An Implicit Generalized Finite-DifferenceTime-Domain …phys.lsu.edu/~fmoxley/chapter.pdf · An...

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This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. Contemporary Mathematics Volume 618, 2014 http://dx.doi.org/10.1090/conm/618/12325 An Implicit Generalized Finite-Difference Time-Domain Scheme for Solving Nonlinear Schr¨ odinger Equations Frederick Ira Moxley III, David T. Chuss, and Weizhong Dai This chapter is dedicated to Dr. Ronald E. Micken’s 70th birthday. Abstract. In this chapter, we develop the linearized generalized FDTD im- plicit scheme for solving time-dependent nonlinear Schr¨odinger equations in 1D. Using the discrete energy method, the G-FDTD scheme is shown to sat- isfy the discrete analogous form of a conservation law. The new scheme is tested by two examples of soliton propagation and collision. Compared with other popular existing methods, numerical results demonstrate that the present scheme provides a more accurate solution. 1. Introduction The nonlinear Schr¨ odinger equation (NLSE) is one of the most widely applicable equations in physical science, and is used to characterize nonlinear dispersive waves, plasmas, nonlinear optics, water waves, and the dynamics of molecules. The NLSE can be expressed as (1) i ∂ψ(x, t) ∂t 2 ψ(x, t) ∂x 2 + λ |ψ(x, t)| p1 ψ(x, t)=0, t> 0,x R n , where ψ(x, t) is a complex valued function that governs the evolution of a weakly nonlinear, strongly dispersive, almost monochromatic wave [4]. The integer p 3 determines the nature of the nonlinear term, and i = 1. In addition, λ is a positive or negative constant corresponding to the focusing, or defocusing NLSE, respectively. The NLSE permits single and multiple hyperbolic secant solutions known as bright solitons, or hyperbolic tangent solutions known as dark solitons [19]. The behavior of solutions depends considerably on the sign of λ, the parameter p, and the spatial dimension n. We now focus on Eq.(1) in the 1D case. Multiplying Eq.(1) by the conjugate function, ψ(x, t), integrating over the space R, and taking the imaginary part, then multiplying Eq.(1) by ψ(x,t) ∂t , integrating over the space R and taking the real part, 2010 Mathematics Subject Classification. Primary 65M12; Secondary 65M06. Key words and phrases. Finite-difference time-domain (FDTD) scheme, nonlinear Schr¨odinger equation, soliton. The research was supported by a grant from NASA EPSCoR & LaSPACE, Louisiana. c 2014 American Mathematical Society 181

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Contemporary MathematicsVolume 618, 2014http://dx.doi.org/10.1090/conm/618/12325

An Implicit Generalized Finite-Difference Time-DomainScheme for Solving Nonlinear Schrodinger Equations

Frederick Ira Moxley III, David T. Chuss, and Weizhong Dai

This chapter is dedicated to Dr. Ronald E. Micken’s 70th birthday.

Abstract. In this chapter, we develop the linearized generalized FDTD im-plicit scheme for solving time-dependent nonlinear Schrodinger equations in1D. Using the discrete energy method, the G-FDTD scheme is shown to sat-isfy the discrete analogous form of a conservation law. The new scheme istested by two examples of soliton propagation and collision. Compared withother popular existing methods, numerical results demonstrate that the presentscheme provides a more accurate solution.

1. Introduction

The nonlinear Schrodinger equation (NLSE) is one of the most widely applicableequations in physical science, and is used to characterize nonlinear dispersive waves,plasmas, nonlinear optics, water waves, and the dynamics of molecules. The NLSEcan be expressed as

(1) i∂ψ(x, t)

∂t− ∂2ψ(x, t)

∂x2+ λ |ψ(x, t)|p−1 ψ(x, t) = 0, t > 0, x ∈ Rn,

where ψ(x, t) is a complex valued function that governs the evolution of a weaklynonlinear, strongly dispersive, almost monochromatic wave [4]. The integer p ≥ 3determines the nature of the nonlinear term, and i =

√−1. In addition, λ is a

positive or negative constant corresponding to the focusing, or defocusing NLSE,respectively. The NLSE permits single and multiple hyperbolic secant solutionsknown as bright solitons, or hyperbolic tangent solutions known as dark solitons[19]. The behavior of solutions depends considerably on the sign of λ, the parameterp, and the spatial dimension n.

