Improved High-Order Difference Scheme for the Conservation of Mass and Energy in the Two-Dimensional Spatial Fractional Schrödinger Equation

Abstract

In this paper, our primary objective is to develop a robust and efficient higher-order structure-preserving algorithm for the numerical solution of the two-dimensional nonlinear spatial fractional Schrödinger equation. This equation, which incorporates fractional derivatives, poses significant challenges due to its non-local nature and nonlinearity, making it essential to design numerical methods that not only achieve high accuracy but also preserve the intrinsic physical and mathematical properties of the system. To address these challenges, we employ the scalar auxiliary variable (SAV) method, a powerful technique known for its ability to maintain energy stability and simplify the treatment of nonlinear terms. Combined with the composite Simpson’s formula for numerical integration, which ensures high precision in approximating integrals, and a fourth-order numerical differential formula for discretizing the Riesz derivative, we construct a highly effective finite difference scheme. This scheme is designed to balance computational efficiency with numerical accuracy, making it suitable for long-time simulations. Furthermore, we rigorously analyze the conserving properties of the numerical solution, including mass and energy conservation, which are critical for ensuring the physical relevance and stability of the results.

1. Introduction

In this paper, our main concern is to use efficient numerical methods to deal with the following two-dimensional nonlinear space-fractional Schrödinger equation

$$\begin{array}{c}\hfill \mathrm{i}{\displaystyle \frac{\partial u}{\partial t}}+\mu {\mathcal{L}}_{\alpha ,\beta }u+\rho {\left|u\right|}^{2}u=0,(x,y)\in \mathsf{\Omega },t\in (0,T],\end{array}$$

(1)

with the initial condition

$${\displaystyle u(x,y,0)=u_0(x,y),(x,y)\in \mathsf{\Omega },}$$

where $\mathsf{\Omega }=(a,b)\times (c,d)$, $\mathrm{i}=\sqrt{-1}$, $u(x,y,t)$ is a complex-valued wave function with periodic boundary conditions, and $u_0(x,y)$ is a given smooth function. $\mu$ and $\rho$ are real constants. The fractional operator ${\mathcal{L}}_{\alpha ,\beta }$ is defined by ${\mathcal{L}}_{\alpha ,\beta }:={\partial }_x^{\alpha }+{\partial }_y^{\beta }$ with $1<\alpha ,\beta \le 2$, and ${\partial }_x^{\alpha }$ and ${\partial }_y^{\beta }$ denote the Riesz fractional derivative operators in the x and y direction, respectively, which are defined by

$${\displaystyle {\partial }_x^{\alpha }u=-\frac{1}{2cos\left(\frac{\pi }{2}\alpha \right)\mathsf{\Gamma }(2-\alpha )}\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}{\int }_a^b{|x-s_1|}^{1-\alpha }u\left(s_1,y,t\right)\mathrm{d}{s}_1,1<\alpha \le 2,}$$

and

$${\displaystyle {\partial }_y^{\beta }u=-\frac{1}{2cos\left(\frac{\pi }{2}\beta \right)\mathsf{\Gamma }(2-\beta )}\frac{{\mathrm{d}}^2}{\mathrm{d}{y}^2}{\int }_c^d{|y-s_2|}^{1-\beta }u\left(x,s_2,t\right)\mathrm{d}{s}_2,1<\beta \le 2,}$$

where $\mathsf{\Gamma }(·)$ stands for the gamma function, which is defined as $\mathsf{\Gamma }\left(z\right)={\int }_0^{\infty }{t}^{z-1}{e}^{-t}\mathrm{d}t,$$\Re \left(z\right)>0$.

Here, it is worth mentioning that compared with other fractional operators, the Riesz derivative has the following remarkable characteristics. Firstly, the Riesz derivative exhibits symmetry in space, capturing both forward and backward non-local effects, whereas other fractional operators (such as Riemann–Liouville or Caputo derivatives) typically describe unidirectional non-local behavior. Secondly, the Riesz derivative behaves consistently in isotropic spaces, making it suitable for multi-dimensional problems and preserving the geometric symmetry of physical systems, while some fractional operators may exhibit direction-dependent properties.

As we all know, the most important physical feature of Equation (1) is to maintain the conservation of mass and energy, as follows:

$${\displaystyle \mathcal{M}\left(t\right)=\mathcal{M}\left(0\right),\mathcal{E}\left(t\right)=\mathcal{E}\left(0\right),0\le t\le T,}$$

where

$${\displaystyle \mathcal{M}\left(t\right)={\int }_{\mathsf{\Omega }}{\left|u(x,y,t)\right|}^2\mathrm{d}x\mathrm{d}y,}$$

and

$${\displaystyle \mathcal{E}\left(t\right)={\int }_{\mathsf{\Omega }}-\left[\mu u^{*}(x,y,t){\mathcal{L}}_{\alpha ,\beta }u(x,y,t)+\frac{\rho }{2}{\left|u(x,y,t)\right|}^4\right]\mathrm{d}x\mathrm{d}y,}$$

where $u^{*}(x,y,t)$ denotes the conjugate of the complex-valued function $u(x,y,t)$.

