Maximum-norm error analysis of a conservative scheme for the damped nonlinear fractional Schrödinger equation

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ScienceDirect Mathematics and Computers in Simulation 166 (2019) 206–223 www.elsevier.com/locate/matcom

Original articles

Maximum-norm error analysis of a conservative scheme for the damped nonlinear fractional Schrödinger equation Yayun Fu, Yongzhong Song, Yushun Wang ∗ Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China Received 3 January 2019; received in revised form 26 April 2019; accepted 4 May 2019 Available online 16 May 2019

Abstract This paper aims to construct a numerical scheme for the damped nonlinear space fractional Schrödinger equation. First, the conservation laws of mass and energy for the continuous equation are derived. Then, based on the fractional centered difference formula, a semi-discrete scheme, which preserves the semi-discrete mass and energy conservation laws is proposed. Further applying the Crank–Nicolson method on the temporal direction gives a fully-discrete conservative scheme. Furthermore, the solvability, boundedness and convergence in the maximum norm of the numerical solutions are given. Some numerical examples are displayed to confirm the theoretical results. c 2019 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights ⃝ reserved. Keywords: Damped nonlinear fractional Schrödinger equation; Conservative difference scheme; Stability and convergence analysis

1. Introduction Fractional order differential equations are often used for modeling many scientific and engineering problems. This kind of mathematical models is widely applied in automatic control, electronic circuits, biological materials and other fields [26,33]. In physics, especially quantum mechanics, the fractional order Schr¨odinger equation is more accurate than the integer order in describing how the quantum state of a physical system changes in time. Discussions regarding this equation have received much attention in recent years. For instance, Laskin [15,16] extended the fractality concept in quantum physics and obtained the space fractional Schr¨odinger equation via extending the Feynman path integral to the L´evy integral. The well-posedness and existence of the global smooth solution of the space fractional Schr¨odinger equation were considered in Refs. [10,11]. A slice of physical applications of this equation can be found in Refs. [12,14]. The damped nonlinear fractional Schr¨odinger (DNFS) equation is a generalization of the standard damped nonlinear Schr¨odinger equation, which can describe the resonant phenomena in viscous–elastic materials with long-range dispersion processes and memory effects. Nowadays, the investigation for the DNFS equation has captured researchers’ increasing attention [19,27]. In this paper, we numerically consider the system of DNFS equation α

iψt − (−∆) 2 ψ + β|ψ|2 ψ + iωψ = 0,

x ∈ R, 0 ≤ t ≤ T,

(1.1)

∗ Corresponding author.

E-mail address: [email protected] (Y. Wang). https://doi.org/10.1016/j.matcom.2019.05.001 c 2019 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights 0378-4754/⃝ reserved.

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with the initial condition ψ(x, 0) = ψ0 (x),

(1.2)

x ∈ R,

and the boundary condition ψ(x, t) = 0,

x ∈ R \ Ω,

0 ≤ t ≤ T,

(1.3)

where Ω = (a, b), 1 < α ≤ 2, i = −1, β is a real constant, ω > 0 represents dissipation, the initial condition ψ0 (x) is a given smooth function vanishing in R \ Ω for a ≪ 0 and b ≫ 0. When α = 2, the DNFS equation (1.1) reduces to the classical damped nonlinear Schr¨odinger equation, which has been studied by a host of mathematicians and physicists [2,29]. The fractional Laplacian is defined as a pseudo-differential operator with the symbol |θ |α in the Fourier space [3] 2

α

ˆ − (−∆) 2 ψ(x, t) = −F −1 (|θ |α ψ(θ, t)),

(1.4)

ˆ where F is the Fourier transform and ψ(θ, t) = F[ψ(x, t)]. Yang et al. [34] showed that the fractional Laplacian is equivalent to the Riesz fractional derivative in one dimension ∂ α ψ(x, t) 1 α α − (−∆) 2 ψ(x, t) = [ −∞ Dxα ψ(x, t) +x D+∞ ψ(x, t)], (1.5) =− α ∂|x| 2 cos απ 2 α where −∞ Dxα ψ(x, t) and x D+∞ ψ(x, t) are the left and right side Riemann–Liouville fractional derivatives [5], respectively. Some numerical schemes have been devoted to approximate the Riesz and Riemann–Liouville fractional derivative. For example, Meerschaert and Tadjeran proposed the shifted Gr¨unwald formula [22,23]. Ortigueira developed the fractional centered difference formula [24]. Following a similar Lawson transformation in Refs. [17], we define the change of variables

u(x, t) = eωt ψ(x, t),

(1.6)

and arrange the system (1.1) as α

iu t − (−∆) 2 u + βe−2ωt |u|2 u = 0,

x ∈ R, 0 ≤ t ≤ T,

(1.7)

with the initial condition u(x, 0) = u 0 (x) = ψ0 (x),

(1.8)

x ∈ R,

and the boundary condition u(x, t) = 0,

x ∈ R \ Ω,

0 ≤ t ≤ T,

(1.9) α 2

α 4

α 4

Taking the inner products of (1.7) with u t and considering the fact that ((−∆) u, u) = ((−∆) u, (−∆) u) [31], we can deduce the real part to an energy conservation law d E(t) = 0, (1.10) dt where the energy function E(t) is given as ∫ ∫ β −2ωt α 2 4 E(t) = |(−∆) u(x, t)| d x − e |u(x, t)|4 d x 2 Ω Ω ∫ ∫ t − ωβ e−2ων |u(x, ν)|4 d xdν. (1.11) 0

Ω

Then, taking the inner products of (1.7) with u, and we can deduce the image part to a mass conservation law d M(t) = 0, (1.12) dt where the mass function M(t) is defined as ∫ M(t) = |u(x, t)|2 d x. (1.13) Ω

As we all know, the exact solutions of fractional differential equation contain some special functions, such as Mittag-Leffler function, Wright function, which yields some difficulties in numerical computations. In recent years,