We now focus on Eq.(1) in the 1D case. Multiplying Eq.(1) by the conjugatefunction, ψ(x, t), integrating over the space R, and taking the imaginary part, then

multiplying Eq.(1) by ∂ψ(x,t)∂t , integrating over the space R and taking the real part,

2010 Mathematics Subject Classification. Primary 65M12; Secondary 65M06.Key words and phrases. Finite-difference time-domain (FDTD) scheme, nonlinear

Schrodinger equation, soliton.The research was supported by a grant from NASA EPSCoR & LaSPACE, Louisiana.

c©2014 American Mathematical Society

181

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182 FREDERICK IRA MOXLEY III, DAVID T. CHUSS, AND WEIZHONG DAI

one may see that the NLSE satisfies the mass conservation law,

(2) M(ψ) =

∫R

|ψ(x, t)|2 dx = constant,

and the energy conservation law,

(3) E (ψ) =

∫R

{1

2|∇ψ(x, t)|2 + λ

p+ 1|ψ(x, t)|p+1

}dx = constant.

In general, the NLSE requires a numerical solution since it is nonlinear. Spec-tral and pseudospectral methods have been a very popular choice for solving theNLSE where codes are obtained by the fast Fourier transform [1,2]. A split-stepFourier pseudospectral method was studied in [26]. Another spectral method withliberal stability restriction is the integrating-factor method [3,14]. Finite differencemethods [4,8,9,16,17,20,22] are typically more flexible, and easier to use than thespectral methods, particularly for systems with complex boundary conditions [23].The finite element method for the one dimensional NLSE was studied in [10]. Animproved method is the quadrature discretization method [7,13]. Several numeri-cal methods for the one dimensional NLSE were compared in [6,21]. Other popularnumerical investigations of the NLSE include that of Karpman & Krushkal [12],Yajima & Outi [27], Satsuma & Yajima [18], and Hardin & Tappert [11]. Morerecently, Lanczos’ Tau method was investigated in [5], and the Adomian decompo-sition method with variation iteration was examined in [25]. An advantageouslyin-depth, and detailed summary of numerical methods for solving the NLSE can beseen in Yang’s book [28].

In this chapter, we present an implicit generalized finite-difference time-domain(G-FDTD) scheme for solving the NLSE. In this method, the function ψ(x, t) is firstsplit into real and imaginary components resulting in two coupled equations. Thereal and imaginary components are then approximated using higher-order Taylorseries expansions in time and then the derivatives in time are substituted into thederivatives in space via the coupled equations. Finally, the derivatives in space areapproximated using higher-order finite difference methods. The obtained schemewill be shown to satisfy the discrete analogous form of the mass conservation law,Eq.(2). The new scheme is then tested by two examples of bright and dark solitonpropagation. In addition, comparisons are made with the popular spectral methodsfor convenience.

2. G-FDTD Scheme

To obtain the generalized FDTD scheme, we first assume that ψ(x, t) be asufficiently smooth function which vanishes for sufficiently large |x|. Using theidea of the generalized FDTD method in [15] (the primitive idea dates back toVisscher [24]), we split the variable ψ(x, t) into real and imaginary components,ψ(x, t) = ψreal(x, t) + iψimag(x, t). Inserting it into Eq.(1) and then separating thereal and imaginary parts result in the following coupled set of equations:

(4a)∂ψreal(x, t)

∂t=

∂2ψimag(x, t)

∂x2+ λ[ψ2

real(x, t) + ψ2imag(x, t)]

p−12 ψimag(x, t),

(4b)∂ψimag(x, t)

∂t= −∂2ψreal(x, t)

∂x2− λ[ψ2

real(x, t) + ψ2imag(x, t)]

p−12 ψreal(x, t).