In recent years, efficient numerical algorithms for constructing fractional differential equations have attracted great interest among scholars. Common methods include the finite difference method [1,2,3,4,5], finite element method [6,7], spectral method [8], and other methods [9,10,11,12,13]. In particular, some numerical methods have been developed to solve the nonlinear spatial fractional Schrödinger equation in a one-dimensional case. For example, Wang and Huang constructed a structure-preserving difference scheme with convergence order of $\mathcal{O}\left({\tau }^2+h^2\right)$ in [14]. Then, a new finite difference scheme with the same convergence order as the previous algorithm was proposed by them in [15], and the conservation properties of this scheme were studied in detail, including the conservation of mass and energy. Based on the scalar auxiliary variable method proposed by Shen and Xu [16] and the existing fractional center difference formula, Li et al. [17] gave a a fully discrete structure-preserving scheme with convergence order $\mathcal{O}\left({\tau }^2+h^2\right)$. For other related important and meaningful work, please refer to [18,19,20] and the references therein. Next, let’s review the case of two-dimensional equations. At present, there are actually some effective numerical schemes. For example, Zhao [21] and his collaborators proposed a kind of linearized energy conservation finite difference algorithm. A split-step spectral Galerkin scheme for this equation was proposed by Wang et al. [22], and its convergence was strictly proved. Based on the compact implicit integration factor method, an efficient finite difference scheme was constructed by Zhang et al. [23] and carefully studied.

Up to now, although the numerical methods for the nonlinear spatial fractional Schrödinger equation have made great progress, the convergence order of these methods is still relatively low. Up to now, as far as we know, except for [24], the authors have established two linearly finite difference schemes with second-order accuracy in time and spectral accuracy in space, and almost no further work has focused on the scalar auxiliary variable method for high-dimensional nonlinear space fractional Schrödinger equations.

Therefore, in this paper, our main contributions are as follows: inspired by the the scalar auxiliary variable method, we will establish a higher-order numerical method to solve the two-dimensional nonlinear space fractional Schrödinger equations, whose convergence order can reach $\mathcal{O}\left({\tau }^2+h_x^4+h_y^4\right)$. Furthermore, after a detailed theoretical analysis, it is not difficult to find that the difference scheme we established has the same physical characteristics as the original equation, that is, it maintains both energy conservation and mass conservation.

The outline of this paper is arranged as follows: In Section 2, a high-order implicit conservative difference scheme for a two-dimensional nonlinear space-fractional Schrödinger equation is established. In Section 3, we study the conservation properties of the established numerical algorithm. In Section 4, we present numerical examples to validate the proposed scheme. Finally, we conclude this paper with a comprehensive summary of our theoretical findings and discussions on potential extensions.

2. Construction of the High-Order Numerical Scheme

Let $x_i=i{h}_1,i=0,1,\dots ,M_1$, $y_j=j{h}_2,j=0,1,\dots ,M_2$, and $t_k=k\tau ,k=0,1,\dots ,N$, where $h_1={\displaystyle \frac{b-a}{M_1}},h_2={\displaystyle \frac{d-c}{M_2}},\tau ={\displaystyle \frac{T}{N}}$ are spatial and temporal step sizes, respectively. Denoting any given grid function as

$$u=\left\{u_{ij}^{k}0\le i\le M_1,0\le j\le M_2,0\le k\le N\right\},$$

we introduce the following notations:

$${\delta }_t{u}_{ij}^k={\displaystyle \frac{u_{ij}^{k+1}-u_{ij}^k}{\tau }},f_{ij}^{k+{\displaystyle \frac{1}{2}}}={\displaystyle \frac{1}{2}}\left(f_{ij}^{k+1}+f_{ij}^k\right),$$

$${\delta }_x{u}_{i-{\displaystyle \frac{1}{2}}}^n={\displaystyle \frac{1}{h}}\left(u_i^n-u_{i-1}^n\right),{\delta }_x^2{u}_i^n={\displaystyle \frac{1}{h}}\left({\delta }_x{u}_{i+{\displaystyle \frac{1}{2}}}^n-{\delta }_x{u}_{i-{\displaystyle \frac{1}{2}}}^n\right).$$

For any grid functions $u=\left\{u_{ij}\right\}$ and $v=\left\{v_{ij}\right\}$, we define the discrete inner product and its corresponding norm as

$$\begin{array}{c}{\displaystyle \left(u,v\right)=h_1{h}_2\sum _{i=1}^{M_1-1}\sum _{j=1}^{M_2-1}{u}_{ij}{v}_{ij}^{*},{∥u∥}^2=\left(u,v\right).}\hfill \end{array}$$