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the numerical schemes related to the fractional differential equation exerted a tremendous fascination on researchers, and huge fruits have been reaped, which can be referred to Refs. [6,7,35]. The prior researches substantiated that the structure-preserving algorithms are the numerical method that can inherit the intrinsic geometric properties of a given dynamical system [9,13]. They are more superior than traditional methods in long time numerical simulations. At present, structure-preserving methods for fractional differential equations, especially for fractional Schr¨odinger equations, have attracted the researchers’ increasing attention and many significant achievements have been made [8,18,20,21,25,30–32]. However, few researches have focused on developing structure-preserving algorithms for the DNFS equation. To the best of our knowledge, only study of structure-preserving algorithms for this equation is a conformal mass conservation linearized difference scheme which proposed by Liang and Song [19]. The scheme is of second order convergence in space and first order convergence in time, and only preserves the discrete mass. In this paper, we propose a new scheme which is of the second order convergence in both space and time, and can preserve both mass and energy in discrete sense. The error estimate of conformal conservative scheme in Ref. [19] is in the discrete l 2 -norm, whereas the error estimate of our new scheme is in the discrete maximum norm. The paper is arranged as follows. In Section 2, the semi-discrete finite difference scheme and fully-discrete Crank–Nicolson finite difference scheme are established and proved to preserve the discrete global mass and energy. In Section 3, the existence, uniqueness and boundedness of the difference solutions, and the convergence of order O(τ 2 + h 2 ) in the maximum norm are discussed. The numerical examples are presented in Section 4 to demonstrate the theoretical results. Some conclusions are drawn in Section 5. 2. Construction of the energy-preserving scheme We choose two positive integers K and N , let h := b−a and τ := NT be the space size and time step, respectively. K Denoting Ωh = {x j | x j = a + j h, j = 0, 1, . . . , K }, Ωτ = {tn | tn = nτ, n = 0, 1, . . . , N }. For a grid function w = {wnj |wnj = w(x j , tn ), (x j , tn ) ∈ Ωh × Ωτ }, we introduce the following notations n+ 21

δt w j

n+ 1 δtˆw j 2

=

w n+1 − wnj j τ

n+ 12

,

wj ω

ω

− e− 2 τ wnj e 2 τ wn+1 j

=

wn+1 + wnj j 2 n+ 1 wˆ j 2

, ω

ω

+ e− 2 τ wnj e 2 τ wn+1 j

, . = τ 2 We define a space of grid functions on Ωh as V˚ h := {w|(w0 , w1 , . . . , w K )T } and Vh := {w|w ∈ V˚ h , w0 = w K = 0}. For any two grid functions w, v ∈ Vh , we define the discrete inner product and the associated l 2 -norm as =

(w, v) = h

K −1 ∑

w j v¯ j ,

∥w∥2 = (w, w).

j=1

We also define the discrete l p -norm as ∥w∥ pp

=h

K −1 ∑

|w j | p ,

1 ≤ p < ∞,

j=1

and the discrete maximum norm (l ∞ -norm) as ∥w∥∞ =

sup

|w j |.

1≤ j≤K −1

Provided the constant 0 ≤ σ ≤ 1, the fractional Sobolev norm ∥w∥ H σ and semi-norm |w| H σ can be defined as [14] ∫ π ∫ π 2 2 ∥w∥2H σ = h (1 + h −2σ |k|2σ )|w(k)| ˆ dk, |w|2H σ = h h −2σ |k|2σ |w(k)| ˆ dk, −π

where 1 ∑ n −i jk wˆ n (k) = √ wj e . 2π j∈Z

−π

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Obviously, we can obtain ∥w∥2H σ = ∥w∥2 + |w|2H σ .

(2.1)

2.1. Structure-preserving spatial discretization Fractional centered difference scheme [24] has second order accuracy, and can preserve the symmetric property of Riesz fractional derivative very well, which plays an essential role in numerical analysis. In this paper, we employ the fractional centered difference to approximate the Riesz fractional derivative. Lemma 2.1 ([20,24]). Let u(x) ∈ C 5 (R) and all derivatives up to order five belong to L 1 (R). Then, for 1 < α ≤ 2, we have +∞ ∂ α u(x) 1 ∑ (α) g u(x − lh) + O(h 2 ), =− α ∂|x|α h l=−∞ l

where the coefficients gl(α) := g0(α) > 0,

(−1)l Γ (α+1) , Γ ( α2 −l+1)Γ ( α2 +l+1)

(α) gl(α) = g−l ≤ 0,

(2.2)

and have the following properties for 1 < α ≤ 2

l = ±1, ±2, . . . .

(2.3)

Since u(x, t)=0 for x ∈ R \ Ω , we can obtain ∂ α u(x, t) 1 =− α α ∂|x| h

−(a−x)/ h

∑

gl(α) u(x − lh, t) + O(h 2 ).

(2.4)

l=−(b−x)/ h

We denote U nj and u nj as the numerical approximation and the exact solution of u(x, t) at the points (x j , tn ), respectively, then it follows from (1.9) and (2.4) that α

(−∆) 2 u nj =

1 hα

j ∑

gl(α) u nj−l + O(h 2 ) =

l=−K + j

K −1 1 ∑ (α) n g u + O(h 2 ). h α l=1 j−l l

(2.5)

Denote △αh U nj :=

K −1 1 ∑ (α) n g U , h α l=1 j−l l

(2.6)

G as

We denote matrix ⎛ (α) g0 ⎜ g (α) ⎜ 1 G=⎜ . ⎝ .. g (α) M−2

1 ≤ j ≤ K − 1, 0 ≤ n ≤ N .

(α) g−1 g0(α) .. .

··· ··· .. .

g (α) M−3

···

⎞ (α) g−M+2 (α) ⎟ g−M+3 ⎟ .. ⎟ . . ⎠ g0(α)

Let λ = (λ1 , λ2 , . . . , λ K −1 ), where λi (i = 1, 2, . . . , K − 1) is the eigenvalue of matrix G, and satisfies [4] 0 < λi < 2g0(α) ,

i = 1, 2, . . . , K − 1.

