An Explicit–Implicit Spectral Element Scheme for the Nonlinear Space Fractional Schrödinger Equation

Abstract

In this paper, we solve the space fractional nonlinear Schrödinger equation (SFNSE) by developing an explicit–implicit spectral element scheme, which is formulated based on the Legendre spectral element approximation in space and the Crank–Nicolson leap frog (CNLF) difference discretization in time. Both mass and energy conservative properties are discussed for the spectral element scheme. Numerical stability and convergence of the scheme are proved. Numerical experiments are performed to confirm the high accuracy and efficiency of the proposed numerical scheme.

1. Introduction

Fractional differential equations (FDEs) have played a very important role in the simulation of practical problems in science and engineering. Because of the existing fractional derivatives, it is difficult to solve most FDEs by making use of analytical methods for researchers. So, more and more scholars have begun to pay attention to numerical methods for solving different FDEs, such as (discontinuous) Galerkin finite element methods [1,2,3,4,5], finite difference methods [6,7], meshless methods [8], finite volume methods [9,10] and spectral methods [11,12,13,14]. Based on the literature, scholars can see that the studies of different numerical methods of fractional equations have been much concerned.

The fractional Schrödinger equation is yielded by the classical Schrödinger equation, which plays an important role in quantum mechanics and has many applications in the field, such as in nonlinear optics, plasma physics and fluid dynamics [15,16,17,18,19,20]. Naber [21] first deduced the time fractional Schrödinger equation [22,23], where the first-order time derivative was substituted by the Caputo fractional derivative. Laskin created the spatial fractional Schrödinger equation, an extension of the traditional Schrödinger equation. This is achieved by replacing the second-order spatial derivative with a Riesz fractional derivative [24,25]. Guo [26] proved the existence of a global smooth solution of the fractional nonlinear Schrödinger equation with periodic boundary conditions.

In this paper, we consider the following SFNSE:

$$\begin{array}{c}\hfill \mathrm{i}{\partial }_{t}u(x,t)-{(-\Delta )}^{\frac{\alpha }{2}}u(x,t)+\beta {\left|u(x,t)\right|}^{2}u(x,t)=0,\end{array}$$

(1)

with the initial and Dirichlet boundary conditions given by

$$\begin{array}{ccc}& & u(x,0)=u_0\left(x\right),\hfill \end{array}$$

(2)

$$\begin{array}{ccc}& & {u(x,t)|}_{\partial \mathsf{\Omega }}=0,\hfill \end{array}$$

(3)

where $x\in \mathsf{\Omega }=(a,b),t\in (0,T]$, $\alpha \in (1,2]$ and $\beta$ is a real constant. Suppose that the solution of the system (1)–(3) is small outside of the interval $\mathsf{\Omega }$, which can be neglected, i.e., ${u|}_{x\in R\setminus \mathsf{\Omega }}=0.$

The Riesz space fractional derivative is defined as [27]

$$\begin{array}{c}\hfill -{(-\Delta )}^{\frac{\alpha }{2}}u(x,t)=\frac{{\partial }^{\alpha }}{{\partial \left|x\right|}^{\alpha }}u(x,t)=-\frac{1}{2 \cos \left(\frac{\alpha \pi }{2}\right)}\left[{}_{RL}{D}_{a,x}^{\alpha }u(x,t)+{}_{RL}{D}_{x,b}^{\alpha }u(x,t)\right],\end{array}$$

(4)

where ${}_{RL}{D}_{a,x}^{\alpha }u(x,t)$ and ${}_{RL}{D}_{x,b}^{\alpha }u(x,t)$ are the left and right Riemann–Liouville fractional derivatives defined as follows:

$${}_{RL}{D}_{a,x}^{\alpha }u(x,t)=\left\{\begin{array}{ccc}& \frac{1}{\mathsf{\Gamma }(2-\alpha )}\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}{\int }_a^x{(x-s)}^{1-\alpha }u(s,t)\mathrm{d}s,\hfill & \hfill 1<\alpha <2,\\ & \frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}u(x,t),\hfill & \hfill \alpha =2,\end{array}\right.$$

$${}_{RL}{D}_{x,b}^{\alpha }u(x,t)=\left\{\begin{array}{ccc}& \frac{1}{\mathsf{\Gamma }(2-\alpha )}\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}{\int }_x^b{(s-x)}^{1-\alpha }u(s,t)\mathrm{d}s,\hfill & \hfill 1<\alpha <2,\\ & \frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}u(x,t),\hfill & \hfill \alpha =2,\end{array}\right.$$

Due to the existence of the nonlinear term, it is difficult to obtain the solution of the SFNSE exactly; thus, numerical methods for solving the SFNSE have been studied by many authors, such as finite difference method [6,28,29,30,31,32,33], finite element method [34,35,36,37], spectral method [22,38,39,40] and other methods [41]. Li and Wang [42] and Li et al. [43] utilized this method for solving the integer-order nonlinear Schrödinger equation. However, we have not seen any relevant reports on the spectral element method for solving SFNSE.

As is known to all, the spectral element method was yielded by the combination of the finite element method and the spectral method, which means that it incorporates the higher-order accuracy of the spectral method and the geometric flexibility of the finite element scheme. Recently, the spectral element method has been applied for solving FDEs such as the neutral time delay distributed-order fractional damped diffusion-wave equation [44] and the 2D multiterm time-fractional diffusion-wave equation [45]. In 2018, Mao and Shen developed the spectral element method for solving two-sided space FDEs in 1D [46] with geometric mesh.

According to the references above, one can see that the spectral element method is mainly solving integer DEs, time FDEs and 1D linear space FDEs. It is urgent to develop this method to nonlinear cases. To our knowledge, the spectral element method for solving the SFNSE is still blank. This is why we choose the spectral element method, as it is a starting point for solving the one-dimensional SFNSE, and we will extend this method to solve this type of equation on irregular domains in the future.

This study introduces a fresh numerical method for solving spatial fractional Schrödinger equations aimed at narrowing the gap in this area of research. We use Legendre spectral element approximation in space following the idea proposed by Mao and Shen [46], which considered the linear space FDEs in 1D, as well as CNLF difference discretization in time just like Refs. [33,47], as we know the CNLF scheme can reach the accuracy ${\tau }^2$.

Our major contributions are highlighted as follows:

• We create a fully discrete explicit–implicit CNLF-type spectral element scheme for solving the SFNSE.
• The spectral element scheme satisfies both the mass conservation and energy conservation laws, which is compatible with the property of the SFNSE itself.
• Stability and convergence theorems of the spectral element scheme are rigorously proved; we obtain an optimal error estimation and present numerical experiments to confirm our theoretical result.

The outline is organized as follows. Some preliminaries and notations are shown in Section 2. In Section 3, we build a fully discrete spectral element scheme for the SFNSE, analyze the scheme that satisfies both the mass conservation and energy conservation, and also prove the stability and convergence of the fully discrete scheme. Numerical experiments evaluate the effectiveness and the efficiency of the CNLF-type spectral element scheme in Section 4. Finally, some conclusions are made in Section 5.