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AN IMPLICIT GENERALIZED FINITE-DIFFERENCE TIME-DOMAIN SCHEME 183

We let ψnreal(k) be the approximation of ψreal(kΔx, nΔt) and so on. Using

Taylor series expansion at t = (n−1/2)Δt, and using the above equations repeatedly

where t in [ψ2real(x, t) + ψ2

imag(x, t)]p−12 is fixed at (n− 1/2)Δt, we obtain

(5a)

ψnreal(k)− ψn−1

real (k)

= Δt∂ψ

n−1/2real (k)

∂t+

Δt3

24

∂3ψn−1/2real (k)

∂t3+O(Δt5)

= Δt

[∂2

∂x2+∣∣∣ψn−1/2

∣∣∣p−1]ψn−1/2imag (k)− Δt3

24

[∂2

∂x2+∣∣∣ψn−1/2

∣∣∣p−1]3

ψn−1/2imag (k)

+O(Δt5)

and

(5b)

ψnimag(k)− ψn−1

imag(k)

= Δt∂ψ

n−1/2imag (k)

∂t+

Δt3

24

∂3ψn−1/2imag (k)

∂t3+O(Δt5)

= −Δt

[∂2

∂x2+∣∣∣ψn−1/2

∣∣∣p−1]ψn−1/2real (k) +

Δt3

24

[∂2

∂x2+∣∣∣ψn−1/2

∣∣∣p−1]3

ψn−1/2real (k)

+O(Δt5),

where∣∣ψn−1/2

∣∣p−1= λ[ψ2

real(kΔx, tn−1/2) + ψ2imag(kΔx, tn−1/2)]

p−12 . To evaluate

ψn−1/2imag (k) in Eq.(5a), we further use the Taylor series expansion as

ψnimag(k) = ψ

n−1/2imag (k) +

Δt

2

∂ψn−1/2imag (k)

∂t+

Δt2

8

∂2ψn−1/2imag (k)

∂t2

+Δt3

48

∂3ψn−1/2imag (k)

∂t3+ O(Δt4),(6a)

ψn−1imag(k) = ψ

n−1/2imag (k)− Δt

2

∂ψn−1/2imag (k)

∂t+

Δt2

8

∂2ψn−1/2imag (k)

∂t2

− Δt3

48

∂3ψn−1/2imag (k)

∂t3+ O(Δt4).(6b)

Taking an average of Eqs.(6a) and (6b), we obtain

ψn−1/2imag (k) =

ψnimag(k) + ψn−1

imag(k)

2− Δt2

8

∂2ψn−1/2imag (k)

∂t2+O(Δt4)

=ψnimag(k) + ψn−1

imag(k)

2+

Δt2

8

[∂2

∂x2+∣∣∣ψn−1/2

∣∣∣p−1]2

ψn−1/2imag (k)

+O(Δt4).(7a)

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184 FREDERICK IRA MOXLEY III, DAVID T. CHUSS, AND WEIZHONG DAI

Similarly, we have

ψn−1/2real (k) =

ψnreal(k) + ψn−1

real (k)

2− Δt2

8

∂2ψn−1/2real (k)

∂t2+O(Δt4)

=ψnreal(k) + ψn−1

real (k)

2− Δt2

8

[∂2

∂x2+∣∣∣ψn−1/2

∣∣∣p−1]2

ψn−1/2real (k)

+O(Δt4).(7b)

Substituting Eqs.(7a) and (7b) into Eqs.(5a) and (5b), respectively, and keepingthe terms up to O(Δt3), we obtain

ψnreal(k)− ψn−1

real (k) = {Δt[∂2

∂x2+∣∣∣ψn−1/2

∣∣∣p−1

] +Δt3

12[∂2

∂x2+∣∣∣ψn−1/2

∣∣∣p−1

]3}

·ψnimag(k) + ψn−1

imag(k)

2+O(Δt5),(8a)