One can easily verify that the G is a real-value symmetric positive definite Toeplitz matrix. α

1

Lemma 2.2 ([21,30]). For any two grid functions U, V ∈ Vh , then there exists a linear operator Λα = h − 2 G 2 , such that (△αh U, V ) = (Λα U, Λα V ), 1

(2.7) 1

where the G 2 is the unique positive definite square root of G, that is, (G 2 )2 = G.

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Lemma 2.3 ([21,30]). For any complex grid function U n ∈ Vh , 0 ≤ n ≤ N , we have 1

1

Im(△αh U n+ 2 , U n+ 2 ) = 0, 1

1

Re(△αh U n+ 2 , δt U n+ 2 ) =

(2.8) 1 (∥Λα U n+1 ∥2 − ∥Λα U n ∥2 ). 2τ

(2.9)

2.2. Conservative semi-discrete scheme By using the fractional centered difference scheme to discrete (1.7) in space direction, a semi-discrete scheme is given as d i U j − ∆αh U j + βe−2ωt |U j |2 U j = 0. (2.10) dt The conservation properties of above semi-discrete scheme are given in the following theorem. Theorem 2.1. The scheme (2.10) satisfies the semi-discrete mass conservation law ˜ ˜ M(t) = M(0), where ˜ M(t) = ∥U ∥2 . Proof. By taking the inner products of the above with U j , and taking the imaginary part in the resulting formula leads to d ∥U ∥2 = 0, (2.11) dt where (2.8) is used. This completes the proof. □ Theorem 2.2. The semi-discrete scheme (2.10) possesses the following energy conservation law ˜ = E(0), ˜ E(t) where ˜ = ∥Λα U ∥2 − β e−2ωt ∥U ∥44 − ωβ E(t) 2

∫

t

e−2ων ∥U (ν)∥44 dν.

0

Proof. Computing the inner product of (2.10) with dtd U j , and taking the real part, we obtain ∫ t ) d ( α 2 β −2ωt ∥Λ U ∥ − e ∥U ∥44 − ωβ e−2ων ∥U (ν)∥44 dν = 0 dt 2 0 where (2.9) is used, which completes the proof.

(2.12)

2.3. A fully-discrete energy-preserving scheme For the semi-discrete conservative scheme (2.10), we apply the Crank–Nicolson method on the temporal direction and obtain a scheme in the fully discrete version. Then, the fully-discrete Crank–Nicolson (CN) finite difference scheme for the system (1.7)–(1.9) can be written as n+ 1 iδt U j 2

−

n+ 1 △αh U j 2

+

βe

−2ωt

n+ 1 2

2 1 ≤ j ≤ K − 1, 0 ≤ n ≤ N − 1,

U 0j = u 0 (x j ),

0 ≤ j ≤ K,

2

n+ 21

(|U n+1 | + |U nj |2 )U j j

= 0, (2.13) (2.14)

Y. Fu, Y. Song and Y. Wang / Mathematics and Computers in Simulation 166 (2019) 206–223

0 ≤ n ≤ N.

U0n = 0, U Kn = 0,

211

(2.15)

The conservation properties of the CN finite scheme (2.13)–(2.15) are given in the following theorems. Theorem 2.3. The CN finite difference scheme (2.13)–(2.15) satisfies the mass conservation law in discrete sense M n+1 = M n ,

M n = ∥U n ∥2 ,

n = 0, 1, . . . , N − 1. 1

Proof. Computing the discrete inner product of (2.13) with U n+ 2 , and taking the imaginary part in the resulting formula leads to 1 (∥U n+1 ∥2 − ∥U n ∥2 ) = 0, (2.16) 2τ where (2.8) is used. The result indicates that ∥U n+1 ∥2 = ∥U n ∥2 , This gives M

n+1

n = 1, 2, . . . , N − 1.

n

(2.17)

□

=M .

Theorem 2.4. The CN finite difference scheme (2.13)–(2.15) possesses the following energy conservation law in the discrete sense E n = E 0,

n = 0, 1, . . . , N ,

where E n := ∥Λα U n ∥2 −

n β −2ωtn− 1 β ∑ −2ωtl− 3 2 ∥U n ∥4 − 2 (1 − e −2ωτ )∥U l−1 ∥4 . e e 4 4 2 2 l=1 1

Proof. Computing the discrete inner product of (2.13) with δt U n+ 2 , and taking the real part, we obtain the following energy equality β −2ωtn+ 1 β −2ωtn+ 1 2 ∥U n+1 ∥4 + 2 ∥U n ∥4 = 0, ∥Λα U n+1 ∥2 − ∥Λα U n ∥2 − e e (2.18) 4 4 2 2 where (2.9) is used. Summing up for n from 1 to m and then replacing m by n, we complete the proof. Remark 2.1. Based on Theorem 2.3, it is easy to show that the numerical solution of the CN finite difference scheme (2.13)–(2.15) is bounded in the discrete l 2 -norm, i.e., there exists a constant C > 0, such that ∥U n ∥2 = ∥U n−1 ∥2 = · · · = ∥U 0 ∥2 < C.

(2.19)

Here and subsequent theoretical analysis, C is a general positive constant which is independent of τ and h. Note that C may vary in different circumstances. 3. Numerical analysis In this section, we will discuss the boundedness, solvability and convergence of the solutions of the fully-discrete CN finite difference scheme (2.13)–(2.15). Some lemmas for subsequent theoretical analysis are given. Lemma 3.1 (Discrete Sobolev inequality [14]). Note that ∥u n ∥ H σ ≤ ∥u n ∥ H ρ for 0 ≤ σ ≤ ρ ≤ 1. Then for any 1 < σ ≤ 1, there exists a constant C = C(σ ) > 0 independent of h > 0, such that 2 ∥u n ∥∞ ≤ C∥u n ∥ H σ ,

(3.1)

for all u n ∈ lh2 . Lemma 3.2 (Gagliardo–Nirenberg inequality [14]). For any independent of h > 0, such that σ /σ

∥u n ∥4 ≤ C∥u n ∥ H0σ ∥u n ∥1−σ0 /σ , for any σ0 ≤ σ ≤ 1.