2. Preliminaries and Notations

Let $(·,·)$ be the inner product on the space $L^2\left(\mathsf{\Omega }\right)$ with the $L^2$ norm $\Vert ·\Vert$—that is,

$$(u,v)={\int }_{\mathsf{\Omega }}u\overline{v}\mathrm{d}x,\parallel u\parallel ={\left({\int }_{\mathsf{\Omega }}{\left|u\right|}^2\mathrm{d}x\right)}^{1/2}.$$

Let $\mu$ be a non-negative real number; we use $H^{\mu }\left(\mathsf{\Omega }\right)$ and $H_0^{\mu }\left(\mathsf{\Omega }\right)$ as the usual Sobolev spaces with the norm ${\parallel ·\parallel }_{\mu }$ and the seminorm ${|·|}_{\mu }$. c is consistently used to stand for a generic positive constant independent of N.

In the sequel, we list some fractional spaces $J_L^{\mu }\left(\mathsf{\Omega }\right),J_R^{\mu }\left(\mathsf{\Omega }\right)$ and $J_S^{\mu }\left(\mathsf{\Omega }\right)$, which we will use hereafter (see [48,49] for more detail).

Definition 1. 

Let $\mu >0$; we define the seminorm

$${\left|u\right|}_{J_L^{\mu }\left(\mathsf{\Omega }\right)}=\parallel {}_{RL}{D}_{a,x}^{\mu }u\parallel$$

and the norm

$${\parallel u\parallel }_{J_L^{\mu }\left(\mathsf{\Omega }\right)}={\left({\parallel u\parallel }^2+{\left|u\right|}_{J_L^{\mu }\left(\mathsf{\Omega }\right)}\right)}^{\frac{1}{2}}$$

and denote $J_L^{\mu }\left(\mathsf{\Omega }\right)$ $(or{J}_{L,0}^{\mu }\left(\mathsf{\Omega }\right))$ as the closure of $C^{\infty }\left(\mathsf{\Omega }\right)$ $(or{C}_0^{\infty }\left(\mathsf{\Omega }\right))$ with respect to ${\parallel ·\parallel }_{J_L^{\mu }\left(\mathsf{\Omega }\right)}$, where $C_0^{\infty }\left(\mathsf{\Omega }\right)$ is the space of smooth functions with compact support in $\mathsf{\Omega }.$

Definition 2. 

Let $\mu >0$; we define the seminorm

$${\left|u\right|}_{J_R^{\mu }\left(\mathsf{\Omega }\right)}=\parallel {}_{RL}{D}_{x,b}^{\mu }u\parallel$$

and the norm

$${\parallel u\parallel }_{J_R^{\mu }\left(\mathsf{\Omega }\right)}={\left({\parallel u\parallel }^2+{\left|u\right|}_{J_R^{\mu }\left(\mathsf{\Omega }\right)}\right)}^{\frac{1}{2}}$$

and denote $J_R^{\mu }\left(\mathsf{\Omega }\right)$ $(or{J}_{R,0}^{\mu }\left(\mathsf{\Omega }\right))$ as the closure of $C^{\infty }\left(\mathsf{\Omega }\right)$ $(or{C}_0^{\infty }\left(\mathsf{\Omega }\right))$ with respect to ${\parallel ·\parallel }_{J_R^{\mu }\left(\mathsf{\Omega }\right)}$.

Definition 3. 

Let $\mu >0$ and $\mu \ne n-\frac{1}{2},n\in N$; we define the seminorm

$${\left|u\right|}_{J_S^{\mu }\left(\mathsf{\Omega }\right)}=|({}_{RL}{D}_{a,x}^{\mu }u,{}_{RL}{D}_{x,b}^{\mu }u){|}^{\frac{1}{2}}$$

and the norm

$${\parallel u\parallel }_{J_S^{\mu }\left(\mathsf{\Omega }\right)}={\left({\parallel u\parallel }^2+{\left|u\right|}_{J_S^{\mu }\left(\mathsf{\Omega }\right)}\right)}^{\frac{1}{2}}$$

and denote $J_S^{\mu }\left(\mathsf{\Omega }\right)$ $(or{J}_{S,0}^{\mu }\left(\mathsf{\Omega }\right))$ as the closure of $C^{\infty }\left(\mathsf{\Omega }\right)$ $(or{C}_0^{\infty }\left(\mathsf{\Omega }\right))$ with respect to ${\parallel ·\parallel }_{J_S^{\mu }\left(\mathsf{\Omega }\right)}$.

Definition 4. 

For $\mu >0,$ we define the seminorm

$${\left|u\right|}_{H^{\mu }\left(\mathsf{\Omega }\right)}=\parallel {\left|\xi \right|}^{\mu }\widehat{u}\left(\xi \right)\parallel$$

and the norm

$${\parallel u\parallel }_{H^{\mu }\left(\mathsf{\Omega }\right)}={\left({\parallel u\parallel }^2+{\left|u\right|}_{H^{\mu }\left(\mathsf{\Omega }\right)}\right)}^{\frac{1}{2}}$$

where ξ and $\widehat{u}$ stand for the Fourier transform parameter and the Fourier transform of u, respectively. Denote $H^{\mu }\left(\mathsf{\Omega }\right)$ $(or{H}_0^{\mu }\left(\mathsf{\Omega }\right))$ as the closure of $C^{\infty }\left(\mathsf{\Omega }\right)$ $(or{C}_0^{\infty }\left(\mathsf{\Omega }\right))$ with respect to ${\parallel ·\parallel }_{H^{\mu }\left(\mathsf{\Omega }\right)}$.

For the properties of the norms and seminorms defined in the above definitions, we have the following lemma (see [48,49] for more detail).

Lemma 1. 

Suppose $\mu >0$ and $\mu \ne n-\frac{1}{2},n\in N$; then, $J_L^{\mu }\left(\mathsf{\Omega }\right),$ $J_R^{\mu }\left(\mathsf{\Omega }\right)$, $J_S^{\mu }\left(\mathsf{\Omega }\right)$ and $H^{\mu }\left(\mathsf{\Omega }\right)$ are equal with equivalent norms and seminorms. $J_{L,0}^{\mu }\left(\mathsf{\Omega }\right),$ $J_{R,0}^{\mu }\left(\mathsf{\Omega }\right),$ $J_{S,0}^{\mu }\left(\mathsf{\Omega }\right)$ and $H_0^{\mu }\left(\mathsf{\Omega }\right)$ are equal with equivalent norms and seminorms.

The following Gronwall inequality will be used.