First, it is assumed that there is a constant $C_0$ such that ${\int }_{\mathsf{\Omega }}F\left(\left|u\right(x,y,t{|}^2\right)\mathrm{d}x\mathrm{d}y+C_0>0$, where $F^{\prime }\left(u\right)=f\left(u\right)=u$, and we introduce a scalar auxiliary variable

$${\displaystyle r\left(t\right):=\sqrt{{\int }_{\mathsf{\Omega }}F\left({\left|u(x,y,t)\right|}^2\right)\mathrm{d}x\mathrm{d}y+C_0},}$$

which was introduced by Shen et al. [16] to solve gradient flows. Then, it holds that

$${\displaystyle \frac{\mathrm{d}r\left(t\right)}{\mathrm{d}t}=\Re {\int }_{\mathsf{\Omega }}Q\left(u\right)u_t^{*}\mathrm{d}x\mathrm{d}y,}$$

where $\Re \left(z\right)$ means to take the real part of a complex number z, and

$${\displaystyle Q\left(u\right):=\frac{{\left|u(x,y,t)\right|}^{2}u(x,y,t)}{\sqrt{{\int }_{\mathsf{\Omega }}F\left(\left|u\right(x,y,t{|}^2\right)\mathrm{d}x\mathrm{d}y+C_0}}.}$$

At this point, Equation (1) can be written in the following equivalent form

$$\begin{array}{c}\hfill \mathrm{i}{\displaystyle \frac{\partial u}{\partial t}}+\mu {\mathcal{L}}_{\alpha ,\beta }u+\rho r\left(t\right)Q\left(u\right)=0,(x,y)\in \mathsf{\Omega },t\in (0,T],\end{array}$$

(2)

$$\begin{array}{c}\hfill {\displaystyle \frac{\mathrm{d}r\left(t\right)}{\mathrm{d}t}}=\Re {\int }_{\mathsf{\Omega }}Q\left(u\right)u_t^{*}\mathrm{d}x\mathrm{d}y,t\in (0,T].\end{array}$$

(3)

Moreover, by applying analytical methods similar to the one-dimensional equation case in [17], it is not difficult to claim that the new systems (2) and (3) have the same mass and energy as the original Equation (1).

Subsequently, we direct our attention to the construction of a conservative numerical scheme for systems (2) and (3). The discretization of the Riesz fractional derivative requires particular consideration, as its non-local characteristics demand specialized approximation techniques. To establish a high-accuracy discrete operator, we propose a novel generating function characterized by a quartic polynomial structure as follows:

$$G\left(z\right)={\left(\sum _{k=0}^4{\theta }_k{z}^k\right)}^{\gamma },\gamma \in (1,2),$$

where the optimized coefficients ${\left\{{\theta }_k\right\}}_{k=0}^4$ are specifically designed to achieve fourth-order approximation accuracy as follows:

$$\begin{array}{c}{\displaystyle {\theta }_0=\frac{280{\gamma }^3-347{\gamma }^2+144\gamma -20}{120{\gamma }^3},}\hfill \\ {\displaystyle {\theta }_1=-\frac{294{\gamma }^3-517{\gamma }^2+258\gamma -40}{60{\gamma }^3},}\hfill \\ {\displaystyle {\theta }_2=\frac{42{\gamma }^3-95{\gamma }^2+57\gamma -10}{10{\gamma }^3},}\hfill \\ {\displaystyle {\theta }_3=-\frac{122{\gamma }^3-283{\gamma }^2+198\gamma -40}{60{\gamma }^3},}\hfill \\ {\displaystyle {\theta }_4=\frac{48{\gamma }^3-113{\gamma }^2+84\gamma -20}{120{\gamma }^3}.}\hfill \end{array}$$

This polynomial basis selection ensures enhanced approximation properties compared to conventional lower-degree generating functions. Through meticulous Taylor expansion analysis, we derive the fourth-order numerical differentiation formula as follows:

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle {\mathcal{H}}_z^{\gamma }{\partial }_z^{\gamma }u(z,t)={\delta }_z^{\gamma }u(z,t)+\mathcal{O}\left(h_z^4\right),(z,\gamma )=(x,\alpha )\mathrm{or}(y,\beta ),}\hfill \end{array}\end{array}$$

(4)

where the composite operator ${\mathcal{H}}_z^{\gamma }$ and discrete differential operator ${\delta }_z^{\gamma }$ are, respectively, defined as

$$\begin{array}{c}{\displaystyle {\mathcal{H}}_z^{\gamma }u(z,t)=\left(1+\frac{2\gamma +1}{20}{h}_z^2{\delta }_z^2\right)u(z,t),}\hfill \end{array}$$

and

$$\begin{array}{c}{\displaystyle {\delta }_z^{\gamma }u(z,t)=-\frac{1}{2cos\left(\frac{\pi }{2}\gamma \right)}\left({}^L\mathcal{A}_{h_z}^{\gamma }+{}^R\mathcal{A}_{h_z}^{\gamma }\right)u(z,t),}\hfill \end{array}$$

with the left/right discrete fractional operators expressed as follows:

$$\begin{array}{c}{\displaystyle {}^L\mathcal{A}_{h_z}^{\gamma }u(z,t)=\frac{1}{h_z^{\gamma }}\sum _{m=0}^{\left[\frac{z-S_1}{h_z}\right]+1}{\kappa }_m^{\left(\gamma \right)}u\left(z-(m-1)h_z,t\right),{}^R\mathcal{A}_{h_z}^{\gamma }u(z,t)=\frac{1}{h_z^{\gamma }}\sum _{m=0}^{\left[\frac{S_2-z}{h_z}\right]+1}{\kappa }_m^{\left(\gamma \right)}u\left(z+(m-1)h_z,t\right).}\hfill \end{array}$$