1 4

< σ0 ≤ 1, there exists a constant C = C(σ ) > 0 (3.2)

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Lemma 3.3 (Uniform norm equivalence [32]). For any 1 < α ≤ 2, we have ∞ ∑ 2 α | | |u n |2H α/2 ≤ h (△αh u nj u nj ) ≤ |u n |2H α/2 , π j=−∞

(3.3)

∞ ∑ 2 α | | |u n | H α/2 |v n | H α/2 ≤ h (△αh u nj v nj ) ≤ |u n | H α/2 |v n | H α/2 . π j=−∞

(3.4)

and

3.1. Boundedness and solvability Remark 2.1 shows that the numerical solution of CN finite difference scheme (2.13)–(2.15) is bounded in the discrete l 2 -norm. Then we can get the following theorem. Theorem 3.1. The numerical solution of CN finite difference scheme (2.13)–(2.15) satisfies that ∥U n ∥∞ ≤ C, 1 ≤ n ≤ N .

(3.5) 1

Proof. When β > 0, multiplying both sides of (2.13) with e−(n+ 2 )ωτ , we obtain 1

i

1

e−(n+ 2 )ωτ U n+1 − e−(n+ 2 )ωτ U nj j

+

βe

−2ωt

τ n+ 1 2

2

1

1

−

(|U n+1 | + |U nj |2 ) j

e

△αh

+ e−(n+ 2 )ωτ U nj e−(n+ 2 )ωτ U n+1 j

−(n+ 21 )ωτ

2 U n+1 j

2 According to (1.6), we have

+e

−(n+ 12 )ωτ

U nj

2

= 0.

U nj = enωτ ψ nj ,

(3.6)

(3.7)

and substitute it into (3.6), we can deduce ω

ω

i

− e− 2 τ ψ nj e 2 τ ψ n+1 j τ

ω

−

△αh

ω

e 2 τ ψ n+1 + e− 2 τ ψ nj j ωτ 2

2 ω ψ n+1 + e− 2 τ ψ nj j

β ω τ n+1 2 ω 2 e (|e 2 ψ j | + |e− 2 τ ψ nj | ) = 0, 2 2 which can be written as β ω ω 2 2 n+ 1 n+ 1 n+ 1 | + |e− 2 τ ψ nj | )ψˆ j 2 = 0. iδtˆψ j 2 − △αh ψˆ j 2 + (|e 2 τ ψ n+1 j 2 +

n+ 21

We make the discrete inner product of (3.9) with −2τ δtˆψ j energy equality

β − 3ω τ n 4 e 2 ∥ψ ∥4 , n = 0, 1, . . . , N , 2 and using inequality 1 + x ≤ e x , ∀x ∈ R, we can obtain ω ω (1 + τ )∥Λα ψ n ∥2 ≤ e 2 τ ∥Λα ψ n ∥2 . 2 χ n = ∥Λα ψ n ∥2 −

(3.9)

and take the real part, one can obtain the following

β ω τ n+1 4 β ω ω ∥e 2 ψ ∥4 = ∥e− 2 τ Λα ψ n ∥2 − ∥e− 2 τ ψ n ∥44 . 2 2 Then we rewrite (3.10) as ω ω 3ω β 3ω β e 2 τ ∥Λα ψ n+1 ∥2 − e 2 τ ∥ψ n+1 ∥44 = e−ωτ (e− 2 τ ∥Λα ψ n ∥2 − e− 2 τ ∥ψ n ∥44 ). 2 2 Introducing the function ω

∥e 2 τ Λα ψ n+1 ∥2 −

(3.8)

(3.10)

(3.11)

(3.12)

(3.13)

Y. Fu, Y. Song and Y. Wang / Mathematics and Computers in Simulation 166 (2019) 206–223

213

Noting the inequality e x ≤ 1 + xe x , ∀x ∈ R, we have e

3ω τ 2

∥ψ n+1 ∥44 ≤ (e−

3ω τ 2

+ 3ωτ e

3ω τ 2

)∥ψ n+1 ∥44 .

(3.14)

Since 1 < α ≤ 2, according to (3.2) and (3.11), then we can choose σ0 < α2 and deduce that there exists a constant C = Cα > 0 independent of h and τ , such that ωτ α n+1 2 3ωβ 3ω τ n+1 4 χ n+1 + ∥Λ ψ ∥ ≤ e−ωτ χ n + τ e 2 ∥ψ ∥4 2 2 3ωβ 3ω τ σ /σ ≤ e−ωτ χ n + τ e 2 (Cα ∥ψ n+1 ∥ H0σ ∥ψ n+1 ∥1−σ0 /σ )4 2 3ωβ 3ω τ 4σ0 /α ≤ e−ωτ χ n + τ e 2 C∥ψ n+1 ∥ H α/2 2 3ωβ 3ω τ τ e 2 C∥ψ n+1 ∥2H α/2 ≤ e−ωτ χ n + 2 3ωβ 3ω τ 2 ≤ e−ωτ χ n + τ e 2 C(∥ψ n+1 ∥2 + |ψ n+1 | H α/2 ) 2 ωτ n+1 2 ≤ e−ωτ χ n + C |ψ | H α/2 + Cτ e−ωτ ∥eωτ ψ n+1 ∥2 , (3.15) 2 where Lemma 3.2, the inequality (3.13) and the inequality (3.14) are used. With Lemma 3.3, (3.15) reduces to χ n ≤ e−ωτ χ n−1 + Ce−ωτ ∥eωτ ψ n ∥2 ≤ · · · ≤ (e

) χ + Ce

−ωτ n

0

−ωτ

n−1 ∑ [(e−ωτ )l (eωτ ∥ψ n−l ∥2 )].

