# A Conservative Crank-Nicolson Fourier Spectral Method for the Space Fractional Schrödinger Equation with Wave Operators

Authors: Lei Zhang; Rui Yang; Li Zhang; Lisha Wang
Journal: Journal of Function Spaces
 (2021)

---

## Abstract

In this paper, the Crank-Nicolson Fourier spectral method is proposed for solving the space fractional Schrödinger equation with wave operators. The equation is treated with the conserved Crank-Nicolson Fourier Galerkin method and the conserved Crank-Nicolson Fourier collocation method, respectively. In addition, the ability of the constructed numerical method to maintain the conservation of mass and energy is studied in detail. Meanwhile, the convergence with spectral accuracy in space and second-order accuracy in time is verified for both Galerkin and collocation approximations. Finally, the numerical experiments verify the properties of the conservative difference scheme and demonstrate the correctness of theoretical results.


---

## Body

## 1. Introduction

The Schrödinger equation is one of the most basic equations in quantum mechanics, which was proposed by Austrian physicist Schrödinger in 1926. The equation can correctly describe the quantum behaviors of wave function, which has made great contributions to the study of quantum mechanics. Since then, the Schrödinger system has attracted a large number of mathematicians and physicists to explore the characteristics of its solution and physical applications. The study of conservative methods for the Schrödinger equation is one of the most popular research fields.Over the past decades, most of the researches on the conservative method of the Schrödinger equation focus on the integer-order Schrödinger equation (e.g., see Refs. [1–7]). As models of science and engineering are needed to be more realistic, the fractional-order Schrödinger equation becomes one of the most important models in the fields of Bose-Einstein condensation, plasma, nonlinear optics, fluid dynamics [8, 9], etc. However, few studies have been investigated on conservative methods for the fractional Schrödinger equation. Besides that, most of the existing fractional-order conservative methods are finite element and finite difference methods [10, 11].From the viewpoint of mathematics, the solution of the Schrödinger system has important geometric structures such as energy conservation and multisymplectic structure. Therefore, these properties should be maintained as much as possible in the construction of numerical methods. In this paper, we consider the following nonlinear fractional Schrödinger equation:(1)ϕtty,t+−Δα/2ϕy,t+iκϕty,t+βϕy,t2ϕy,t=0,y∈a,b,0<t≤T,subject to the boundary condition
(2)ϕa,t=ϕb,t,0<t≤T,and the initial conditions
(3)ϕy,0=ϕ0y,ϕty,0=ϕ1y,y∈a,b,where β and κ are positive real constants, 1<α≤2, and i2=−1. ϕ0y and ϕ1y are given real functions. The fractional Laplacian operator −Δα/2 can be defined as a pseudo-differential operator with the symbol −ξα:
(4)−−Δα/2ux,t≔−F−1ξαu^ξ,t,where F is the Fourier transform and u^ is the Fourier transform of u.The spectral method is a generalization of a standard separation variable method, for which Chebyshev polynomials and Legendre polynomials are generally used as the basic functions of approximate expansions. And the Fourier series is convenient to deal with the periodic boundary conditions. Bridges and Reich [12] first put forward the Hamiltonian system using the Fourier spectrum discrete method in 2001. Based on their theoretical ideas, Chen and Qin [13] in the same year proposed the Fourier pseudo-spectral method for the Hamiltonian partial differential equation and used it to integrate the nonlinear Schrödinger equation with periodic boundary conditions. For more comprehensive work on the different conservative Fourier pseudo-spectral methods, refer to [2, 14–16] and their references.Since the equation is calculated on a finite intervala,b, it is converted into periodic boundary conditions in this paper and studied on Ω=0,2π and I=0,T below. Let
(5)x=2y−a−bπb−a+π,x∈0,2π.Denoteux,t=ϕy,t, u0,t=u2π,t, u0x=ϕ0y, and u1x=ϕ1y. Thus, (1)–(3) become
(6)uttx,t+Mα−Δα/2ux,t+iκutx,t+βux,t2ux,t=0,x∈0,2π,0<t≤T,(7)u0,t=u2π,t,(8)ux,0=u0x,utx,0=u1x,x∈0,2π,whereM=2π/b−a.In fact, the nonlinear fractional Schrödinger equation ((6)–(8)) has two conserved quantities:
(9)Qt=Q0,Et=E0,0≤t≤T,where
(10)Qt=κ2u·,tL22+Imut,u,Et=ut·,tL22+Mα−Δα/4u·,tL22+β2u·,tL44,with
(11)u·,tLpp=∫Ωux,tpdx,p=2,4.The outline of the remainder of this paper is as follows. In Section2, a conserved Crank-Nicolson Fourier Galerkin method and a conserved Crank-Nicolson Fourier collocation method are constructed to discrete time variables and spatial variables. Energy-preserving and mass-preserving properties of the new method are investigated, and the error estimate is derived in Section 3. In Section 4, numerical experiments are presented to illustrate the theoretical results. Finally, the conclusions are given in Section 5.