Lemma 2 

(see [50]). Assume that $y_1\ge 0$, $h_n,{\phi }_n$ are non-negative sequences and ${\phi }_n$ satisfies

$$\left\{\begin{array}{c}{\phi }_0\le y_0\hfill \\ {\phi }_n\le y_0+\tau {\displaystyle \sum _{j=0}^{n-1}}{h}_j{\phi }_j,n\ge 1.\hfill \end{array}\right.$$

Then it follows that

$${\phi }_n\le y_{0}exp\left(\tau \sum _{j=0}^{n-1}{h}_j\right),n\ge 1.$$

3. The Fully Discrete Scheme

In this section, we mainly give the fully discrete scheme for (1)–(3) and discuss the conservation laws of the scheme. In order to obtain the variational formulation of (1)–(3), we first introduce the following lemma.

Lemma 3 

(See [51]). For $1<\alpha \le 2,$ if $u,v\in J_L^{\alpha }\left(or{J}_R^{\alpha }\right),$ ${u|}_{\partial \mathsf{\Omega }}=0$ and ${v|}_{\partial \mathsf{\Omega }}=0,$ then

$${(}_{RL}{D}_{a,x}^{\alpha }u,v)=\left({}_{RL}{D}_{a,x}^{\frac{\alpha }{2}}u{,}_{RL}{D}_{x,b}^{\frac{\alpha }{2}}v\right),{(}_{RL}{D}_{x,b}^{\alpha }u,v)=\left({}_{RL}{D}_{x,b}^{\frac{\alpha }{2}}u{,}_{RL}{D}_{a,x}^{\frac{\alpha }{2}}v\right).$$

According to (4) and the above lemma, we have

$$\begin{array}{c}\hfill B(u,v):=\left({(-\Delta )}^{\frac{\alpha }{2}}u,v\right)=\frac{1}{2cos\left(\frac{\alpha \pi }{2}\right)}\left[\left({}_{RL}{D}_{a,x}^{\frac{\alpha }{2}}u,{}_{RL}{D}_{x,b}^{\frac{\alpha }{2}}v\right)+\left({}_{RL}{D}_{x,b}^{\frac{\alpha }{2}}u,{}_{RL}{D}_{a,x}^{\frac{\alpha }{2}}v\right)\right].\end{array}$$

(5)

Then, the variational form of (1)–(3) is

$$\begin{array}{c}\hfill \mathrm{i}(u_t,v)-B(u,v)+{\beta \left(\right|u|}^{2}u,v)=0,\forall v\in H_0^{\frac{\alpha }{2}}\left(\mathsf{\Omega }\right).\end{array}$$

(6)

For convenience, we define the following seminorm and norm:

$${\left|u\right|}_{\frac{\alpha }{2}}:=B{(u,u)}^{\frac{1}{2}},{\parallel u\parallel }_{\frac{\alpha }{2}}:={\left({\parallel u\parallel }^2+{\left|u\right|}_{\frac{\alpha }{2}}^2\right)}^{\frac{1}{2}}$$

As is known, the standard spectral Galerkin method is not very suitable for fractional partial differential equations with singularities due to the high requirement of smoothness of the equation. However, the spectral element method is qualified to deal with singularities at endpoints effectively.

Divide the domain $\mathsf{\Omega }$ into $N_x$ non-overlapping uniform elements:

$${\mathsf{\Omega }}_j=(x_{j-1},x_j),j=1,2,\cdots ,N_x,$$

where h is the length of the element, N is a positive integer and ${\mathcal{P}}_N\left(\mathsf{\Omega }\right)$ stands for the space of all polynomials with a degree no greater than N. $V_N$ represents the approximation space, which can be expressed as

$$V_N=\{u\in H^1{\left(\mathsf{\Omega }\right)\left|u\right|}_{{\mathsf{\Omega }}_j}\in {\mathcal{P}}_N\left({\mathsf{\Omega }}_j\right),j=1,2,\cdots ,N_x\},$$

and

$$V_N^0=V_N\cap H_0^1\left(\mathsf{\Omega }\right).$$

For the interior unknowns in each subdomain, we adopt the base functions ${\left\{{\varphi }_j^k\left(x\right)\right\}}_{j=0,1,\cdots ,N-2}^{k=1,2,\cdots ,N_x}$, which are composed of Legendre polynomials ${\left\{L_j^k\right\}}_{j=0,1,\cdots ,N-2}^{k=1,2,\cdots ,N_x}$, in space $V_N^0$ as follows:

$$\begin{array}{c}\hfill {\varphi }_j^k\left(x\right)=\left\{\begin{array}{c}{L}_j^k\left(\widehat{x}\right)-L_{j+2}^k\left(\widehat{x}\right),x=\frac{h\widehat{x}+x_{k-1}+x_k}{2}\in {\mathsf{\Omega }}_k,\hfill \\ 0,else,\hfill \end{array}\right.\end{array}$$

(7)

where $\widehat{x}\in [-1,1]$. For the unknowns at the nodes ${\left\{x_j\right\}}_{j=1,2,\cdots ,N_x-1},$ we define the function $h_k\left(x\right)$:

$$\begin{array}{c}\hfill h_k\left(x\right)=\left\{\begin{array}{cc}\frac{x-x_{k-1}}{h},\hfill & \hfill x\in {\mathsf{\Omega }}_k,\\ \frac{x_{k+1}-x}{h},\hfill & \hfill x\in {\mathsf{\Omega }}_{k+1},\\ 0,\hfill & \hfill else.\end{array}\right.\end{array}$$

(8)

Then, we can write

$$\begin{array}{c}\hfill u_N\left(x\right)=\sum _{k=1}^{N_x}\sum _{j=0}^{N-2}{u}_{kj}^{\left(1\right)}{\varphi }_j^k\left(x\right)+\sum _{k=1}^{N_x-1}{u}_k^{\left(2\right)}{h}_k\left(x\right).\end{array}$$

(9)

Let $\tau$ represent the time step size and M be a positive integer with $\tau =T/M$ and $t_k=k\tau$ for $k=0,1,\cdots ,M$. Denote $u^k=u(·,t_k)$ and

$$u_{\widehat{t}}^k=\frac{u^{k+1}-u^{k-1}}{2\tau },u^{\overline{k}}=\frac{u^{k+1}+u^{k-1}}{2}.$$

Now, we present the CNLF-type spectral method for (1)–(3).

$$\left\{\begin{array}{}\hfill \mathrm{(10)}& \mathrm{i}\left(u_{N\widehat{t},}^k,v_N\right)-B\left(u_N^{\overline{k}},v_N\right)+\beta \left({\left|u_N^k\right|}^2{u}_N^{\overline{k}},v_N\right)=0,\forall v\in V_N^0,\hfill \hfill \mathrm{(11)}& u_N^0=P_N{u}_0,u_N^1=P_N\left(u_0+\tau {\partial }_{t}u\left(0\right)\right),\hfill \end{array}\right.$$

where $P_N$ is an appropriate projection operator and its relevant properties are shown in Section 3.2.

3.1. Mass Conservation and Energy Conservation

The fully discrete scheme (10) satisfies both the mass conservation and energy conservation laws. Now, we consider these two conservation laws carefully.

Lemma 4. 