(3.16)

l=0

Using the following fact m ∑ (e−ωτ )l ≤ l=0

1 eωτ eωτ = ≤ , ∀ 0 < m < ∞, 1 − e−ωτ eωτ − 1 ατ

(3.17)

and (2.19), we can deduce from (3.16) that χ n ≤ (e−ωτ )n χ 0 + C,

(3.18)

which indicates that eωτ χ n ≤ C, n = 1, 2, . . . , N . Returning to the definition of χ n and noticing Lemmas 3.2 and 3.3, we have β 3ω |ψ n |2H α/2 ≤ χ n + e− 2 τ ∥ψ n ∥44 2 β − 3ω τ σ /σ n ≤ χ + τ e 2 (C∥ψ n+1 ∥ H0σ ∥U n+1 ∥1−σ0 /σ )4 2 β 3ω ≤ χ n + e− 2 τ C∥ψ n+1 ∥2H α/2 2 β − 3ω τ 2 n ≤ χ + e 2 C(∥ψ n+1 ∥2 + |ψ n+1 | H α/2 ) 2 2 ≤ χ n + C|ψ n+1 | H α/2 + Cτ e−ωτ ∥eωτ ψ n+1 ∥2 .

(3.19)

(3.20)

According to (3.20), we can get eωτ |ψ n |2H α/2 ≤ 2eωτ χ n + C∥eωτ ψ n+1 ∥2 ≤ C, n = 1, 2, . . . , N ,

(3.21)

which implies that ω

2

|e 2 τ ψ n | H α/2 ≤ C, n = 1, 2, . . . , N .

(3.22)

It is easy to see that |ψ n |2H α/2 ≤ C,

∥ψ n ∥∞ ≤ C, n = 1, 2, . . . , N .

(3.23)

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This, together with (3.7), gives ∥U n ∥∞ = ∥enωτ ψ n ∥∞ ≤ ∥eωT ψ n ∥∞ ≤ C,

(3.24)

where T is a fixed constant. For case of β ≤ 0, using the uniform norm equivalence (see Lemma 3.3), we have ∥Λα U n ∥2 = Cα |U n |2H α/2 ,

(3.25)

where ( π2 )α ≤ Cα ≤ 1. By the Gagliardo–Nirenberg inequality (see Lemma 3.2) and Remark 2.1, we have β −2ωtn− 1 8σ /α 2 ∥U n ∥4 ≤ C∥U n ∥ α0 . e 4 H /2 2 From the definition of E n and β ≤ 0, one gets β −2ωtn− 1 2 ∥U n ∥4 . E n ≥ ∥Λα U n ∥2 − e 4 2 Since 1 < α ≤ 2, from (3.2) and (3.26), we can choose σ0 <

(3.26)

(3.27) α 4

and deduce that

8σ /α

0 E n ≥ Cα ∥U n ∥2H α/2 − C∥U n ∥ H α/2 .

(3.28)

This, together with Lemma 3.1, we can derive E n → +∞ as ∥U n ∥2H α/2 → +∞. Using the discrete conservation law of energy, we have E n < +∞. It is easy to see that the numerical solution of the CN finite scheme is bounded in ∥ · ∥2H α/2 . According to Lemma 3.1, the numerical solution is also bounded in the l ∞ -norm, i.e., there exists a constant C > 0, such that ∥U n ∥∞ ≤ C.

(3.29)

Finally, for β ∈ R, we can deduce that ∥U n ∥∞ ≤ C. □

(3.30)

Lemma 3.4 (Brouwer fixed point theorem [1]). Let (P, ⟨·, ·⟩) be a finite dimensional space with inner product space, ∥ · ∥ be the associated norm, and f : P → P be continuous mapping. If ∃ φ > 0, ∀z ∈ P, ∥z∥ = φ, Re⟨ f (z), z⟩ ≥ 0,

(3.31)

then, there exists a zˆ ∈ P, ∥ˆz ∥ ≤ φ, such that f (ˆz ) = 0. Theorem 3.2. The solution of the CN finite difference scheme (2.13)–(2.15) exists. Proof. For a fixed n, we rewrite (2.13) as −2ωt

1

n+ 2 2 τ βe n+ 1 n+ 1 n+ 1 = − i [△αh U j 2 − (|U nj |2 + |2U j 2 − U nj | )U j 2 ]. 2 2 We define a mapping H : Vh → Vh as follows

n+ 1 Uj 2

U nj

−2ωt

1

(3.32)

n+ 2 τ βe H(ξ ) = ξ + i [△αh ξ − (|U nj |2 + |2ξ − U nj |2 )ξ ], (3.33) 2 2 which is obviously continuous. Computing the inner product of (3.33) with ξ and taking the real part, we have τ Re(H(ξ ), ξ ) = ∥ξ ∥2 − Re(U n , ξ ) − Im∥Λα ξ ∥2 2 −2ωt 1 K −1 n+ ∑ 2 τ βe + [Im h (|U nj |2 + |2ξ j − U nj |2 )|ξ j |2 ] 2 2 j=1

− U nj

= ∥ξ ∥2 − Re(U n , ξ )