## 2. Crank-Nicolson Fourier Spectral Method and Conservation Laws

LetCper∞Ω be the set of all complex-valued and 2π-periodic C∞-functions on Ω. Denote ·,· as the inner product on the space Lper2Ω with the L2 norm ·Lper2Ω (abbreviated as ·):
(12)ux,vx=∫Ωuxv¯xdx,uxLper2Ω=ux,ux.Forμ as a nonnegative real number, let HperμΩ be the closure of Cper∞Ω. Note that Hper0Ω=Lper2Ω. For any function ux∈Lper2Ω, the following equations [17] can be developed easily:
(13)ux=∑ω∈ℤu^ωeiωx,where the Fourier coefficients are arranged as
(14)u^ω=u,eiωx=12π∫Ωue−iωxdx.For the Fourier transform of fractional Laplacian−−Δα/2, we have
(15)F−−Δα/2ux,tω=−ωαFux,tω.In order to discretize the equation in the temporal direction, the time step is defined byτ=T/Nt. Denote difference operator
(16)δt2un=un+1−2un+un−1τ2,δtun=un+1−unτ,δt^un=un+1−un−12τ,u~n=un−1+un+12,where n is a positive integer (0≤n≤Nt). Therefore, the Crank-Nicolson method was used to discretize equation (6) in the time axis.
(17)iκδt^un=−δt2un−Mα−Δα/2u~n−β2un−12+un+12u~n+Rn,whereRn=Oτ2.


### 2.1. Crank-Nicolson Fourier Galerkin Method

For positive even numberN, the basis function space can be constructed as
(18)SN=spaneiωx−N2≤ω≤N2−1,where the norm and seminorm of HperαΩ are characterized by
(19)uα≜∑ω=−N/2N/2−11+ω2αu^ω21/2,uα≜∑ω=−N/2N/2−1ω2αu^ω21/2.Let(20)uNt=∑ω=−N/2N/2−1u^ωteiωx.The orthogonal operatorsPN:Lper2Ω→SN are defined as follows:
(21)PNu−u,v=0,∀v∈SN.Lemma 1 [18, 19].
Suppose thatu∈HpersΩ for all 0≤μ≤s; it holds that
(22)u−PNuμ≤CNμ−sus.
Denote(23)uNn=∑ω=−N/2N/2−1u^ωneiωx,n=0,1,⋯,Nt.
The time variables of equation (6) are discretized by the Crank-Nicolson method. And the discrete Fourier Galerkin approximation for equation (6) has a modified scheme as follows:
(24)τκuNn+1−τ2Mαi−Δα/2uNn+1−2iuNn+1,v=τκuNn−1+τ2Mαi−Δα/2uNn−1+2iuNn−1,v−4iuNn,v+βτ2i2uNn+12uNn+1+uNn−12uNn−1+uNn+12uNn−1+uNn−12uNn+1,v,(25)uN0=PNu0,δt^uN0=PNu1,where uNn+1∈SN, ∀v∈SN.