Here, the spatial domain partitions satisfy $\left(S_1,S_2\right)=(a,b)$ for $z=x$ and $\left(S_1,S_2\right)=(c,d)$ for $z=y$. The critical coefficients ${\kappa }_m^{\left(\gamma \right)}$ are uniquely determined through the generating function relationship

$$G\left(z\right)=\sum _{m=0}^{\infty }{\kappa }_m^{\left(\gamma \right)}{z}^m,\left|z\right|<1.$$

Hence, following (4), we have

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle {\mathcal{H}}_x^{\alpha }{\mathcal{H}}_y^{\beta }{\mathcal{L}}_{\alpha ,\beta }u(x,y,t)={\mathcal{H}}_y^{\beta }{\delta }_x^{\alpha }u(x,y,t)+{\mathcal{H}}_x^{\alpha }{\delta }_y^{\beta }u(x,y,t)+\mathcal{O}\left(h_x^4+h_y^4\right).}\hfill \end{array}\end{array}$$

(5)

Considering the systems (2) and (3) at the grid point $\left(x_i,y_j,t_{k+{\displaystyle \frac{1}{2}}}\right)$, for integral ${\int }_{\mathsf{\Omega }}F\left(\left|u\right(x,y,t{|}^2\right)\mathrm{d}x\mathrm{d}y$ and following the composite Simpson formula, we obtain

$$\begin{array}{c}\hfill {\int }_{\mathsf{\Omega }}F\left({\left|u\left(x_i,y_j,t_{k+{\displaystyle \frac{1}{2}}}\right)\right|}^2\right)\mathrm{d}x\mathrm{d}y={\displaystyle \frac{h_x{h}_y}{36}}\sum _{i=0}^{M_1-1}\sum _{j=0}^{M_2-1}S\left(x_i,y_j,t_{k+{\displaystyle \frac{1}{2}}}\right)+\mathcal{O}\left(h_x^4+h_y^4\right),\end{array}$$

where

$$\begin{array}{cc}\hfill S\left(x_i,y_j,t_{k+{\displaystyle \frac{1}{2}}}\right)=& F\left({\left|u\left(x_i,y_j,t_{k+{\displaystyle \frac{1}{2}}}\right)\right|}^2\right)+F\left({\left|u\left(x_{i+1},y_j,t_{k+{\displaystyle \frac{1}{2}}}\right)\right|}^2\right)+F\left({\left|u\left(x_i,y_{j+1},t_{k+{\displaystyle \frac{1}{2}}}\right)\right|}^2\right)\hfill \\ \hfill & +F\left({\left|u\left(x_{i+1},y_{j+1},t_{k+{\displaystyle \frac{1}{2}}}\right)\right|}^2\right)+4\left[F\left({\left|u\left(x_{i+{\displaystyle \frac{1}{2}}},y_j,t_{k+{\displaystyle \frac{1}{2}}}\right)\right|}^2\right)+F\left({\left|u\left(x_i,y_{j+{\displaystyle \frac{1}{2}}},t_{k+{\displaystyle \frac{1}{2}}}\right)\right|}^2\right)\right.\hfill \\ \hfill & \left.+F\left({\left|u\left(x_{i+1},y_{j+{\displaystyle \frac{1}{2}}},t_{k+{\displaystyle \frac{1}{2}}}\right)\right|}^2\right)+F\left({\left|u\left(x_{i+{\displaystyle \frac{1}{2}}},y_{j+1},t_{k+{\displaystyle \frac{1}{2}}}\right)\right|}^2\right)\right]+16F\left({\left|u\left(x_{i+{\displaystyle \frac{1}{2}}},y_{j+{\displaystyle \frac{1}{2}}},t_{k+{\displaystyle \frac{1}{2}}}\right)\right|}^2\right).\hfill \end{array}$$

By using (5) for spatial discretization and applying the Crank–Nicolson method in time, we obtain

$$\begin{array}{cc}\hfill & \mathrm{i}{\mathcal{H}}_x^{\alpha }{\mathcal{H}}_y^{\beta }{\delta }_{t}u\left(x_i,y_j,t_{k+{\displaystyle \frac{1}{2}}}\right)+\mu \left({\mathcal{H}}_y^{\beta }{\delta }_x^{\alpha }+{\mathcal{H}}_x^{\alpha }{\delta }_y^{\beta }\right)u\left(x_i,y_j,t_{k+{\displaystyle \frac{1}{2}}}\right)+\hfill \\ \hfill & \rho {\mathcal{H}}_x^{\alpha }{\mathcal{H}}_y^{\beta }r\left(t_{k+{\displaystyle \frac{1}{2}}}\right)Q\left(u\left(x_i,y_j,t_{k+{\displaystyle \frac{1}{2}}}\right)\right)=\mathcal{O}\left({\tau }^2+h_x^4+h_y^4\right),\hfill \end{array}$$

(6)

$$\begin{array}{c}\hfill {\delta }_{t}r\left(t_{k+{\displaystyle \frac{1}{2}}}\right)=\Re \left(Q\left(u\left(x_i,y_j,t_{k+{\displaystyle \frac{1}{2}}}\right)\right),{\delta }_{t}u\left(x_i,y_j,t_{k+{\displaystyle \frac{1}{2}}}\right)\right)+\mathcal{O}\left({\tau }^2+h_x^4+h_y^4\right),\end{array}$$

