ2=∑j=1Nψj2h1/2.
We consider that the problem is solved in [-15,15] till time t=1 for accuracy test. Note that in Table 1 the spatial error (N=100) 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.
Time accuracy of new scheme (10) with initial condition (57) (N=100).
τ
L2 error
Order
0.004
1.1351e-004
—
0.002
2.8338e-005
2.0020
0.001
7.0510e-006
2.0068
0.0005
1.7305e-006
2.0266
0.00025
4.0445e-007
2.0972
0.000125
9.4398e-008
2.0991
We also test our new scheme on the following initial condition ψ(x,0)=0.5+0.025cos(μx) with the periodic boundary condition ψ(0,t)=ψ(42π,t). We take L=42π, μ=2π/L. The initial condition is in the vicinity of the homoclinic orbit in [19].
In this case, we also take N=200 and the time step τ=10-3 for new scheme (10). The corresponding waveforms at different time levels and the changes of errors of discrete conservation law F 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.
The numerical solutions at t=1,10,14,50 and the changes of ErrF with time.
t=1
t=10
t=14
t=50
0≤t≤1
0≤t≤10
0≤t≤14
0≤t≤50
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 ·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.
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).
MenyukC. R.Stability of solitons in birefringent optical fibersWadatiM.IizukaT.HisakadoM.A coupled nonlinear Schrödinger equation and optical solitonsFengK.QinM. Z.QinM. Z.WangY. S.CaiJ.-X.WangY.-S.A conservative Fourier pseudospectral algorithm for a coupled nonlinear Schrödinger systemAkrivisG. D.Finite difference discretization of the cubic Schrödinger equationJiangC.-L.SunJ.-Q.A high order energy preserving scheme for the strongly coupled nonlinear Schrödinger systemLvZ.-Q.WangY.-S.SongY.-Z.A new multi-symplectic integration method for the nonlinear schrödinger equationDehghanM.TaleeiA.A compact split-step finite difference method for solving the nonlinear Schrödinger equations with constant and variable coefficientsChangQ.JiaE.SunW.Difference schemes for solving the generalized nonlinear Schrödinger equationWangT. C.GuoB. L.Unconditional convergence of two conservative compact difference schemes for non-linear Schrödinger equation in one dimensionWangY. S.LiQ. H.SongY. Z.Two new simple multi-symplectic schemes for the nonlinear Schrödinger equationWangY. S.LiS. T.New schemes for the coupled nonlinear Schrödinger equationZhangR.-P.YuX.-J.FengT.Solving coupled nonlinear Schrödinger equations via a direct discontinuous Galerkin methodQuispelG. R.McLarenD. I.A new class of energy-preserving numerical integration methodsCelledoniE.GrimmV.McLachlanR. I.McLarenD. I.O'NealeD.OwrenB.QuispelG. R.Preserving energy resp. dissipation in numerical PDEs using the ‘average vector field’ methodCanutoC.QuarteroniA.Approximation results for orthogonal polynomials in Sobolev spacesZhouY. L.ChenJ.-B.QinM.-Z.Multi-symplectic Fourier pseudospectral method for the nonlinear Schrödinger equation