### 2.2. Crank-Nicolson Fourier Collocation Method

For positive even numberN, consider the points xj=2πj/N, j=0,1,⋯,N−1, as collocation nodes. The discrete Fourier coefficients [18] of a function u on 0,2π with respect to the collocation points are the following form:
(26)u^ω=1N∑j=0N−1uxje−iωxj,ω=−N2,⋯,N2−1.Using the inversion formula, we have(27)uxj=∑ω=−N/2N/2−1u^ωeiωxj,j=0,1,⋯,N−1.Define the interpolation operatorIN [18] at the collocation points:
(28)INuxj=uxj,j=0,1,⋯,N−1.According to (27),
(29)INux=∑ω=−N/2N/2−1u^ωeiωx.Lemma 2 [18, 19].
For anyu∈HpersΩ, s≥1, the estimate
(30)u−INu≤CN−susin the sense of the Sobolev norm.Combining Lemma2 and the triangle inequality, Corollary 3 is drawn.Corollary 3.
For anyu∈HpersΩ, s≥1, there exists a constant C independent of u and N, such that
(31)INu≤CN−sus+u.
Using the Fourier collocation method to discrete the spatial variables of the equation, we get the fully discrete scheme for equations (6)–(8) as the following forms:
(32)τκuNn+1xj−τ2Mαi−Δα/2uNn+1xj−2iuNn+1xj=τκuNn−1xj+τ2Mαi−Δα/2uNn−1xj+2iuNn−1xj−4iuNnxj+βτ2i2INuNn+1xj2uNn+1xj+uNn−1xj2uNn−1xj+βτ2i2INuNn+1xj2uNn−1xj+uNn−1xj2uNn+1xj,(33)uN0xj=u0xj,δt^uN0xj=u1xj.
Applying the Fourier transformation to (24), we get the following form:
(34)τκ−τ2Mαiωα−2iu^Nωn+1=τκ+τ2Mαiωα+2iu^Nωn−1−4iu^Nωn+βτ2i2F^Nωn,uN0^ω=u0^ω,δt^uN0^ω=u1^ω,where FNn=uNn+12uNn+1+uNn−12uNn−1+uNn+12uNn−1+uNn−12uNn+1,ω=−N/2,⋯N/2−1.


### 2.3. Theory Analysis of Conservation

Theorem 4.
The Crank-Nicolson Fourier Galerkin method (24) of solving equations (6)–(8) preserves the discrete mass and discrete energy:
(35)Qn=Q0,0≤n≤Nt,En=E0,0≤n≤Nt,where
(36)Qn=ImδtuNn,uNn+κ4uNn+12+uNn2,En=δtuNn2+Mα2−Δα/4uNn+12+−Δα/4uNn2+β4uNn+1L4Ω4+uNnL4Ω4.Proof.
We derive the full discrete Fourier Galerkin method:(37)iκδt^uNn,v=−δt2uNn,v−Mα−Δα/2u~Nn,v−β2uNn−12+uNn+12u~Nn,v.
Letv=u~Nn in equation (37); it holds that
(38)iκδt^uNn,u~Nn=−δt2uNn,u~Nn−Mα−Δα/2u~Nn,u~Nn−β2uNn−12+uNn+12u~Nn,u~Nn.
Taking the imaginary part of equation (38), due to
(39)Imiκδt^uNn,u~Nn=κReδt^uNn,u~Nn=κReuNn+1−uNn−12τ,uNn−1+uNn+12=κ4τuNn+12−uNn−12,Imδt2uNn,u~Nn=12τImδtuNn,uNn−1+uNn+1−ImδtuNn−1,uNn−1+uNn+1=12τ2ImδtuNn,uNn+ImδtuNn,uNn+1−uNn−ImδtuNn,uNn−uNn−1−ImδtuNn−1,uNn+1−uNn−ImδtuNn−1,uNn−uNn−1−2ImδtuNn−1,uNn−1=1τImδtuNn,uNn−ImδtuNn−1,uNn−1,Im−Δα/2u~Nn,u~Nn=0,ImuNn−12+uNn+12u~Nn,u~Nn=0.
Therefore,(40)ImδtuNn,uNn−ImδtuNn−1,uNn−1τ+κ4τuNn+12−uNn−12=0,thus,
(41)ImδtuNn,uNn+κ4uNn+12+uNn2=ImδtuNn−1,uNn−1+κ4uNn2+uNn−12.
The above equality indicates that the method (24) maintains the conservation of the discrete mass. The following items consider the conservation of the discrete energy.
Letv=δt^uNn; according to equation (37), we also get
(42)iκδt^uNn,δt^uNn=−δt2uNn,δt^uNn−Mα−Δα/2u~Nn,δt^uNn−β2uNn−12+uNn+12u~Nn,δt^uNn.
Taking the real part of (42), due to
(43)Reiκδt^uNn,δt^uNn=0,(44)Reδt2uNn,δt^uNn=12τReδtuNn−δtuNn−1,δtuNn+δtuNn−1=δtuNn2−δtuNn−122τ,(45)Re−Δα/2u~Nn,δt^uNn=Re−Δα/2uNn−1+uNn+12,uNn+1−uNn−12τ=14τ−Δα/4uNn+1,−Δα/4uNn+1−−Δα/4uNn−1,−Δα/4uNn−1=−Δα/4uNn+12−−Δα/4uNn−124τ,(46)ReuNn−12+uNn+12u~Nn,δt^uNn=ReuNn−12+uNn+12uNn−1+uNn+12,uNn+1−uNn−12τ=14τuNn+1L4Ω4−uNn−1L4Ω4,therefore, using (43)–(46), we obtain
(47)δtuNn2−δtuNn−122τ+Mα4τ−Δα/4uNn+12−−Δα/4uNn−12+β8τuNn+1L4Ω4−uNn−1L4Ω4=0,thus,
(48)δtuNn2+Mα2−Δα/4uNn+12+−Δα/4uNn2+β4uNn+1L4Ω4+uNnL4Ω4=δtuNn−12+Mα2−Δα/4uNn2+−Δα/4uNn−12+β4uNnL4Ω4+uNn−1L4Ω4.
Based on the above analysis, the method (24) also maintains the conservation of the discrete energy.☐