(7)

where

$$\begin{array}{c}\hfill r\left(t_{k+{\displaystyle \frac{1}{2}}}\right)=\sqrt{{\displaystyle \frac{h_x{h}_y}{36}}\sum _{i=0}^{M_1-1}\sum _{j=0}^{M_2-1}S\left(x_i,y_j,t_{k+{\displaystyle \frac{1}{2}}}\right)+C_0},\end{array}$$

and

$$\begin{array}{c}\hfill Q\left(u\left(x_i,y_j,t_{k+{\displaystyle \frac{1}{2}}}\right)\right)={\displaystyle \frac{{\left|u\left(x_i,y_j,t_{k+\frac{1}{2}}\right)\right|}^{2}u\left(x_i,y_j,t_{k+\frac{1}{2}}\right)}{\sqrt{\frac{h_x{h}_y}{36}{\sum }_{i=0}^{M_1-1}{\sum }_{j=0}^{M_2-1}S\left(x_i,y_j,t_{k+\frac{1}{2}}\right)+C_0}}}.\end{array}$$

Omitting the high-order terms in (6) and (7) and replacing the exact solutions $u\left(x_i,y_j,t_{k+{\displaystyle \frac{1}{2}}}\right)$, $Q\left(u\left(x_i,y_j,t_{k+{\displaystyle \frac{1}{2}}}\right)\right)$, and $r\left(t_{k+{\displaystyle \frac{1}{2}}}\right)$ with their numerical solutions $u_{ij}^{k+{\displaystyle \frac{1}{2}}}$, $Q\left(u_{ij}^{k+{\displaystyle \frac{1}{2}}}\right)$, and $r^{k+{\displaystyle \frac{1}{2}}}$, we arrive at

$$\begin{array}{c}\hfill \mathrm{i}{\mathcal{H}}_x^{\alpha }{\mathcal{H}}_y^{\beta }{\delta }_t{u}_{ij}^{k+{\displaystyle \frac{1}{2}}}+\mu \left({\mathcal{H}}_y^{\beta }{\delta }_x^{\alpha }+{\mathcal{H}}_x^{\alpha }{\delta }_y^{\beta }\right)u_{ij}^{k+{\displaystyle \frac{1}{2}}}+\rho {\mathcal{H}}_x^{\alpha }{\mathcal{H}}_y^{\beta }{r}^{k+{\displaystyle \frac{1}{2}}}Q\left(u_{ij}^{k+{\displaystyle \frac{1}{2}}}\right)=0,\end{array}$$

(8)

$$\begin{array}{c}\hfill {\delta }_t{r}^{k+{\displaystyle \frac{1}{2}}}=\Re \left(Q\left(u^{k+{\displaystyle \frac{1}{2}}}\right),{\delta }_t{u}^{k+{\displaystyle \frac{1}{2}}}\right).\end{array}$$

(9)

$${\displaystyle u_{ij}^0=u_0(x_i,y_j),(x_i,y_j)\in \mathsf{\Omega }.}$$

3. Theoretical Analysis of the Constructed Difference Scheme

In this section, we explore the conservation properties of the schemes (8) and (9). First, we establish several lemmas that will be instrumental in deriving the main results and supporting the analysis that follows.

Lemma 1. 

Let ${\mathbf{C}}_p(wherep=1,2)$ represent a symmetric tridiagonal matrix with dimensions $(M_p-1)\times (M_p-1)$, denoted as

$${\mathbf{C}}_p={\left(\begin{array}{ccccc}{\displaystyle \frac{2}{3}}& {\displaystyle \frac{1}{6}}& 0& \cdots & 0\\ {\displaystyle \frac{1}{6}}& {\displaystyle \frac{2}{3}}& {\displaystyle \frac{1}{6}}& 0& \cdots \\ ⋮& ⋮& \ddots & \ddots & \ddots \\ 0& \dots & {\displaystyle \frac{1}{6}}& {\displaystyle \frac{2}{3}}& {\displaystyle \frac{1}{6}}\\ 0& 0& \dots & {\displaystyle \frac{1}{6}}& {\displaystyle \frac{2}{3}}\end{array}\right)}_{(M_p-1)\times (M_p-1)},$$

then, ${\mathbf{C}}_p$ is positive definite.

Proof. 

It is straightforward to determine that the eigenvalues of matrix ${\mathbf{C}}_p$ are

$$\begin{array}{c}{\displaystyle {\lambda }_j\left({\mathbf{C}}_p\right)=\frac{2}{3}+\frac{1}{3}cos\left(\frac{j\pi }{M_p}\right)=\frac{1}{3}+\frac{2}{3}{cos}^2\left(\frac{j\pi }{2M_p}\right)>0,j=1,2,\dots ,M_p-1.}\hfill \end{array}$$

As a result, we can confirm that matrix ${\mathbf{C}}_p$ is positive definite. The proof is hereby completed. □

Lemma 2. 