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215

≥ ∥ξ ∥2 − ∥ξ ∥∥U n ∥ 1 (3.34) ≥ (∥ξ ∥2 − ∥U n ∥2 ), 2 where (2.7) and inequality are used. √ the Cauchy–Schwarz √ Let µ = ∥U n ∥2 + 1 = ∥U 0 ∥2 + 1, we have Re(H(ξ ), ξ ) ≥ 21 , for ∥ξ ∥ = µ. Thus, there exists a solution 1 ξˆ ∈ Vh satisfying H(ξˆ ) = 0, and then the existence of U n+ 2 follows from Lemma 3.4, which completes the proof. □ Lemma 3.5 ([28]). For any complex functions U, V, u, v, we have ||U |2 V − |u|2 v| ≤ (max{|U |, |V |, |u|, |v|})2 (2|U − u| + |V − v|). Theorem 3.3. The solution of the CN finite difference scheme (2.13)–(2.15) is unique. Proof. Let U nj and V jn be the numerical solutions of the scheme (2.13)–(2.15) and ηnj = U nj − V jn . From (2.13), we have n+ 12

n+ 12

− △αh η j

iδt η j

n+ 12

+ aj

= 0,

(3.35)

where n+ 1 aj 2

=

βe

−2ωt

n+ 1 2

n+ 12

2

[(|U n+1 | + |U nj |2 )U j j

2 Combining Lemma 3.5 and (3.29), we have −2ωt

|βe 2

n+ 1 2

|

n+ 21

2

[(|U n+1 | + |U nj |2 )U j j

n+ 12

2

− (|V jn+1 | + |V jn |2 )V j n+ 12

−2ωt

n+ 1 2

2 n |Cmax (|ηn+1 j | + |η j | + 2|η j

−2ωt

n+ 1 2

2 n |Cmax (2|ηn+1 j | + 2|η j |),

≤ |βe ≤ |βe

n+ 12

2

− (|V jn+1 | + |V jn |2 )V j

].

(3.36)

]

|) (3.37)

where Cmax = max{|U n+1 |, |U nj |, |V jn+1 |, |V jn |}. Thus, there exists a constant C > 0, such that j n+ 12

|a j

n | ≤ C(|ηn+1 j | + |η j |).

(3.38)

Computing the discrete inner product of (3.35) with ηn+1 − ηn , taking the imaginary part, and using Lemma 2.3, we have 2

∥ηn ∥2 + ∥ηn+1 ∥ ≤ τ h

K −1 ∑

n+ 21

(|a j

n |(|ηn+1 j | + |η j |)).

(3.39)

j=1

Thus there exists a constant C > 0 such that ∥ηn ∥2 + ∥ηn+1 ∥2 ≤ τ C(∥ηn ∥2 + ∥ηn+1 ∥2 ). When τ C < 1, we have ∥η ∥ + ∥η n 2

n+1 2

∥ = 0. This completes the proof.

(3.40) □

3.2. Convergence analysis Lemma 3.6 (Gronwall inequality [36]). Assume that the discrete function {wn |n = 0, 1, . . . , N : N τ = T } satisfies the following recurrence formula ˜ w n + Bτ ˜ w n−1 + C˜n τ, w n − wn−1 ≤ Aτ

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˜ B, ˜ C˜n (n = 1, 2, . . . , N ) are nonnegative constants. Then where A, max |wn | ≤ (w 0 + τ

1≤n≤N

N ∑

˜ ˜ C˜k )e2( A+ B) ,

k=1

˜ ≤ where τ is a sufficiently small positive constant with ( A˜ + B)

N −1 , 2N

N > 1.

Next, we will analyse the convergence of the CN finite difference scheme in the l ∞ -norm. First, we have following lemma. Lemma 3.7. Let u(x, t) ∈ C 4 ([0, T ]; C 5 (R) ∩ L 1 (R)) be the exact solution of the original problem (1.7)–(1.9), then the numerical solution of the CN finite difference scheme (2.13)–(2.15) is unconditionally convergent with order O(τ 2 + h 2 ) in discrete l 2 -norm, where 0 < τ < τ0 , and τ0 is a constant. Proof. Define the truncation errors of the scheme (2.13)–(2.15) as follows n+ 1 iδt u j 2

−

n+ 1 △αh u j 2

+

βe

−2ωt

n+ 1 2

n+ 21

2

n 2 (|u n+1 j | + |u j | )u j

2 1 ≤ j ≤ K − 1, 0 ≤ n ≤ N − 1.

= q nj , (3.41)

By the Taylor expansion and (2.5), we can obtain |q nj | ≤ C(τ 2 + h 2 ),

1 ≤ j ≤ K − 1, 0 ≤ n ≤ N − 1.

(3.42)

This implies that ∥q n ∥ ≤ C(τ 2 + h 2 ),

0 ≤ n ≤ N − 1.

(3.43)

0 ≤ j ≤ K, 0 ≤ n ≤ N.

(3.44)

Let ε nj = u nj − U nj ,

Subtracting (2.13) from (3.41) yields that n+ 21

iδt ε j

n+ 12

− △αh ε j

n+ 12

+ Fj

= q nj ,

(3.45)

where n+ 21

βe

−2ωt

n+ 1 2

n+ 12

2

n 2 [(|u n+1 j | + |u j | )u j

2

n+ 21

− (|U n+1 | + |U nj |2 )U j j

]. (3.46) 2 Based on the smoothness assumption of the exact solution, together with Theorem 3.1 and Lemma 3.5, we have Fj

=

n+ 21

|F j

| ≤ C(|ε nj | + |ε n+1 j |).

(3.47)

Then we can obtain 1

∥F n+ 2 ∥ ≤ C(∥ε nj ∥ + ∥ε n+1 j ∥).

(3.48)

Computing the inner product of (3.45) with ε n+ 21 t

Im(iδ ε

,ε

n+ 12

) + Im(F

n+ 21

,ε

n+ 12

n+ 12

and taking the imaginary part, we obtain 1

) = Im(q n , εn+ 2 ),

(3.49)

where 1 (∥ε n+1 ∥2 − ∥ε n ∥2 ). 2τ By the Young’s inequality and (3.48), we can get 1

1

1

1

Im(iδt ε n+ 2 , εn+ 2 ) = Re(δt ε n+ 2 , εn+ 2 ) =

1

Im(q n , εn+ 2 ) ≤ C(∥ε n ∥2 + ∥ε n+1 ∥2 + ∥q n ∥2 ),

(3.50)

(3.51)

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217

and 1

1

Im(F n+ 2 , εn+ 2 ) ≤ C(∥ε n ∥2 + ∥ε n+1 ∥2 ).