## 2.1. Crank-Nicolson Fourier Galerkin Method

For positive even numberN, the basis function space can be constructed as
(18)SN=spaneiωx−N2≤ω≤N2−1,where the norm and seminorm of HperαΩ are characterized by
(19)uα≜∑ω=−N/2N/2−11+ω2αu^ω21/2,uα≜∑ω=−N/2N/2−1ω2αu^ω21/2.Let(20)uNt=∑ω=−N/2N/2−1u^ωteiωx.The orthogonal operatorsPN:Lper2Ω→SN are defined as follows:
(21)PNu−u,v=0,∀v∈SN.Lemma 1 [18, 19].
Suppose thatu∈HpersΩ for all 0≤μ≤s; it holds that
(22)u−PNuμ≤CNμ−sus.
Denote(23)uNn=∑ω=−N/2N/2−1u^ωneiωx,n=0,1,⋯,Nt.
The time variables of equation (6) are discretized by the Crank-Nicolson method. And the discrete Fourier Galerkin approximation for equation (6) has a modified scheme as follows:
(24)τκuNn+1−τ2Mαi−Δα/2uNn+1−2iuNn+1,v=τκuNn−1+τ2Mαi−Δα/2uNn−1+2iuNn−1,v−4iuNn,v+βτ2i2uNn+12uNn+1+uNn−12uNn−1+uNn+12uNn−1+uNn−12uNn+1,v,(25)uN0=PNu0,δt^uN0=PNu1,where uNn+1∈SN, ∀v∈SN.