For the discrete solution $u_N^k$, $k=0,1,\cdots ,M$, we have

$$\mathrm{Re}B(u_N^{\overline{k}},u_{N\widehat{t}}^k)=\frac{1}{4\tau }\left(|u_N^{k+1}{|}_{\frac{\alpha }{2}}^2-{|u_N^{k-1}|}_{\frac{\alpha }{2}}^2\right)$$

Proof. 

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill B(u_N^{\overline{k}},u_{N\widehat{t}}^k)=& \frac{1}{2cos\left(\frac{\alpha }{2}\pi \right)}\left[\left({}_{RL}{D}_{a,x}^{\frac{\alpha }{2}}{u}_N^{\overline{k}},{}_{RL}{D}_{x,b}^{\frac{\alpha }{2}}{u}_{N\widehat{t}}^k\right)+\left({}_{RL}{D}_{x,b}^{\frac{\alpha }{2}}{u}_N^{\overline{k}},{}_{RL}{D}_{a,x}^{\frac{\alpha }{2}}{u}_{N\widehat{t}}^k\right)\right]\hfill \\ \hfill =& \frac{1}{8\tau cos\left(\frac{\alpha }{2}\pi \right)}[\left({}_{RL}{D}_{a,x}^{\frac{\alpha }{2}}{u}_N^{k+1},{}_{RL}{D}_{x,b}^{\frac{\alpha }{2}}{u}_N^{k+1}\right)-\left({}_{RL}{D}_{a,x}^{\frac{\alpha }{2}}{u}_N^{k+1},{}_{RL}{D}_{x,b}^{\frac{\alpha }{2}}{u}_N^{k-1}\right)\hfill \\ & +\left({}_{RL}{D}_{a,x}^{\frac{\alpha }{2}}{u}_N^{k-1},{}_{RL}{D}_{x,b}^{\frac{\alpha }{2}}{u}_N^{k+1}\right)-\left({}_{RL}{D}_{a,x}^{\frac{\alpha }{2}}{u}_N^{k-1},{}_{RL}{D}_{x,b}^{\frac{\alpha }{2}}{u}_N^{k-1}\right)\hfill \\ & +\left({}_{RL}{D}_{x,b}^{\frac{\alpha }{2}}{u}_N^{k+1},{}_{RL}{D}_{a,x}^{\frac{\alpha }{2}}{u}_N^{k+1}\right)-\left({}_{RL}{D}_{x,b}^{\frac{\alpha }{2}}{u}_N^{k+1},{}_{RL}{D}_{a,x}^{\frac{\alpha }{2}}{u}_N^{k-1}\right)\hfill \\ & +\left({}_{RL}{D}_{x,b}^{\frac{\alpha }{2}}{u}_N^{k-1},{}_{RL}{D}_{a,x}^{\frac{\alpha }{2}}{u}_N^{k+1}\right)-\left({}_{RL}{D}_{x,b}^{\frac{\alpha }{2}}{u}_N^{k-1},{}_{RL}{D}_{a,x}^{\frac{\alpha }{2}}{u}_N^{k-1}\right)]\hfill \\ \hfill =& \frac{1}{4\tau }\left(|u_N^{k+1}{|}_{\frac{\alpha }{2}}^2-{\left|u_N^{k-1}\right|}_{\frac{\alpha }{2}}^2\right)+\frac{1}{4\tau }\left(B(u_N^{k-1},u_N^{k+1})-B(u_N^{k+1},u_N^{k-1})\right).\hfill \end{array}\end{array}$$

Taking the real part of the above inequality, we obtain the desired result. □

Theorem 1. 

The fully discrete scheme (10) satisfies the following two conservation laws:

$$\mathit{mass}\mathit{conservation}:{∥u_N^{k+1}∥}^2=\left\{\begin{array}{cc}{∥u_N^0∥}^2,\hfill & k\mathit{is}\mathit{odd},\hfill \\ {∥u_N^1∥}^2,\hfill & k\mathit{is}\mathit{even}\hfill \end{array}\right.$$

and

$$energyconservation:Q^k=Q^0,k=0,1,\cdots ,M,$$

where

$$Q^k=|u_N^{k+1}{|}_{\frac{\alpha }{2}}^2+|u_N^k{|}_{\frac{\alpha }{2}}^2-\beta \left(|u_N^{k+1}{|}^2,{\left|u_N^k\right|}^2\right).$$

Proof. 

Setting $v_N=u_N^{\overline{k}}$ in (10), we obtain

$$\begin{array}{c}\hfill \mathrm{i}\left(u_{N\widehat{t}}^k,u_N^{\overline{k}}\right)-B\left(u_N^{\overline{k}},u_N^{\overline{k}}\right)+\beta \left(|u_N^k{|}^2{u}_N^{\overline{k}},u_N^{\overline{k}}\right)=0.\end{array}$$

(12)

Taking the imaginary part of Equation (12), we have

$$\mathrm{Im}\left\{\mathrm{i}\left(u_{N\widehat{t}}^k,u_N^{\overline{k}}\right)\right\}=\parallel u_N^{k+1}{\parallel }^2-{\parallel u_N^{k-1}\parallel }^2=0.$$

Thus, we obtain the mass conservation law. Now, we prove the energy conservation law. Letting $v_N=u_{N\widehat{t}}^k$, we have

$$\begin{array}{c}\hfill \mathrm{i}\left(u_{N\widehat{t}}^k,u_{N\widehat{t}}^k\right)-B\left(u_N^{\overline{k}},u_{N\widehat{t}}^k\right)+\beta \left(|u_N^k{|}^2{u}_N^{\overline{k}},u_{N\widehat{t}}^k\right)=0.\end{array}$$

(13)

Taking the real part of Equation (13), we obtain

$$|u_N^{k+1}{|}_{\frac{\alpha }{2}}^2-|u_N^{k-1}{|}_{\frac{\alpha }{2}}^2=\beta \left(|u_N^k{|}^2,{\left|u_N^{k+1}\right|}^2\right)-\beta \left(|u_N^k{|}^2,{\left|u_N^{k-1}\right|}^2\right),$$

Thus, we have

$$|u_N^{k+1}{|}_{\frac{\alpha }{2}}^2+|u_N^k{|}_{\frac{\alpha }{2}}^2-\beta \left(|u_N^{k+1}{|}^2,{\left|u_N^k\right|}^2\right)=|u_N^k{|}_{\frac{\alpha }{2}}^2+|u_N^{k-1}{|}_{\frac{\alpha }{2}}^2-\beta \left(|u_N^k{|}^2,{\left|u_N^{k-1}\right|}^2\right).$$

Finally, we obtain the desired result. □

Remark 1. 

Theorem 1 indicates that the fully discrete spectral element scheme (10) is unconditionally stable in the $L^2$ sense.