ψnimag(k)− ψn−1

imag(k) = −{Δt[

∂2

∂x2+∣∣∣ψn−1/2

∣∣∣p−1

] +Δt3

12[∂2

∂x2+∣∣∣ψn−1/2

∣∣∣p−1

]3}

· ψnreal(k) + ψn−1

real (k)

2+O(Δt5).(8b)

Noting that the term∣∣ψn−1/2

∣∣p−1in Eq.(8) needs to be evaluated, we use a similar

argument for Eq.(8) and obtain

ψn+1/2real (k)− ψ

n−1/2real (k) =

{Δt[

∂2

∂x2+ |ψn|p−1] +

Δt3

12[∂2

∂x2+ |ψn|p−1]3

}

·ψn+1/2imag (k) + ψ

n−1/2imag (k)

2+ O(Δt5),(9a)

ψn+1/2imag (k)− ψ

n−1/2imag (k) = −

{Δt[

∂2

∂x2+ |ψn|p−1

] +Δt3

12[∂2

∂x2+ |ψn|p−1

]3}

· ψn+1/2real (k) + ψ

n−1/2real (k)

2+O(Δt5),(9b)

where |ψn|p−1 = λ[ψ2real(kΔx, tn) + ψ2

imag(kΔx, tn)]p−12 . Next, we couple Eqs.(8)

and (9) together, dropping out the truncation error O(Δt5), and replacing ∂2

∂x2 by a

fourth-order accurate central difference operator, 1Δx2D

2xu(k) =

112Δx2 [−u(k+2)+

16u(k+1)− 30u(k) + 16u(k− 1)− u(k− 2)]. This results in our implicit G-FDTDscheme for solving the NLSE as follows:

ψnreal(k)− ψn−1

real (k) =

{σD2

x +Δt∣∣∣ψn−1/2

∣∣∣p−1

+1

12[σD2

x +Δt∣∣∣ψn−1/2

∣∣∣p−1

]3}

·ψnimag(k) + ψn−1

imag(k)

2,(10a)

ψnimag(k)− ψn−1

imag(k) = −{σD2

x +Δt∣∣∣ψn−1/2

∣∣∣p−1

+1

12[σD2

x +Δt∣∣∣ψn−1/2

∣∣∣p−1

]3}

· ψnreal(k) + ψn−1

real (k)

2;(10b)

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AN IMPLICIT GENERALIZED FINITE-DIFFERENCE TIME-DOMAIN SCHEME 185

ψn+1/2real (k)− ψ

n−1/2real (k) =

{σD2

x +Δt |ψn|p−1+

1

12[σD2

x +Δt |ψn|p−1]3}

·ψn+1/2imag (k) + ψ

n−1/2imag (k)

2,(11a)

ψn+1/2imag (k)− ψ

n−1/2imag (k) = −

{σD2

x +Δt |ψn|p−1 +1

12[σD2

x +Δt |ψn|p−1]3}

· ψn+1/2real (k) + ψ

n−1/2real (k)

2.(11b)

Here, σ = Δt/Δx2. The truncation error of the above scheme is O(Δx4 +Δt5), as

compared with Eqs.(8) and (9). It should be noted that the partial derivative ∂2

∂x2

can alternatively be approximated using a spectral or other higher-order method.In this study, we confine our attention to the finite difference method with a fourth-order central difference approximation.

3. Discrete Conservation Law

To show that the present scheme satisfies the mass conservation law, Eq.(2),we introduce some finite difference operators, δ2x, δ

22x, ∇x, ∇x, ∇2x, and ∇2x as

δ2xu(k) = u(k+1)−2u(k)+u(k−1), δ22xu(k) = u(k+2)−2u(k)+u(k−2),∇xu(k) =u(k+1)−u(k), ∇xu(k) = u(k)−u(k−1), ∇2xu(k) = u(k+2)−u(k), ∇2xu(k) =u(k) − u(k − 2). It can be seen that these finite difference operators satisfy therelations: D2