## 2.2. Crank-Nicolson Fourier Collocation Method

For positive even numberN, consider the points xj=2πj/N, j=0,1,⋯,N−1, as collocation nodes. The discrete Fourier coefficients [18] of a function u on 0,2π with respect to the collocation points are the following form:
(26)u^ω=1N∑j=0N−1uxje−iωxj,ω=−N2,⋯,N2−1.Using the inversion formula, we have(27)uxj=∑ω=−N/2N/2−1u^ωeiωxj,j=0,1,⋯,N−1.Define the interpolation operatorIN [18] at the collocation points:
(28)INuxj=uxj,j=0,1,⋯,N−1.According to (27),
(29)INux=∑ω=−N/2N/2−1u^ωeiωx.Lemma 2 [18, 19].
For anyu∈HpersΩ, s≥1, the estimate
(30)u−INu≤CN−susin the sense of the Sobolev norm.Combining Lemma2 and the triangle inequality, Corollary 3 is drawn.Corollary 3.
For anyu∈HpersΩ, s≥1, there exists a constant C independent of u and N, such that
(31)INu≤CN−sus+u.
Using the Fourier collocation method to discrete the spatial variables of the equation, we get the fully discrete scheme for equations (6)–(8) as the following forms:
(32)τκuNn+1xj−τ2Mαi−Δα/2uNn+1xj−2iuNn+1xj=τκuNn−1xj+τ2Mαi−Δα/2uNn−1xj+2iuNn−1xj−4iuNnxj+βτ2i2INuNn+1xj2uNn+1xj+uNn−1xj2uNn−1xj+βτ2i2INuNn+1xj2uNn−1xj+uNn−1xj2uNn+1xj,(33)uN0xj=u0xj,δt^uN0xj=u1xj.
Applying the Fourier transformation to (24), we get the following form:
(34)τκ−τ2Mαiωα−2iu^Nωn+1=τκ+τ2Mαiωα+2iu^Nωn−1−4iu^Nωn+βτ2i2F^Nωn,uN0^ω=u0^ω,δt^uN0^ω=u1^ω,where FNn=uNn+12uNn+1+uNn−12uNn−1+uNn+12uNn−1+uNn−12uNn+1,ω=−N/2,⋯N/2−1.


## 2.3. Theory Analysis of Conservation

Theorem 4.
The Crank-Nicolson Fourier Galerkin method (24) of solving equations (6)–(8) preserves the discrete mass and discrete energy:
(35)Qn=Q0,0≤n≤Nt,En=E0,0≤n≤Nt,where
(36)Qn=ImδtuNn,uNn+κ4uNn+12+uNn2,En=δtuNn2+Mα2−Δα/4uNn+12+−Δα/4uNn2+β4uNn+1L4Ω4+uNnL4Ω4.Proof.
We derive the full discrete Fourier Galerkin method:(37)iκδt^uNn,v=−δt2uNn,v−Mα−Δα/2u~Nn,v−β2uNn−12+uNn+12u~Nn,v.
Letv=u~Nn in equation (37); it holds that
(38)iκδt^uNn,u~Nn=−δt2uNn,u~Nn−Mα−Δα/2u~Nn,u~Nn−β2uNn−12+uNn+12u~Nn,u~Nn.
Taking the imaginary part of equation (38), due to
(39)Imiκδt^uNn,u~Nn=κReδt^uNn,u~Nn=κReuNn+1−uNn−12τ,uNn−1+uNn+12=κ4τuNn+12−uNn−12,Imδt2uNn,u~Nn=12τImδtuNn,uNn−1+uNn+1−ImδtuNn−1,uNn−1+uNn+1=12τ2ImδtuNn,uNn+ImδtuNn,uNn+1−uNn−ImδtuNn,uNn−uNn−1−ImδtuNn−1,uNn+1−uNn−ImδtuNn−1,uNn−uNn−1−2ImδtuNn−1,uNn−1=1τImδtuNn,uNn−ImδtuNn−1,uNn−1,Im−Δα/2u~Nn,u~Nn=0,ImuNn−12+uNn+12u~Nn,u~Nn=0.
Therefore,(40)ImδtuNn,uNn−ImδtuNn−1,uNn−1τ+κ4τuNn+12−uNn−12=0,thus,
(41)ImδtuNn,uNn+κ4uNn+12+uNn2=ImδtuNn−1,uNn−1+κ4uNn2+uNn−12.
The above equality indicates that the method (24) maintains the conservation of the discrete mass. The following items consider the conservation of the discrete energy.
Letv=δt^uNn; according to equation (37), we also get
(42)iκδt^uNn,δt^uNn=−δt2uNn,δt^uNn−Mα−Δα/2u~Nn,δt^uNn−β2uNn−12+uNn+12u~Nn,δt^uNn.
Taking the real part of (42), due to
(43)Reiκδt^uNn,δt^uNn=0,(44)Reδt2uNn,δt^uNn=12τReδtuNn−δtuNn−1,δtuNn+δtuNn−1=δtuNn2−δtuNn−122τ,(45)Re−Δα/2u~Nn,δt^uNn=Re−Δα/2uNn−1+uNn+12,uNn+1−uNn−12τ=14τ−Δα/4uNn+1,−Δα/4uNn+1−−Δα/4uNn−1,−Δα/4uNn−1=−Δα/4uNn+12−−Δα/4uNn−124τ,(46)ReuNn−12+uNn+12u~Nn,δt^uNn=ReuNn−12+uNn+12uNn−1+uNn+12,uNn+1−uNn−12τ=14τuNn+1L4Ω4−uNn−1L4Ω4,therefore, using (43)–(46), we obtain
(47)δtuNn2−δtuNn−122τ+Mα4τ−Δα/4uNn+12−−Δα/4uNn−12+β8τuNn+1L4Ω4−uNn−1L4Ω4=0,thus,
(48)δtuNn2+Mα2−Δα/4uNn+12+−Δα/4uNn2+β4uNn+1L4Ω4+uNnL4Ω4=δtuNn−12+Mα2−Δα/4uNn2+−Δα/4uNn−12+β4uNnL4Ω4+uNn−1L4Ω4.
Based on the above analysis, the method (24) also maintains the conservation of the discrete energy.☐