Denote

$${\mathbf{G}}_p^{\left(\gamma \right)}={\displaystyle \frac{1}{h_z^{\gamma }}}{\left(\begin{array}{ccccc}{\kappa }_1^{\left(\gamma \right)}& {\kappa }_0^{\left(\gamma \right)}& 0& \cdots & 0\\ {\kappa }_2^{\left(\gamma \right)}& {\kappa }_1^{\left(\gamma \right)}& {\kappa }_0^{\left(\gamma \right)}& 0& \cdots \\ ⋮& ⋮& \ddots & \ddots & \ddots \\ {\kappa }_{M_p-2}^{\left(\gamma \right)}& \dots & {\kappa }_2^{\left(\gamma \right)}& {\kappa }_1^{\left(\gamma \right)}& {\kappa }_0^{\left(\gamma \right)}\\ {\kappa }_{M_p-1}^{\left(\gamma \right)}& {\kappa }_{M_p-2}^{\left(\gamma \right)}& \dots & {\kappa }_2^{\left(\gamma \right)}& {\kappa }_1^{\left(\gamma \right)}\end{array}\right)}_{(M_p-1)\times (M_p-1)},$$

which is an associate matrix of fractional difference quotient operator ${}^L\mathcal{A}_{h_z}^{\gamma }$. Then, the matrix

$$\begin{array}{c}{\displaystyle {\mathbf{D}}_p^{\left(\gamma \right)}=-\frac{1}{2cos\left(\frac{\pi }{2}\gamma \right)}\left({\mathbf{G}}_p^{\left(\gamma \right)}+{{\mathbf{G}}_p^{\left(\gamma \right)}}^T\right),1<\gamma <2,}\hfill \end{array}$$

is the associate matrix of fractional difference quotient operator ${\delta }_z^{\gamma }$ and it is negative semi-definite.

Lemma 3. 

For any two grid functions u and v, there is a linear difference operator, denoted by ${\delta }^{\alpha ,\beta }$, such that

$$\begin{array}{c}\hfill \left({\mathcal{H}}_y^{\beta }{\delta }_x^{\alpha }+{\mathcal{H}}_x^{\alpha }{\delta }_y^{\beta }u,v\right)=-\left({\delta }^{\alpha ,\beta }u,{\delta }^{\alpha ,\beta }v\right).\end{array}$$

Proof. 

Denote

$$\begin{array}{c}u={(u_{1,1},u_{2,1},\dots ,u_{M_1-1,1},\dots ,u_{1,M_2-1},u_{2,M_2-1},\dots ,u_{M_1-1,M_2-1})}^T\hfill \end{array}$$

and

$$\begin{array}{c}v={(v_{1,1},v_{2,1},\dots ,v_{M_1-1,1},\dots ,v_{1,M_2-1},v_{2,M_2-1},\dots ,v_{M_1-1,M_2-1})}^T.\hfill \end{array}$$

Then, we know that the following holds

$$\begin{array}{c}{\displaystyle \left({\mathcal{H}}_y^{\beta }{\delta }_x^{\alpha }+{\mathcal{H}}_x^{\alpha }{\delta }_y^{\beta }u,v\right)=h_1{h}_2{v}^T\mathbf{H}u,}\hfill \end{array}$$

where

$$\begin{array}{c}{\displaystyle \mathbf{H}=\left({\mathbf{C}}_2\otimes {\mathbf{I}}_1\right)\left({\mathbf{I}}_2\otimes {\mathbf{D}}_1^{\left(\alpha \right)}\right)+\left({\mathbf{I}}_2\otimes {\mathbf{C}}_1\right)\left({\mathbf{D}}_2^{\left(\beta \right)}\otimes {\mathbf{I}}_1\right).}\hfill \end{array}$$

Here, ${\mathbf{I}}_p$ represents the unit matrices of size $M_p-1$ $(forp=1,2)$, and the symbol ⊗ refers to the Kronecker product of any two matrices. Consequently, using the properties of the Kronecker product [25], we derive

$$\begin{array}{cc}\hfill {\mathbf{H}}^T=& {\left({\mathbf{I}}_2\otimes {\mathbf{D}}_1^{\left(\alpha \right)}\right)}^T{\left({\mathbf{C}}_2\otimes {\mathbf{I}}_1\right)}^T+{\left({\mathbf{D}}_2^{\left(\beta \right)}\otimes {\mathbf{I}}_1\right)}^T{\left({\mathbf{I}}_2\otimes {\mathbf{C}}_1\right)}^T\hfill \\ \hfill =& \left({\mathbf{I}}_2\otimes {\mathbf{D}}_1^{\left(\alpha \right)}\right)\left({\mathbf{C}}_2\otimes {\mathbf{I}}_1\right)+\left({\mathbf{D}}_2^{\left(\beta \right)}\otimes {\mathbf{I}}_1\right)\left({\mathbf{I}}_2\otimes {\mathbf{C}}_1\right)\hfill \\ \hfill =& \left({\mathbf{C}}_2\otimes {\mathbf{I}}_1\right)\left({\mathbf{I}}_2\otimes {\mathbf{D}}_1\right)+\left({\mathbf{I}}_2\otimes {\mathbf{C}}_1\right)\left({\mathbf{D}}_2\otimes {\mathbf{I}}_1\right)=\mathbf{H}.\hfill \end{array}$$