(3.52)

Substituting (3.50)–(3.52) into (3.49), yields that ∥ε n+1 ∥2 − ∥ε n ∥2 ≤ Cτ (∥ε n ∥2 + ∥ε n+1 ∥2 + ∥q n ∥2 ).

(3.53)

From (3.53) and (3.43), we can derive ∥ε n+1 ∥2 − ∥ε n ∥2 ≤ Cτ [(∥ε n ∥2 + ∥ε n+1 ∥2 ) + (τ 2 + h 2 )2 ].

(3.54)

Using Lemma 3.6, we have ∥ε n ∥2 ≤ C(τ 2 + h 2 )2 .

(3.55) 2

This completes the proof of convergence in discrete l -norm.

□

∞

We present the error estimation in the l -norm of the CN finite difference scheme (2.13)–(2.15) in the following theorem. Theorem 3.4. Let u(x, t) ∈ C 4 ([0, T ]; C 5 (R) ∩ L 1 (R)) be the exact solution of the original problem (1.7)–(1.9), then the numerical solution of the CN finite difference scheme (2.13)–(2.15) is unconditionally convergent with order O(τ 2 + h 2 ) in discrete l ∞ -norm, where 0 < τ < τ0 , and τ0 is a constant. 1

Proof. Computing the inner product of (3.45) with δt ε n+ 2 and taking the real part yields 1

1

1

1

1

Re(△αh ε n+ 2 , δt ε n+ 2 ) = Re(F n+ 2 , δt ε n+ 2 ) + Re(−q n , δt ε n+ 2 ).

(3.56)

From (3.41), one has 1

1

1

1

1

Re(F n+ 2 , δt ε n+ 2 ) = Re(F n+ 2 , −i △αh ε n+ 2 + iF n+ 2 − iq n ) 1

1

= Im(F n+ 2 , △αh ε n+ 2 + q n ). By Lemma 2.3 and uniform norm equivalence (see Lemma 3.3), one gets 1 1 1 (∥Λα ε n+1 ∥2 − ∥Λα ε n ∥2 ) Re(△αh ε n+ 2 , δt ε n+ 2 ) = 2τ Cα n+1 2 = (|ε | H α/2 − |ε n |2H α/2 ), 2τ where ( π2 )α ≤ Cα ≤ 1. From Lemma 3.3, one gets Cα 1 1 2 1 2 1 (|F n+ 2 | H α/2 + |ε n+ 2 | H α/2 ). |Im(F n+ 2 , △αh ε n+ 2 )| ≤ 2

(3.57)

(3.58)

(3.59)

1

This, together with the definition of |F n+ 2 | and Lemma 3.5, gives 1 2

2

|F n+ 2 | H α/2 ≤ C(|ε n+1 | H α/2 + |ε n |2H α/2 ).

(3.60)

Noticing the l 2 -norm error estimation (3.55), from (3.43) and (3.48), we can deduce that 1

|Im(F n+ 2 , q n )| ≤ C(h 2 + τ 2 )2 .

(3.61)

Substituting (3.59)–(3.61) into (3.57) yields 1

1

2

Re(F n+ 2 , δt ε n+ 2 ) ≤ C(|ε n+1 | H α/2 + |ε n |2H α/2 ) + C(h 2 + τ 2 )2 .

(3.62)

Then substituting (3.58) and (3.62) into (3.56) gives that Cα n+1 2 2 (|ε | H α/2 − |ε n |2H α/2 ) ≤ C(|ε n+1 | H α/2 + |ε n |2H α/2 ) 2τ 1 + Re(−q n , δt ε n+ 2 ) + C(h 2 + τ 2 )2 .

(3.63)

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Summing up for the superscript n from 0 to m, we obtain that m

∑ Cα m+1 2 2 2 (|ε | H α/2 − |ε 0 | H α/2 ) ≤ C (|ε n+1 | H α/2 + |ε n |2H α/2 ) 2τ n=0 +

m ∑

1

Re(−q n , δt ε n+ 2 ) + mC(h 2 + τ 2 )2 .

(3.64)

n=0

From (3.43), (3.48) and (3.55), we can deduce that |2τ

m ∑

1

Re(−q n , δt ε n+ 2 )| ≤ C[∥ε 0 ∥2 + τ

n=0

m ∑

∥ε m ∥2 + ∥ε m+1 ∥2 + (h 2 + τ 2 )2 ].

(3.65)

n=1

Substituting (3.65) in to (3.64), according to (3.55) and mτ < T , we have 2

Cα |ε m+1 | H α/2 ≤ 2τ C

m ∑ (∥ε n ∥2 + ∥ε n+1 ∥2 ) + C T (h 2 + τ 2 )2 .

(3.66)

n=0

Thus, there exists a constant τ0 : 0 < τ0 ≤ 2

|εm+1 | H α/2 ≤

4τ C Cα − 2τ0 C

m ∑

Cα , 2C

∥ε k ∥2 +

n=0

such that

CT (h 2 + τ 2 )2 . Cα − 2τ0 C

(3.67)

This, together with Lemma 3.5, we can obtain |ε n |2H α/2 ≤ C(h 2 + τ 2 )2 .

(3.68)

According to Lemma 3.1, it holds that ∥ε n ∥2∞ ≤ C∥ε n ∥2H α/2 = C(∥ε n ∥2 + |ε n |2H α/2 ).

(3.69)

Noticing (3.55) and (3.68), we can derive that ∥ε n ∥2∞ ≤ C(h 2 + τ 2 )2 ,

0 ≤ n ≤ N.