## 3. Theory Analysis of Convergence

In order to simplify the notation, we always assume thatC is a positive constant in this article, which might be different in every formula.Lemma 5 [20].
For any discrete functionuNn, it holds that
(49)uNn+12−uNn2≤τδtuNn2+12uNn+12+uNn2.Lemma 6.
ForuN∈HperμΩ, there exists a positive constant C, such that
(50)uNn≤C,−Δα/4uNn≤C,δtuNn≤C,n=0,1,⋯,Nt.Proof.
Using Theorem4, it yields
(51)δtuNn2+Mα2−Δα/4uNn+12+−Δα/4uNn2+β4uNn+1L4Ω4+uNnL4Ω4=En=E0,thus,
(52)δtuNn2+Mα2−Δα/4uNn+12+−Δα/4uNn2=E0−β4uNn+1L4Ω4+uNnL4Ω4.
Because ofβ>0, it satisfies
(53)δtuNn2+Mα2−Δα/4uNn+12+−Δα/4uNn2≤C.
Sum the inequalities of Lemma5 from 0 to n yields
(54)1−τ2uNn+12≤1+τ2uN02+τδtuN02+τ∑k=1nδtuNk2+uNk2.
Adding (53) and (54), we can obtain the following items:
(55)1−τ2uNn+12+δtuNn2+Mα2−Δα/4uNn+12+−Δα/4uNn2≤C+τ∑k=1nδtuNk2+uNk2.
Forτ is sufficiently small (τ<1), this implies
(56)12uNn+12+δtuNn2+Mα2−Δα/4uNn+12+−Δα/4uNn2≤C+2τ∑k=1n12uNk2+δtuNk−12+Mα2−Δα/4uNk2+−Δα/4uNk−12.
According to the discrete Gronwall’s inequality, there is(57)12uNn+12+δtuNn2+Mα2−Δα/4uNn+12+−Δα/4uNn2≤C.
Therefore,(58)uNn≤C,−Δα/4uNn2≤C,δtuNn≤C.
☐Theorem 7.
Ifs≥1, assume that u∈C2I;HperαΩ∩HsΩ is the exact solution of (6)–(8), and uNn is the numerical solution of (24). It possesses the following conclusion:
(59)uNn−ux,tn≤Cτ2+N−sus.Proof.
Letu∗=PNu, e=u−uN, ξ=u−u∗, and η=u∗−uN; then, en=ξn+ηn. From triangle inequality and Lemma 1, it yields
(60)en≤ξn+ηn≤CN−sus+ηn.
According to the orthogonality of the projection operatorPN, we get
(61)iκδt^u∗n,v=−δt2un,v−Mα−Δα/2u~n,v−β2un−12+un+12u~n,v+Rn,v.
The authors derive the full discrete Fourier Galerkin method:(62)iκδt^uNn,v=−δt2uNn,v−Mα−Δα/2u~Nn,v−β2uNn−12+uNn+12u~Nn,v.
Subtracting equation (62) from equation (61), due to
(63)δt^u∗n−δt^uNn=u∗n+1−u∗n−12τ−uNn+1−uNn−12τ=ηn+1−ηn−12τ,−δt2un+δt2uNn=−un+1−2un+un−1τ2+uNn+1−2uNn+uNn−1τ2=−en+1−2en+en−1τ2,−Δα/2u~Nn−−Δα/2u~n=−Δα/2uNn−1+uNn+12−−Δα/2un−1+un+12=−−Δα/2un−1−uNn−1+un+1−uNn+12=−−Δα/2en−1+en+12,thus,