This means that the matrix $\mathbf{H}$ is real symmetric. Moreover, using Lemmas 1 and 2, we can further determine that $\mathbf{H}$ is semi-negative definite because all its eigenvalues are nonpositive. Consequently, there exist an orthogonal matrix $\mathbf{L}$ and a diagonal matrix $\mathbf{\Lambda }$ such that

$$\begin{array}{c}{\displaystyle \mathbf{H}=-{\mathbf{L}}^T\mathbf{\Lambda }\mathbf{L}=-{\left({\mathbf{\Lambda }}^{\frac{1}{2}}\mathbf{L}\right)}^T\left({\mathbf{\Lambda }}^{\frac{1}{2}}\mathbf{L}\right).}\hfill \end{array}$$

Therefore, we have

$$\begin{array}{c}\left({\mathcal{H}}_y^{\beta }{\delta }_x^{\alpha }+{\mathcal{H}}_x^{\alpha }{\delta }_y^{\beta }u,v\right)=h_1{h}_2{v}^T\mathbf{H}u=-h_1{h}_2{\left({\mathbf{\Lambda }}^{{\displaystyle \frac{1}{2}}}\mathbf{L}v\right)}^T{\mathbf{\Lambda }}^{{\displaystyle \frac{1}{2}}}\mathbf{L}u=-\left({\delta }^{\alpha ,\beta }u,{\delta }^{\alpha ,\beta }v\right),\hfill \end{array}$$

where ${\mathbf{\Lambda }}^{{\displaystyle \frac{1}{2}}}\mathbf{L}$ is the matrix associated with the fractional difference quotient operator ${\delta }^{\alpha ,\beta }$, thus concluding the proof. □

Lemma 4. 

For any two grid function u and v, there exists a linear difference operator denoted by ${\mathcal{H}}^{\alpha ,\beta }$ such that

$$\begin{array}{c}\hfill \left({\mathcal{H}}_x^{\alpha }{\mathcal{H}}_y^{\beta }u,v\right)=\left({\mathcal{H}}^{\alpha ,\beta }u,{\mathcal{H}}^{\alpha ,\beta }v\right).\end{array}$$

Proof. 

Using the same method as Lemma 3, we can easily obtain the results. Therefore, the detailed proof process is omitted. □

Below, we give the main results as follows:

Theorem 1. 

Suppose $u_{i,j}^k$ is the solution of the finite difference schemes (8) and (9). Then, these schemes are mass-conserved, namely,

$$\begin{array}{c}\hfill {\mathcal{M}}_{\tau }\left(u^{k+1}\right)={\mathcal{M}}_{\tau }\left(u^k\right),k=0,1,\dots ,N-1,\end{array}$$

(10)

where

$$\begin{array}{c}\hfill {\mathcal{M}}_{\tau }\left(u^k\right)=∥{\mathcal{H}}^{\alpha ,\beta }{u}^k∥\end{array}$$

is the discrete mass of the numerical solution at $t_k$.

Proof. 

Taking the discrete inner product of Equation (8) with $u^{k+{\displaystyle \frac{1}{2}}}$, one has

$$\begin{array}{c}\hfill \mathrm{i}\left({\mathcal{H}}_x^{\alpha }{\mathcal{H}}_y^{\beta }{\delta }_t{u}_{ij}^{k+{\displaystyle \frac{1}{2}}},u^{k+{\displaystyle \frac{1}{2}}}\right)+\mu \left(\left({\mathcal{H}}_y^{\beta }{\delta }_x^{\alpha }+{\mathcal{H}}_x^{\alpha }{\delta }_y^{\beta }\right)u^{k+{\displaystyle \frac{1}{2}}},u^{k+{\displaystyle \frac{1}{2}}}\right)+\rho \left({\mathcal{H}}_x^{\alpha }{\mathcal{H}}_y^{\beta }{r}^{k+{\displaystyle \frac{1}{2}}}Q\left(u^{k+{\displaystyle \frac{1}{2}}}\right),u^{k+{\displaystyle \frac{1}{2}}}\right)=0.\end{array}$$

(11)

Using Lemmas 3 and 4 and taking the imaginary part of (11) leads to

$$\begin{array}{c}\hfill ∥{\mathcal{H}}^{\alpha ,\beta }{u}^{k+1}∥=∥{\mathcal{H}}^{\alpha ,\beta }{u}^k∥,\end{array}$$

which implies that the (10) holds, and the proof is completed. □

Theorem 2. 

Suppose $u_{i,j}^k$ is the solution of the finite difference schemes (8) and (9). Then, these schemes are energy-conserved, that is

$$\begin{array}{c}\hfill {\mathcal{E}}_{\tau }\left(u^{k+1}\right)={\mathcal{E}}_{\tau }\left(u^k\right),k=0,1,\dots ,N-1,\end{array}$$

(12)

where

$$\begin{array}{c}\hfill {\mathcal{E}}_{\tau }\left(u^k\right)=\mu {∥{\delta }^{\alpha ,\beta }{u}^k∥}^2-\rho {\mathcal{H}}_x^{\alpha }{\mathcal{H}}_y^{\beta }{\left|r^k\right|}^2,\end{array}$$

is the discrete energy of the numerical solution at $t_k$.