3.2. Convergence

In this subsection, we prove the convergence for the spectral element scheme (10). Before we prove the convergence theorem, we first introduce the following orthogonal projection operator $P_N$: $H_0^{\frac{\alpha }{2}}\to V_N^0$, which is defined as

$$B(u-P_{N}u,v)=0,\forall v\in V_N^0.$$

(14)

It is obvious that by taking $v=P_{N}u$ and using the properties of B, we have

$$\begin{array}{c}\hfill \parallel P_{N}u{\parallel }_{\frac{\alpha }{2}}\le {\parallel u\parallel }_{\frac{\alpha }{2}}.\end{array}$$

(15)

For the property of the projection $P_N$, we have the following lemma.

Lemma 5 

(see [52]). Let r be a real number satisfying $r>\frac{\alpha }{2}$; if $u\in H_0^{\frac{\alpha }{2}}\left(\mathsf{\Omega }\right)\cap H^r\left(\mathsf{\Omega }\right),$ the following estimate holds:

$$\begin{array}{ccc}& & \parallel u-P_{N}u\parallel \le C{h}^{min\{N+1,r\}}{N}^{-r}{\parallel u\parallel }_r,\alpha \ne \frac{3}{2},\hfill \\ & & \parallel u-P_{N}u\parallel \le C{h}^{min\{N+1,r\}}{N}^{ϵ-r}{\parallel u\parallel }_r,\alpha =\frac{3}{2},0<ϵ<\frac{1}{2},\hfill \end{array}$$

where C is a positive constant that is independent of N.

For the orthogonal projection approximation result of the spectral method, we refer to [39].

Theorem 2. 

Suppose that the exact solution u of the system (1)–(3) satisfies $u\in H^{1,\infty }(0,T;H^r\left(\mathsf{\Omega }\right))$, ${\partial }_t^{2}u\in L^{\infty }(0,T;H^{\alpha }\left(\mathsf{\Omega }\right))$ and ${\partial }_t^{3}u\in L^{\infty }(0,T;L^2\left(\mathsf{\Omega }\right)).$ Then, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \parallel u^k-u_N^k\parallel \le & C({\tau }^2+h^{min\{N+1,r\}}{N}^{-r}),\alpha \ne \frac{3}{2},\hfill \\ \hfill \parallel u^k-u_N^k\parallel \le & C({\tau }^2+h^{min\{N+1,r\}}{N}^{ϵ-r}),\alpha =\frac{3}{2},0<ϵ<\frac{1}{2},\hfill \end{array}\end{array}$$

where C is independent of τ and N.

Proof. 

Denote

$$e^k=u^k-u_N^k=\left(u^k-P_N{u}^k\right)+\left(P_N{u}^k-u_N^k\right)\triangleq {\eta }_N^k+e_N^k.$$

Subtracting (10) from (6) and using the definition of $P_N$, we obtain the error equation as follows:

$$\begin{array}{c}\hfill \mathrm{i}(e_{N\widehat{t}}^k,v)-B(e_N^{\overline{k}},v)=\mathrm{i}\left(P_N{u}_{\widehat{t}}^k-u_t^k,v\right)+B(u^k-u^{\overline{k}},v)+\beta \left(|u_N^k{|}^2{u}_N^{\overline{k}}-{\left|u^k\right|}^2{u}^k,v\right).\end{array}$$

(16)

Letting $v=e_N^{\overline{k}}$ in (16), then taking the imaginary part, we infer that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \parallel e_N^{k+1}{\parallel }^2-{\parallel e_N^{k-1}\parallel }^2=& 2\tau \{\mathrm{Re}\left(P_N{u}_{\widehat{t}}^k-u_t^k,e_N^{\overline{k}}\right)+\mathrm{Im}B(u^k-u^{\overline{k}},e_N^{\overline{k}})\hfill \\ & +\beta \mathrm{Im}\left(|u_N^k{|}^2{u}_N^{\overline{k}}-{\left|u^k\right|}^2{u}^k,e_N^{\overline{k}}\right)\}.\hfill \end{array}\end{array}$$

(17)

Next, we estimate the right term of Equation (17) for the case $\alpha \ne \frac{3}{2}$. According to Hölder’s inequality, Young’s inequality, Lemma 5 and the following equalities estimated by Taylor expansion,

$$u_{\widehat{t}}^k=\frac{u^{k+1}-u^{k-1}}{2\tau }=\frac{1}{2\tau }\left({\int }_{t^k}^{t^{k+1}}{\partial }_{t}u\mathrm{d}t-{\int }_{t^k}^{t^{k-1}}{\partial }_{t}u\mathrm{d}t\right),$$

$$u_{\widehat{t}}^k-u_t=\frac{u^{k+1}-u^{k-1}}{2\tau }-u_t=\frac{1}{4\tau }\left({\int }_{t^k}^{t^{k+1}}{(t-t_{k+1})}^2{\partial }_t^{3}u\mathrm{d}t-{\int }_{t^k}^{t^{k-1}}{(t_{k-1}-t)}^2{\partial }_t^{3}u\mathrm{d}t\right),$$

and

$$u^{\overline{k}}-u^k=\frac{u^{k+1}+u^{k-1}}{2}-u^k=\frac{1}{2}\left({\int }_{t^k}^{t^{k+1}}(t-t_{k+1}){\partial }_t^{2}u\mathrm{d}t+{\int }_{t^k}^{t^{k-1}}(t_{k-1}-t){\partial }_t^{2}u\mathrm{d}t\right),$$

we deduce

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathrm{Re}\left(P_N{u}_{\widehat{t}}^k-u_t^k,e_N^{\overline{k}}\right)=& \mathrm{Re}\left(P_N{u}_{\widehat{t}}^k-u_{\widehat{t}}^k,e_N^{\overline{k}}\right)+\mathrm{Re}(u_{\widehat{t}}^k-u_t^k,e_N^{\overline{k}})\hfill \\ \hfill \le & \parallel P_N{u}_{\widehat{t}}^k-u_{\widehat{t}}^k\parallel \parallel e_N^{\overline{k}}\parallel +\parallel u_{\widehat{t}}^k-u_t^k\parallel \parallel e_N^{\overline{k}}\parallel \hfill \\ \hfill \le & \frac{1}{2}\parallel P_N{u}_{\widehat{t}}^k-u_{\widehat{t}}^k{\parallel }^2+\frac{1}{2}\parallel u_{\widehat{t}}^k-u_t^k{\parallel }^2+{\parallel e_N^{\overline{k}}\parallel }^2\hfill \\ \hfill \le & c{h}^{2min\{N+1,r\}}{N}^{-2r}\parallel {\partial }_t{u\parallel }_{L^{\infty }(0,T;H^r\left(\mathsf{\Omega }\right))}^2+c{\tau }^4\parallel {\partial }_t^3{u\parallel }_{L^{\infty }(0,T;L^2\left(\mathsf{\Omega }\right))}^2+c{\parallel e_N^{\overline{k}}\parallel }^2,\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathrm{Im}B(u^k-u^{\overline{k}},e_N^{\overline{k}})=& \mathrm{Im}\left({(-\Delta )}^{\frac{\alpha }{2}}(u^k-u^{\overline{k}}),e_N^{\overline{k}}\right)\hfill \\ \hfill \le & \parallel {(-\Delta )}^{\frac{\alpha }{2}}{\partial }_t^{2}u\parallel \parallel e_N^{\overline{k}}\parallel \hfill \\ \hfill \le & c{\tau }^4\parallel {(-\Delta )}^{\frac{\alpha }{2}}{\partial }_t^{2}u{\parallel }_{L^{\infty }(0,T;L^2(\mathsf{\Omega }))}^2+c{\parallel e_N^{\overline{k}}\parallel }^2,\hfill \end{array}\end{array}$$