x = 112 [−δ22x + 16δ2x], δ

2x = ∇x · ∇x = ∇x − ∇x, δ

22x = ∇2x · ∇2x =

∇2x−∇2x, ∇2x·∇x = ∇x·∇2x, etc. Furthermore, one may observe that for any meshfunctions, u(k) and v(k), which are assumed to be zero for sufficiently large |k| ,∑

kεZ ∇xu(k) · v(k) = −∑

kεZ u(k) · ∇xv(k),∑

kεZ ∇xu(k) · v(k) = −∑

kεZ u(k) ·∇xv(k),

∑kεZ ∇2xu(k) · v(k) = −

∑kεZ u(k) · ∇2xv(k),

∑kεZ∇2xu(k) · v(k) =

−∑

kεZ u(k) · ∇2xv(k),∑

kεZ δ2xu(k) ·v(k) = −∑

kεZ∇xu(k) ·∇xv(k) =∑

kεZ u(k) ·δ2xv(k), and

∑kεZ δ22xu(k) ·v(k) = −

∑kεZ ∇2xu(k) ·∇2xv(k) =

∑kεZ u(k) ·δ22xv(k),

where Z is the set of all positive and negative integers. The first step is to multiply

Eq.(10a) byψn

real(k)+ψn−1real (k)

2 , Eq.(10b) byψn

imag(k)+ψn−1imag(k)

2 , and then sum k over allintegers Z. This gives

(12)

1

2

∑kεZ

{[ψn

real(k)]2 − [ψn−1

real (k)]2 + [ψn

imag(k)]2 − [ψn−1

imag(k)]2}

=∑kεZ

{[σD2x +Δt

∣∣∣ψn−1/2∣∣∣p−1

]ψnimag(k) + ψn−1

imag(k)

2· ψ

nreal(k) + ψn−1

real (k)

2

− [σD2x +Δt

∣∣∣ψn−1/2∣∣∣p−1

]ψnreal(k) + ψn−1

real (k)

2·ψnimag(k) + ψn−1

imag(k)

2}

+∑kεZ

{ 1

12[σD2

x +Δt∣∣∣ψn−1/2

∣∣∣p−1

]3ψnimag(k) + ψn−1

imag(k)

2· ψ

nreal(k) + ψn−1

real (k)

2

− 1

12[σD2

x +Δt∣∣∣ψn−1/2

∣∣∣p−1

]3ψnreal(k) + ψn−1

real (k)

2·ψnimag(k) + ψn−1

imag(k)

2}.

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186 FREDERICK IRA MOXLEY III, DAVID T. CHUSS, AND WEIZHONG DAI

It can be seen that the first summation on the right-hand-side of Eq.(12) is zero anddisappears. For the second summation on the right-hand-side of Eq.(12), we expand

the operator [σD2x+Δt

∣∣ψn−1/2∣∣p−1

]3 using a binomial formula and obtain that forany mesh functions, u(k) and v(k), which are assumed to be zero for sufficientlylarge |k| ,

∑kεZ

([D2

x]3u(k)

)· v(k)

=1

123

∑kεZ

[∇2xδ22xu(k) · ∇2xδ

22xv(k)− 48∇xδ

22xu(k) · ∇xδ

22xv(k)

+ 3 · 162∇2xδ2xu(k) · ∇2xδ

2xv(k)− 163∇xδ

2xu(k) · ∇xδ

2xv(k)],(13a)

∑kεZ

([D2

x]2∣∣∣ψn−1/2

∣∣∣p−1

u(k)

)· v(k)

=1

122{∑kεZ

δ22x

(∣∣∣ψn−1/2∣∣∣p−1

u(k)

)· δ22xv(k)

− 32∑kεZ

∇2x∇x

(∣∣∣ψn−1/2∣∣∣p−1

u(k)

)· ∇2x∇xv(k)

+ 162∑kεZ

δ2x

(∣∣∣ψn−1/2∣∣∣p−1

u(k)

)· δ2xv(k)},(13b)

∑kεZ

∣∣∣ψn−1/2∣∣∣p−1

[D2x]

2u(k) · v(k)