(64)iκδt^ηn,v=−δt2en,v−Mα−Δα/2e~n,v−β2un−12+un+12u~n−uNn−12+uNn+12u~Nn,v+Rn,v.
According to the orthogonality of operatorPN, i.e., PNu−u,v=0,∀v∈SN. Therefore,
(65)ej,ηk=ξj+ηj,ηk=ξj,ηk+ηj,ηk=ηj,ηk,j,k=0,1,⋯,Nt.
Letv=δt^ηn in (64), and taking the real part, due to
(66)Reiκδt^ηn,δt^ηn=0,(67)Reδt2en,δt^ηn=Reδt2ηn,δt^ηn=12τReδtηn−δtηn−1,δtηn+δtηn−1=δtηn2−δtηn−122τ,(68)Re−Δα/2e~n,δt^ηn=Re−Δα/2η~n,δt^ηn=Re−Δα/2ηn−1+ηn+12,ηn+1−ηn−12τ=14τ−Δα/4ηn+1,−Δα/4ηn+1−−Δα/4ηn−1,−Δα/4ηn−1=−Δα/4ηn+12−−Δα/4ηn−124τ,therefore, using (66)-(68), this implies
(69)ReRn,δt^ηn=δtηn2−δtηn−122τ+Mα4τ−Δα/4ηn+12−−Δα/4ηn−12+ReGn,δt^ηn,where
(70)Gn=β2un−12+un+12u~n−uNn−12+uNn+12u~Nn=β2un−12+un+12un−1+un+12−β2un−1−en−12+un+1−en+12uNn−1+uNn+12=β2un−12e~n+un+12e~n+en−1u¯Nn−1u~Nn+un−1e¯n−1u~Nn+en+1u¯Nn+1u~Nn+un+1e¯n+1u~Nn≤β2maxun−1,un+1,uNn−1,uNn+122en−1+2en+1+2e~n.
Thus, according to Lemma6, we can get
(71)Gn2≤Cen−12+en+12.
Note Lemma1; it gives that
(72)Gn2=∫ΩGn2dx≤Cen−12+en+12≤Cηn−12+ηn+12+CN−2sus2.
Then,(73)ReGn,δt^ηn≤Gn2+δt^ηn22≤12δt^ηn2+Cηn−12+ηn+12+CN−2sus2,ReRn,δt^ηn≤Rn2+δt^ηn22≤12δt^ηn2+Cτ4.
Thus, (69) becomes
(74)δtηn2−δtηn−122τ+Mα4τ−Δα/4ηn+12−−Δα/4ηn−12≤Cηn−12+ηn+12+δt^ηn2+Cτ4+N−2sus2.
Because of(75)δt^ηn=δtηn+δtηn−12,and from Lemma 5, it gives that
(76)ηn2−ηn−12τ≤δtηn2+12ηn2+ηn−12.
Then, combining (74) and (76) leads to
(77)δtηn2−δtηn−122τ+Mα4τ−Δα/4ηn+12−−Δα/4ηn−12+ηn2−ηn−12τ≤Cδtηn2+δtηn−12+ηn2+ηn+12+ηn−12+Cτ4+N−2sus2.
Summing above inequalities (77) from 1 to n yields
(78)12δtηn2+Mα4−Δα/4ηn+12+−Δα/4ηn2+ηn2≤τC∑i=1n12δtηi2+Mα4−Δα/4ηi+12+−Δα/4ηi2+ηi2+Cτ4+N−2sus2.
Hence, using the discrete Gronwall’s inequality gives(79)12δtηn2+Mα4−Δα/4ηn+12+−Δα/4ηn2+ηn2≤Cτ4+N−2sus2,thus,
(80)ηn2≤Cτ4+N−2sus2.
Substituting (80) into (60) can yield
(81)en≤CN−sus+τ2,which immediately gives conclusion.☐Similar to the proof of Theorem7, we can obtain the following theorem.Theorem 8.
Lets≥1; assume that u∈C2I;HperαΩ∩HsΩ is the exact solution of (6)–(8), and uNn is the numerical solution of (32). It possesses the following conclusion:
(82)uNn−ux,tn≤Cτ2+N−sus.