and

$$\begin{array}{c}\hfill \begin{array}{cc}& \mathrm{Im}\left(|u_N^k{|}^2{u}_N^{\overline{k}}-{|u^k|}^2{u}^k,e_N^{\overline{k}}\right)\hfill \\ \hfill =& \mathrm{Im}\left(|u_N^k{|}^2(u_N^{\overline{k}}-u^{\overline{k}})+|u_N^k{|}^2(u^{\overline{k}}-u^k)+u^k(|u_N^k{|}^2-|u^k{|}^2),e_N^{\overline{k}}\right)\hfill \\ \hfill =& \mathrm{Im}\left(-|u_N^k{|}^2{\eta }_N^{\overline{k}}+{|u_N^k|}^2(u^{\overline{k}}-u^k)-u^k\left(u_N^k({\overline{e}}_N^k+{{\overline{\eta }}_N}^k)+{\overline{u}}^k(e_N^k+{\eta }_N^k)\right),e_N^{\overline{k}}\right)\hfill \\ \hfill \le & c{h}^{2min\{N+1,r\}}{N}^{-2r}{\parallel u\parallel }_{L^{\infty }(0,T;H^r\left(\mathsf{\Omega }\right))}^2+c{\tau }^4\parallel {\partial }_t^2{u\parallel }_{L^{\infty }(0,T;L^2\left(\mathsf{\Omega }\right))}^2+c\parallel e_N^k{\parallel }^2+c{\parallel e_N^{\overline{k}}\parallel }^2.\hfill \end{array}\end{array}$$

Substituting the above three estimates into (17), noting that $e_N^0=0$, and summing up k from 1 to $n-1$, we infer that

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \parallel e_N^n{\parallel }^2\le \parallel e_N^1{\parallel }^2+c(h^{2min\{N+1,r\}}{N}^{-2r}+{\tau }^4)+c\tau \sum _{k=0}^n{\parallel e_N^k\parallel }^2.\end{array}\end{array}$$

(18)

For the estimate $\parallel e_N^1\parallel$, by Taloy’s expansion $u^1=u_0+\tau {\partial }_t{u}_0+{\int }_0^{\tau }(\tau -t){\partial }_t^{2}u\mathrm{d}t$, we obtain

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \parallel e_N^1\parallel =\parallel P_N{\int }_0^{\tau }(\tau -t){\partial }_t^{2}u\mathrm{d}t\parallel \le c{\tau }^2{\parallel {\partial }_t^{2}u\parallel }_{L^{\infty }(0,T;H^{\frac{\alpha }{2}}(\mathsf{\Omega }))}.\end{array}\end{array}$$

(19)

With a sufficiently small $\tau$, using Gronwall inequality, we infer that

$$\parallel e_N^n\parallel \le C(h^{min\{N+1,r\}}{N}^{-r}+{\tau }^2).$$

Similar to the procedure above, we obtain the following conclusion when $\alpha =\frac{3}{2}$:

$$\parallel e_N^n\parallel \le C(h^{min\{N+1,r\}}{N}^{ϵ-r}+{\tau }^2),0<ϵ<\frac{1}{2}.$$

By the triangle inequality and Lemma 5, we finally obtain the theoretical result. □

4. Numerical Results

In this section, we carry out numerical experiments by using the spectral element scheme (10) and (11) to verify our theoretical results.

4.1. Numerical Implementation

From Equation (10), we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathrm{i}(u_N^{k+1},v_N)-& \tau B(u_N^{k+1},v_N)+\tau \beta \left(\right|u_N^k{|}^2{u}_N^{k+1},v_N)\hfill \\ & =\mathrm{i}(u_N^{k-1},v_N)+\tau B(u_N^{k-1},v_N)-\tau \beta \left(\right|u_N^k{|}^2{u}_N^{k-1},v_N)\triangleq (f,v_N).\hfill \end{array}\end{array}$$

(20)

Note that it is difficult to calculate the nonlinear terms in a straightforward manner. Thus, we use the Gauss integral formula to calculate the nonlinear terms. Substituting $u_N$ defined by Equation (9) into Equation (20) and letting $v_N$ run through all basis functions of $V_N^0$, we obtain the following system:

$$\left(\mathrm{i}M-\frac{\tau }{2cos(\frac{\alpha \pi }{2})}(S_l^{\frac{\alpha }{2}}+S_r^{\frac{\alpha }{2}})+\tau C\right)U=F,$$

(21)

where

$$U={[u_{10}^{\left(1\right)},u_{11}^{\left(1\right)},\cdots ,u_{1,N-2}^{\left(1\right)};\cdots ;u_{N_{x}0}^{\left(1\right)},u_{N_{x}1}^{\left(1\right)},\cdots ,u_{N_x,N-2}^{\left(1\right)};u_1^{\left(2\right)},u_2^{\left(2\right)},\cdots ,u_{N_x-1}^{\left(2\right)}]}^T,$$

$$F={[\tilde{F},\overline{F}]}^T,$$

$$\tilde{F}=[{\tilde{f}}_{10},{\tilde{f}}_{10},\cdots ,{\tilde{f}}_{1,N-2};\cdots ;{\tilde{f}}_{N_{x}0},{\tilde{f}}_{N_{x}1},\cdots ,{\tilde{f}}_{N_x,N-2}],{\tilde{f}}_{k,j}=(f,{\varphi }_j^k\left(x\right)),$$

$$\overline{F}=[{\overline{f}}_1,{\overline{f}}_2,\cdots ,{\overline{f}}_{N_x-1}],{\overline{f}}_j=(f,h_j\left(x\right)),$$