=1

122{∑kεZ

δ22xu(k) · δ22x(∣∣∣ψn−1/2

∣∣∣p−1

v(k)

)

− 32∑kεZ

∇2x∇xu(k) · ∇2x∇x

(∣∣∣ψn−1/2∣∣∣p−1

v(k)

)

+ 162∑kεZ

δ2xu(k) · δ2x(∣∣∣ψn−1/2

∣∣∣p−1

v(k)

)},(13c)

(13d)∑kεZ

D2x

(∣∣∣ψn−1/2∣∣∣p−1

D2xu(k)

)· v(k) =

∑kεZ

∣∣∣ψn−1/2∣∣∣p−1

D2xu(k) ·D2

xv(k),

∑kεZ

D2x

([∣∣∣ψn−1/2

∣∣∣p−1

]2u(k)

)· v(k)

=1

12{∑kεZ

∇2x

([∣∣∣ψn−1/2

∣∣∣p−1

]2u(k)

)· ∇2xv(k)

− 16∑kεZ

∇x

([∣∣∣ψn−1/2

∣∣∣p−1

]2u(k)

)· ∇xv(k)},(13e)

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AN IMPLICIT GENERALIZED FINITE-DIFFERENCE TIME-DOMAIN SCHEME 187

∑kεZ

∣∣∣ψn−1/2∣∣∣p−1

D2x

(∣∣∣ψn−1/2∣∣∣p−1

u(k)

)· v(k)

=1

12{∑kεZ

∇2x

(∣∣∣ψn−1/2∣∣∣p−1

u(k)

)· ∇2x

(∣∣∣ψn−1/2∣∣∣p−1

v(k)

)

− 16∑kεZ

∇x

(∣∣∣ψn−1/2∣∣∣p−1

u(k)

)· ∇x

(∣∣∣ψn−1/2∣∣∣p−1

v(k)

)},(13f)

∑kεZ

[∣∣∣ψn−1/2

∣∣∣p−1

]2D2xu(k) · v(k)

=1

12{∑kεZ

∇2xu(k) · ∇2x

([∣∣∣ψn−1/2

∣∣∣p−1

]2v(k)

)

− 16∑kεZ

∇xu(k) · ∇x

([∣∣∣ψn−1/2

∣∣∣p−1

]2v(k)

)},(13g)

(13h)∑kεZ

[∣∣∣ψn−1/2

∣∣∣p−1

]3u(k) · v(k) =∑kεZ

u(k) · [∣∣∣ψn−1/2

∣∣∣p−1

]3v(k).

Letting u(k) = 12 [ψ

nimag(k) + ψn−1

imag(k)], v(k) = 12 [ψ

nreal(k) + ψn−1

real (k)], and then

u(k) = 12 [ψ

nreal(k) + ψn−1

real (k)], v(k) =12 [ψ

nimag(k) + ψn−1

imag(k)] in Eq.(13), we obtain

that the second summation on the RHS of Eq.(12) is zero, and hence

(14)∑kεZ

{[ψnreal(k)]

2 + [ψnimag(k)]

2} =∑kεZ

{[ψn−1real (k)]

2 + [ψn−1imag(k)]

2} = constant.

Similarly, we can prove that Eq.(11) also satisfies a similar result as in Eq.(14),implying that the present scheme also satisfies the first conservation law.

4. Numerical Examples

The first example is to consider a single soliton propagation, where λ = −2and p = 3 in Eq.(1) and the analytical solution is ψ(x, t) = sech(x + 10 − 4t) ·exp[−i(2x+20−3t)]. In our computation, the interval was taken to be−20 ≤ x ≤ 20and the boundary condition was based on the analytical solution for simplicity.We chose the number of grid points to be 200, 400, and 800, respectively, andΔt = 0.0001 in order to study the convergence with respect to space x. Thesolution was obtained based on the Jacobi iteration. The maximum error and thecomputational rate of convergence are listed in Table 1, from which one may seethat the rate of convergence is valued as expected. This implies that the numericalaccuracy of the G-FDTD scheme is similar to that of the theoretical analysis. Table2 shows a comparison of the solutions obtained by using the G-FDTD scheme, thepseudospectral method, the split-step with Fourier transform (FT) method, and theintegrating-factor with FT method in the Matlab, respectively. The table indicatesthat the G-FDTD provides much more accurate solutions.