## 4. Numerical Example

Numerical examples will be proposed in this section to verify the correctness of the theoretical analysis, that is, the convergence of the numerical method and its ability to maintain discrete mass and discrete energy.Example 1.
Consider the nonlinear fractional Schrödinger equation with the wave operator:(83)ϕtt+−Δα/2ϕ+iϕt+ϕy,t2ϕy,t=0,y∈−10,10,t∈0,T,ϕ−10,t=ϕ10,t=0,t∈0,T,ϕy,0=1+iyexp−101−y2,ϕty,0=0,y∈−10,10.
Letτ=0.01, N=128, and T=10. Figures 1 and 2 present the numerical solutions for α=1.4 and α=1.6. We can find that the order of α will affect the shape of the solution.Figure 1
Numerical solutions for equation (83) with α=1.4 when τ=0.01 and N=128.Figure 2
Numerical solutions for equation (83) with α=1.6 when τ=0.01 and N=128.There is no exact solution of (83) known for 1<α<2. Therefore, numerical solution calculated by the method (24) with N=1024 and τ=2−10 is taken as the reference solution. Let Φ be the numerical solution, and calculate the error at t=tn in the sense of the discrete L2 norm:
(84)error=ϕn−Φn.The convergence rates in the direction of time and space are calculated as(85)order=logerrorτ1/errorτ2logτ1/τ2,logerrorN1/errorN2logN1/N2.LetT=1. Tables 1 and 2 show that the numerical method is proven to have spectral accuracy in space and second-order accuracy in time for solving equation (83) with α=1.4 and α=1.6.Table 1
Errors and convergence rates in time forN=512 and T=1.
ατErrorOrder1.42−51.9071e−2—2−64.7789e−31.99662−71.1822e−32.01521.62−53.2228e−2—2−67.8070e−31.99522−71.9320e−32.0148Table 2
Errors and convergence rates in space forτ=2−10 and T=1.
αNErrorOrder1.41281.3922e−2—2568.4405e−30.72205123.9789e−31.08501.61281.9930e−2—2561.2081e−20.72255125.6949e−31.0850Figures3–6 show the ability of the numerical method (24) to maintain the discrete mass and discrete energy when α=1.4 and α=1.6. It can be seen from the figure that the numerical method (24) maintains the discrete mass and discrete energy well.Figure 3
Discrete mass error whenτ=0.01, N=128, and α=1.4.Figure 4
Discrete mass error whenτ=0.01, N=128, and α=1.6.Figure 5
Discrete energy error whenτ=0.01, N=128, and α=1.4.Figure 6
Discrete energy error whenτ=0.01, N=128, and α=1.6.


## 5. Conclusion

For the fractional Schrödinger equation with wave operators, we successfully constructed the effective conservative Crank-Nicolson Fourier spectral method for solving this equation, based on the relative theory of a fractional-order derivative and its property. We give the strict theoretical derivation for the convergence rate of the numerical method, i.e.,Oτ2+N−sus. Finally, numerical examples are introduced to verify the correctness of the theoretical results and the validity of our numerical methods. Both theoretical derivation and numerical experiment verify that the numerical method can keep the energy conservation and mass conservation of the original fractional Schrödinger equation. Both environmental noise and regime switching are important factors [21–29]; we will introduce them in the model ((6)–(8)) in the future.



---

*Source: 5137845-2021-06-09.xml*