M is the mass matrix and, by the definition of the basis functions, we obtain

$$M=\left(\begin{array}{ccccccc}{M}_{11}& 0& 0& \cdots & 0& 0& {\overline{M}}_1\\ 0& M_{22}& 0& \cdots & 0& 0& {\overline{M}}_2\\ 0& 0& M_{33}& \cdots & 0& 0& {\overline{M}}_3\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮\\ 0& 0& 0& \cdots & M_{N_x-1,N_x-1}& 0& {\overline{M}}_{N_x-1}\\ 0& 0& 0& \cdots & 0& M_{N_x,N_x}& {\overline{M}}_{N_x}\\ {\tilde{M}}_1& {\tilde{M}}_2& {\tilde{M}}_3& \cdots & {\tilde{M}}_{N_x-1}& {\tilde{M}}_{N_x}& \widehat{M}\end{array}\right),$$

where

$$\begin{array}{c}\hfill \begin{array}{cc}& {\left(M_{pq}\right)}_{ij}=({\varphi }_j^p\left(x\right),{\varphi }_i^q\left(x\right)),1\le q=p\le N_x,0\le i\le N-2,0\le j\le N-2;\hfill \\ & {\left({\tilde{M}}_k\right)}_{ij}=({\varphi }_j^k\left(x\right),h_i\left(x\right)),1\le k\le N_x,1\le i\le N_x-1,0\le j\le N-2;\hfill \\ & {\left({\overline{M}}_k\right)}_{ij}=(h_j\left(x\right),{\varphi }_i^k\left(x\right)),1\le k\le N_x,0\le i\le N-2,1\le j\le N_x-1;\hfill \\ & {\left(\widehat{M}\right)}_{ij}=(h_j\left(x\right),h_i\left(x\right)),1\le i\le N_x-1,1\le j\le N_x-1.\hfill \end{array}\end{array}$$

By the definition of the basis functions, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\left(M_{pq}\right)}_{ij}=& ({\varphi }_j^p\left(x\right),{\varphi }_i^q\left(x\right))={({\varphi }_j^p\left(x\right),{\varphi }_i^p\left(x\right))}_{{\mathsf{\Omega }}_p}\hfill \\ \hfill =& {\int }_{x_{p-1}}^{x_p}{\varphi }_j^p\left(x\right){\varphi }_i^p\left(x\right)\mathrm{d}x\hfill \\ \hfill =& \frac{h}{2}{\int }_{-1}^1(L_j-L_{j+2})(L_i-L_{i+2})\mathrm{d}x.\hfill \end{array}\end{array}$$

According to the orthogonality of Legendre polynomials, we obtain

$$\begin{array}{c}\hfill {\left(M_{pq}\right)}_{ij}=\left\{\begin{array}{cc}\frac{h}{2j+1}+\frac{h}{2j+5},\hfill & \hfill i=j,\\ -\frac{h}{2j+1},\hfill & \hfill i=j-2,\\ -\frac{h}{2j+5},\hfill & \hfill i=j+2,\\ 0,\hfill & \hfill else.\end{array}\right.\end{array}$$

For ${\left({\tilde{M}}_k\right)}_{ij}=({\varphi }_j^k\left(x\right),h_i\left(x\right))$, we calculate it by arguing as follows:

If $k=1$,

$$\begin{array}{cc}\hfill {\left({\tilde{M}}_k\right)}_{ij}=& {\left({\varphi }_j^1\left(x\right),\frac{x-x_0}{h}\right)}_{{\mathsf{\Omega }}_1}={\int }_{x_0}^{x_1}{\varphi }_j^1\left(x\right)\frac{x-x_0}{h}\mathrm{d}x\hfill \\ \hfill =& \frac{h}{4}{\int }_{-1}^1(L_j-L_{j+2})(x+1)\mathrm{d}x.(i=1)\hfill \end{array}$$

If $k=N_x$,

$$\begin{array}{cc}\hfill {\left({\tilde{M}}_k\right)}_{ij}=& {\left({\varphi }_j^{N_x}\left(x\right),\frac{x_{N_x}-x}{h}\right)}_{{\mathsf{\Omega }}_{N_x}}={\int }_{x_{N_x-1}}^{x_{N_x}}{\varphi }_j^{N_x}\left(x\right)\frac{x_{N_x}-x}{h}\mathrm{d}x\hfill \\ \hfill =& \frac{h}{4}{\int }_{-1}^1(L_j-L_{j+2})(1-x)\mathrm{d}x.(i=N_x)\hfill \end{array}$$

If $1<k<N_x$,

$$\begin{array}{cc}\hfill {\left({\tilde{M}}_k\right)}_{ij}=& {\left({\varphi }_j^k\left(x\right),h_i\left(x\right)\right)}_{{\mathsf{\Omega }}_k}=\left\{\begin{array}{cc}{\left({\varphi }_j^k\left(x\right),\frac{x-x_{k-1}}{h}\right)}_{{\mathsf{\Omega }}_k}=\frac{h}{4}{\int }_{-1}^1(L_j-L_{j+2})(x+1)\mathrm{d}x,\hfill & \hfill i=k,\\ {\left({\varphi }_j^k\left(x\right),\frac{x_k-x}{h}\right)}_{{\mathsf{\Omega }}_k}=\frac{h}{4}{\int }_{-1}^1(L_j-L_{j+2})(1-x)\mathrm{d}x,\hfill & \hfill i=k-1,\\ 0,\hfill & \hfill else.\end{array}\right.\hfill \end{array}$$

We can calculate the last integral by the Legendre–Gauss quadrature. By the definition of $\overline{M}$, we also find that $\overline{M}={\tilde{M}}^T$. For the matrix $\widehat{M}$, we obtain

$$\begin{array}{c}\hfill {\left(\widehat{M}\right)}_{ij}=(h_j\left(x\right),h_i\left(x\right))=\left\{\begin{array}{cc}{\int }_{x_{j-1}}^{x_j}{\left(\frac{x-x_{j-1}}{h}\right)}^2\mathrm{d}x+{\int }_{x_j}^{x_{j+1}}{\left(\frac{x_{j+1}-x}{h}\right)}^2\mathrm{d}x=\frac{2}{3}h,\hfill & \hfill i=j,\\ {\int }_{x_j}^{x_{j+1}}\frac{x_{j+1}-x}{h}\frac{x-x_j}{h}\mathrm{d}x=\frac{1}{6}h,\hfill & \hfill i=j+1,\\ {\int }_{x_{j-1}}^{x_j}\frac{x-x_{j-1}}{h}\frac{x_j-x}{h}\mathrm{d}x=\frac{1}{6}h,\hfill & \hfill i=j-1,\\ 0,\hfill & \hfill else.\end{array}\right.\end{array}$$