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188 FREDERICK IRA MOXLEY III, DAVID T. CHUSS, AND WEIZHONG DAI

Figure 1. Simulation of a bright soliton propagating in free space,where the G-FDTD scheme and other numerical methods wereemployed with Δt = 0.0001, Δx = 0.1 at (a) t = 1 and (b) t = 2.

Figure 1 illustrates the bright soliton propagation in free space at t = 1 and2, obtained using N = 400. It can be seen from Figure 1 that there is no sig-nificant difference between the numerical solution and the analytical solution. Inparticular, Figure 2 illustrates the bright soliton propagation in free space near theright boundary. This conveniently demonstrates the soliton propagates out of theboundary without reflection.

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AN IMPLICIT GENERALIZED FINITE-DIFFERENCE TIME-DOMAIN SCHEME 189

Figure 2. Simulation of a bright soliton propagating in free space,where the G-FDTD scheme was employed with Δt = 0.0001, Δx =0.1 at (a) t = 2, (b) t = 7. Figure continues.

TABLE 1. Maximum error for a single soliton propagation when 0 ≤ t ≤ 1 andΔt = 0.0001.

Grid Points Maximum Error Rate of Convergence

200 0.001265517400 8.00973693 × 10−5 3.982800 5.00361649 × 10−6 4.001

TABLE 2. Maximum error for a bright soliton propagation when 0 ≤ t ≤ 1.

Grid Points Pseudospectral Split-step Integrating-factor G-FDTD200 0.06825 0.06825 0.06825 0.00127400 0.04822 0.04822 0.04822 8.00974 × 10−5

800 0.03410 0.03410 0.03410 5.00362 × 10−6

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190 FREDERICK IRA MOXLEY III, DAVID T. CHUSS, AND WEIZHONG DAI

Figure 2. (Continued) Simulation of a bright soliton propagatingin free space, where the G-FDTD scheme was employed with Δt =0.0001, Δx = 0.1 at (c) t = 7.5, (d) t = 8.

The second example is to consider a dark soliton propagation, where λ = 2 andp = 3 in Eq.(1). The initial condition was chosen such that the analytical solutionis ψ(x, t) = [tanh(2x − 10 − 4t) + i] · exp(−i8t). Again, in our computation, theinterval was taken to be −20 ≤ x ≤ 20 and the boundary condition was based onthe analytical solution for simplicity. We chose the number of grid points to be1000, 1500, and 2000, respectively, with Δt = 0.0001. Figure 3 illustrates the darksoliton propagation in free space at t = 1 and 2, obtained using the three differentmeshes. Again, it can be seen from Figure 3 that there is no significant differencebetween the numerical solution and the analytical solution.

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AN IMPLICIT GENERALIZED FINITE-DIFFERENCE TIME-DOMAIN SCHEME 191

Figure 3. Simulation of a dark soliton propagating in free space,where the G-FDTD scheme was employed with Δt = 0.0001 andthree meshes at (a) t = 1 and (b) t = 2.

5. Discussion

We have developed a linearized implicit G-FDTD scheme for solving the NLSEin 1D. The G-FDTD scheme is shown to satisfy the discrete analogous form of themass conservation law. Compared with other popular existing methods, numericalresults demonstrate that the present scheme provides a more accurate solution. Fur-ther research will focus on the stability analysis of the scheme and implementationof an artificial boundary condition to the scheme.

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192 FREDERICK IRA MOXLEY III, DAVID T. CHUSS, AND WEIZHONG DAI

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Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803

E-mail address: [email protected]

NASA Goddard Space Flight Center, Mail Code 665, Greenbelt, Maryland 20771

E-mail address: [email protected]

Mathematics and Statistics, Louisiana Tech University, Ruston, Louisiana 71272

E-mail address: [email protected]