(3.70)

This completes the proof. □ Remark 3.1. If the system (1.1) has no damping item, i.e., ω = 0, it becomes the nonlinear fractional Schr¨odinger equation. The conservation laws obtained are consistent with the ones discussed in Ref. [30]. Moreover, the error estimate of our paper is in the discrete l ∞ -norm, whereas the error estimate of the Ref. [30] is in the discrete l 2 -norm. Remark 3.2. When α = 2, the system (1.1) can be written as the classical damped nonlinear Schr¨odinger equation. Moreover, if ω = 0, it becomes nonlinear Schr¨odinger equation, and we can obtain the similar conclusions. 4. Numerical examples In this section, numerical examples of the CN finite difference scheme (2.13)–(2.15) are given to support our theoretical analysis. We study the system (1.1) with β = 2 and different fractional order α, and damping ω. The initial condition without damping is chosen as u(x, 0) = sech(x) exp(2ix), x ∈ Ω .

(4.1)

Since u(x, t) decays exponentially to zero as the variable x tends to infinity, the wave function can be negligible outside the interval [a, b] for a ≪ 0 and b ≫ 0. In our computation, we take the computation domain Ω = [−20, 20] and the Dirichlet condition as u(−20, t) = u(20, t) = 0, 0 ≤ t ≤ T.

(4.2)

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219

Table 1 The numerical errors and convergence orders for α = 2 and ω = 0 at T = 1. h

τ

∥u − U ∥∞

Order

0.4 0.2 0.1 0.05

0.04 0.02 0.01 0.005

6.8531e−01 1.4439e−01 3.4719e−02 8.5888e−03

– 2.24 2.05 2.02

Table 2 The numerical errors and convergence order for different α with ω = 0.01 at T = 1. h 0.4 0.2 0.1 0.05

α = 1.6

τ 0.04 0.02 0.01 0.005

α = 1.9

e(h, τ )

Order

e(h, τ )

Order

5.9018e−01 1.8616e−01 3.9849e−02 9.5036e−03

– 1.66 2.22 2.06

5.7164e−01 1.2344e−01 2.8725e−02 7.0890e−03

– 2.21 2.10 2.01

Table 3 The numerical errors and convergence order for different α with ω = 0.02 at T = 1. h 0.4 0.2 0.1 0.05

α = 1.5

τ 0.04 0.02 0.01 0.005

α = 1.8

e(h, τ )

Order

e(h, τ )

Order

5.7791e−01 2.0151e−01 4.1637e−02 9.7891e−03

– 1.52 2.27 2.08

5.5533e−01 1.3671e−01 3.1679e−02 7.7595e−03

– 2.02 2.10 2.02

Without loss of generality, we take α=1.5, 1.6, 1.8, 1.9, 2, and ω=0, 0.01, 0.02, 0.5. When α = 2, ω = 0, the system reduces to the standard nonlinear Schr¨odinger equation with the exact solution given by u(x, t) = sech(x − 4t) exp(i(2x − 3t)).

(4.3)

First, to test the accuracy of the our scheme, let U (x, t) be the numerical solution and u(x, t) be the exact solution. When α = 2, ω = 0, the exact solution u can be calculated precisely, then we compute the l ∞ -norm error ∥u − U ∥∞ at T = 1. When 1 < α < 2 and ω > 0, the exact solution is not given, to obtain the numerical errors, we take ω = 0.01 with α = 1.6, 1.9, and ω = 0.02 with α = 1.5, 1.8, and use the error function defined as follows e(h, τ ) = ∥Uin (h, τ ) − Uin (h/2, τ/2)∥∞ , and the convergence orders can be calculated by order = log2 (e(h, τ )/e(h/2, τ/2)). Table 1 shows the numerical errors and convergence orders for the standard nonlinear Schr¨odinger equation. Tables 2 and 3 show the numerical errors and convergence orders for the DNFS equation with different α and ω. The convergence orders indicate that our scheme is of second order accuracy in both space and time, which verifies the theoretical analysis in Theorem 3.4. Second, we verify the conservative properties of the difference scheme. The relative errors of mass M and energy E are defined as R M n = |(M n − M 0 )/M 0 |,

R E n = |(E n − E 0 )/E 0 |,

where M n and E n denote the mass and energy at t = nτ , respectively. We take h = τ = 0.05, and compute the discrete conservation laws. Figs. 1 and 2 reveal the relative errors of mass M and energy E for different values of fractional order α and damping coefficient ω, which indicate that our scheme can preserve the mass and energy in discrete sense.

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Fig. 1. The relative mass and energy errors with different α when ω = 0.01.

Fig. 2. The relative mass and energy errors with different α when ω = 0.02.

Finally, we take h = τ = 0.05, the evolution of solitons for different values of fractional order α and damping coefficients ω are presented in Figs. 3–5. In Fig. 3, we select ω = 0 with different fractional order α, the results are qualitatively similar to those obtained in Ref. [30]. In order to study the evolution of soliton affected by the fractional order and damping coefficient, we select α = 1.6 and 1.9, ω = 0.01 and 0.5, the results are presented in Figs. 4 and 5, which demonstrate that the fractional order α will affect the shape of the soliton, when α becomes smaller, the shape of the soliton will change more quickly. In addition, the bigger the damping coefficient ω is, the faster the soliton dissipates.

5. Conclusions In this paper, we derive an energy conservation law of the damped nonlinear fractional Schr¨odinger equation. Subsequently, a new conservative finite difference scheme is developed to solve this equation, and we prove the new scheme is unconditionally stable and is convergent with order O(h 2 + τ 2 ) in the discrete maximum norm. Numerical experiments demonstrate that the new scheme has desirable conservation properties, and is efficient for solving the damped nonlinear fractional Schr¨odinger equation.

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221

Fig. 3. Evolution of the solitons with ω = 0 and different fractional order α.

Fig. 4. Evolution of the solitons with ω = 0.01 and different fractional order α.

Acknowledgments This work is supported by the National Natural Science Foundation of China (Grant No. 11771213, 61872422), the National Key Research and Development Project of China (Grant No. 2016YFC0600310, 2018YFC0603500, 2018YFC1504205), the Major Projects of Natural Sciences of University in Jiangsu Province of China (Grant No.

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Fig. 5. Evolution of the solitons with ω = 0.5 and different fractional order α.

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