$S_l^{\frac{\alpha }{2}}$ and $S_r^{\frac{\alpha }{2}}$ are the corresponding left and right fractional Riemann–Liouville stiff matrices, respectively. It is easy to check that

$$S_l^{\frac{\alpha }{2}}={\left(S_r^{\frac{\alpha }{2}}\right)}^T.$$

$S_l^{\frac{\alpha }{2}}$ is a block matrix defined by

$$S_l^{\frac{\alpha }{2}}=\left(\begin{array}{ccccccc}{S}_{11}& 0& 0& \cdots & 0& 0& {\overline{S}}_1\\ S_{21}& S_{22}& 0& \cdots & 0& 0& {\overline{S}}_2\\ S_{31}& S_{32}& S_{33}& 0& \cdots & 0& {\overline{S}}_3\\ ⋮& ⋮& ⋮& \ddots & 0& 0& ⋮\\ S_{N_x-1,1}& S_{N_x-1,2}& S_{N_x-1,3}& \cdots & S_{N_x-1,N_x-1}& 0& {\overline{S}}_{N_x-1}\\ S_{N_x,1}& S_{N_x,2}& S_{N_x,3}& \cdots & S_{N_x,N_x-1}& S_{N_x,N_x}& {\overline{S}}_{N_x}\\ {\tilde{S}}_1& {\tilde{S}}_2& {\tilde{S}}_3& \cdots & {\tilde{S}}_{N_x-1}& {\tilde{S}}_{N_x}& \widehat{S}\end{array}\right),$$

where

$$\begin{array}{c}\hfill \begin{array}{cc}& {\left(S_{pq}\right)}_{ij}=\left({}_{RL}{D}_{a,x}^{\frac{\alpha }{2}}{\varphi }_j^p\left(x\right),{}_{RL}{D}_{x,b}^{\frac{\alpha }{2}}{\varphi }_i^q\left(x\right)\right),1\le q\le p\le N_x,0\le i\le N-2,0\le j\le N-2;\hfill \\ & {\left({\overline{S}}_{pq}\right)}_{ij}=\left({}_{RL}{D}_{a,x}^{\frac{\alpha }{2}}{h}_j\left(x\right),{}_{RL}{D}_{x,b}^{\frac{\alpha }{2}}{\varphi }_i^k\left(x\right)\right),1\le k\le N_x,0\le i\le N-2,1\le j\le N_x-1;\hfill \\ & {\left({\tilde{S}}_k\right)}_{ij}=\left({}_{RL}{D}_{a,x}^{\frac{\alpha }{2}}{\varphi }_j^k\left(x\right),{}_{RL}{D}_{x,b}^{\frac{\alpha }{2}}{h}_i\left(x\right)\right),1\le k\le N_x,1\le i\le N_x-1,0\le j\le N-2;\hfill \\ & {\left(\widehat{S}\right)}_{ij}=\left({}_{RL}{D}_{a,x}^{\frac{\alpha }{2}}{h}_j\left(x\right),{}_{RL}{D}_{x,b}^{\frac{\alpha }{2}}{h}_i\left(x\right)\right),1\le i\le N_x-1,1\le j\le N_x-1.\hfill \end{array}\end{array}$$

are corresponding modal–modal, nodal–nodal, modal–nodal and nodal–nodal block stiff matrices [46]. Comparing the trivial computation of mass matrix M and C, it is non-trivial to compute $S_l^{\frac{\alpha }{2}}$ efficiently because of the non-local properties of the fractional derivative. Following the idea of [46], we can use uniform mesh instead of geometric mesh.

4.2. Numerical Results

In this subsection, to validate the effectiveness of the presented algorithm, we choose the following SFNSE (1):

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\mathrm{i}{\partial }_{t}u(x,t)-{(-\Delta )}^{\frac{\alpha }{2}}u(x,t)+\beta {\left|u(x,t)\right|}^{2}u(x,t)=0,x\in (a,b),t\in (0,T]\hfill \\ u(x,0)=sech\left(x\right)exp\left(2ix\right),\hfill \\ u(a,t)=u(b,t)=0.\hfill \end{array}\right.\end{array}$$

(22)

First, we define the convergence rates in time and space in the $L^2$-norm sense as the following:

$$\mathit{Rate}=\left\{\begin{array}{c}\frac{log\left(∥e\left({\tau }_1,N\right)∥/∥e\left({\tau }_2,N\right)∥\right)}{log\left({\tau }_1/{\tau }_2\right)}\mathit{in}\mathit{time},\hfill \\ \frac{log\left(∥e\left(\tau ,N_1\right)∥/∥e\left(\tau ,N_2\right)∥\right)}{log\left(N_1/N_2\right)}\mathit{in}\mathit{space}.\hfill \end{array}\right.$$

Due to there being no exact solution to problem (22) for $1<\alpha <2$, we approximate the error as error ($\tau$) $=\parallel u(x,t,0.001)-u(x,t,\tau )\parallel .$ We take $(a,b)=(-20,20),\beta =1,T=3,$ $\tau =0.005,N=50$ and $N_x=10$; then, the numerical solutions to problem (22) with different $\alpha$ are depicted in Figure 1. We observe that the shape of the solution will be affected by the order $\alpha$. When $\alpha$ becomes smaller, the shape of the solution will change more quickly; when $\alpha$ tends to 2, the numerical solutions are convergent to the solutions of the usual classical integer one. This property of the fractional Schrödinger equation can be used in physics to modify the shape of the wave without changing the nonlinearity and dispersion effects, readers can refer to [29] for more detail.

We also test the time accuracy and space accuracy with the parameters $(a,b)=(-10,10),\beta =1$, respectively.

Table 1. The errors and order versus $\tau$ of $L^2$-norm for different $\alpha$ when $N=50,N_x=5$.

$1/\mathit{\tau }$	$\mathit{\alpha }=1.1$		$\mathit{\alpha }=1.5$		$\mathit{\alpha }=1.9$
	Error	Rate	Error	Rate	Error	Rate
10	4.75 × 10${}^{-2}$	*	1.88 × 10${}^{-1}$	*	5.45 × 10${}^{-1}$	*
20	1.29 × 10${}^{-2}$	1.88	5.67 × 10${}^{-2}$	1.73	1.89 × 10${}^{-1}$	1.52
40	3.31 × 10${}^{-3}$	1.96	1.50 × 10${}^{-2}$	1.92	5.22 × 10${}^{-2}$	1.86
100	5.29 × 10${}^{-4}$	2.00	2.41 × 10${}^{-3}$	1.99	8.61 × 10${}^{-3}$	1.97

* represents empty.

shows that the errors and order versus $\tau$ of $L^2$-norm with $N=50,N_x=5$ for $\alpha =1.1,1.5,1.9$, respectively. We obtain that the second-order accuracy in time is observed for $L^2$-error.

In Figure 2, we plot the $L^2$-errors versus N with $\tau =0.001$ for $\alpha =1.1,1.5,1.9$, respectively. The picture indicates an exponential convergence rate, which is the so-called spectral accuracy. It is obvious that our numerical results are in agreement with the theoretical analysis.

We also test the mass conservation and energy conservation. Figure 3 shows the evolution of mass and energy. We can see that the three curves of the left picture in Figure 3 overlap with each other, which further illustrates that the mass only depends on the initial values and is unaffected by different $\alpha$; thus, we deduce that the numerical experiments coincide with Theorem 1.

All the numerical results are all consistent with our theoretical analysis.

5. Conclusions

In this paper, for solving the SFNSE, we establish an explicit–implicit discrete spectral element scheme formulated by combining the temporal CNLF scheme with the spectral element algorithm. From the numerical results, one can see that the scheme we consider in this paper satisfies both the mass conservation and energy conservation laws. Further, the stability and convergence are also analyzed. Numerical experiments are shown to illustrate the effectiveness of the considered numerical algorithm. In addition to the current study, in the future, this developed method will be applied to high-dimensional cases with irregular domains and coupled space fractional Schrödinger equations [28].