Proof. 

Taking the discrete inner product of Equation (8) with ${\delta }_t{u}^{k+{\displaystyle \frac{1}{2}}}$ and following Lemmas 3 and 4 yields

$$\begin{array}{c}\hfill \mathrm{i}∥{\mathcal{H}}^{\alpha ,\beta }{\delta }_t{u}_{ij}^{k+{\displaystyle \frac{1}{2}}}∥-\mu \left({\delta }^{\alpha ,\beta }{u}^{k+{\displaystyle \frac{1}{2}}},{\delta }^{\alpha ,\beta }{\delta }_t{u}^{k+{\displaystyle \frac{1}{2}}}\right)+\rho \left({\mathcal{H}}_x^{\alpha }{\mathcal{H}}_y^{\beta }{r}^{k+{\displaystyle \frac{1}{2}}}Q\left(u^{k+{\displaystyle \frac{1}{2}}}\right),{\delta }_t{u}^{k+{\displaystyle \frac{1}{2}}}\right)=0.\end{array}$$

(13)

Taking the real part of (13); then, one has

$$\begin{array}{c}\hfill {\displaystyle \frac{\mu }{2\tau }}\left[{∥{\delta }^{\alpha ,\beta }{u}^{k+1}∥}^2-{∥{\delta }^{\alpha ,\beta }{u}^k∥}^2\right]=\rho r^{k+{\displaystyle \frac{1}{2}}}\Re \left\{\left({\mathcal{H}}_x^{\alpha }{\mathcal{H}}_y^{\beta }Q\left(u^{k+{\displaystyle \frac{1}{2}}}\right),{\delta }_t{u}^{k+{\displaystyle \frac{1}{2}}}\right)\right\}.\end{array}$$

(14)

Next, if we multiply both sides of Equation (9) by ${\mathcal{H}}_x^{\alpha }{\mathcal{H}}_y^{\beta }{r}^{k+{\displaystyle \frac{1}{2}}}$, we obtain

$$\begin{array}{c}\hfill {\mathcal{H}}_x^{\alpha }{\mathcal{H}}_y^{\beta }{r}^{k+{\displaystyle \frac{1}{2}}}{\delta }_t{r}^{k+{\displaystyle \frac{1}{2}}}=r^{k+{\displaystyle \frac{1}{2}}}\Re \left\{\left({\mathcal{H}}_x^{\alpha }{\mathcal{H}}_y^{\beta }Q\left(u_{ij}^{k+{\displaystyle \frac{1}{2}}}\right),{\delta }_t{u}_{ij}^{k+{\displaystyle \frac{1}{2}}}\right)\right\}.\end{array}$$

(15)

Combine (14) and (15) to obtain

$$\begin{array}{c}\hfill \mu \left[{∥{\delta }^{\alpha ,\beta }{u}^{k+1}∥}^2-{∥{\delta }^{\alpha ,\beta }{u}^k∥}^2\right]=\rho {\mathcal{H}}_x^{\alpha }{\mathcal{H}}_y^{\beta }\left[{\left|r^{k+1}\right|}^2-{\left|r^k\right|}^2\right],\end{array}$$

in other terms, (12) is satisfied, and this completes the proof. □

4. Numerical Example

In this section, we use numerical results to verify the main findings of this paper. Since the time derivative is standard and discretized by the common Crank–Nicolson method, we do not need to test it again. We only need to check the accuracy of our proposed numerical differentiation formula for approximating the Riesz derivative. For this purpose, we choose the following function:

$$\begin{array}{c}\hfill u\left(x\right)=x^4{(1-x)}^4,x\in (0,1),\end{array}$$

Table 1. Maximum absolute errors and convergence orders of the example by Equation (4).

$\mathit{\gamma }$	h	Maximum Absolute Errors	Convergence Orders
$1.1$	${\displaystyle \frac{1}{100}}$	5.869713 × ${10}^{-009}$	—
	${\displaystyle \frac{1}{120}}$	2.792955 × ${10}^{-009}$	4.0736
	${\displaystyle \frac{1}{140}}$	1.493018 × ${10}^{-009}$	4.0629
	${\displaystyle \frac{1}{160}}$	8.687830 × ${10}^{-010}$	4.0549
	${\displaystyle \frac{1}{180}}$	5.392700 × ${10}^{-010}$	4.0488
$1.2$	${\displaystyle \frac{1}{100}}$	1.497645 × ${10}^{-008}$	—
	${\displaystyle \frac{1}{120}}$	7.384049 × ${10}^{-009}$	3.8786
	${\displaystyle \frac{1}{140}}$	4.047813 × ${10}^{-009}$	3.8997
	${\displaystyle \frac{1}{160}}$	2.399977 × ${10}^{-009}$	3.9146
	${\displaystyle \frac{1}{180}}$	1.511485 × ${10}^{-009}$	3.9256
$1.3$	${\displaystyle \frac{1}{100}}$	3.356549 × ${10}^{-008}$	—
	${\displaystyle \frac{1}{120}}$	1.654048 × ${10}^{-008}$	3.8815
	${\displaystyle \frac{1}{140}}$	9.063573 × ${10}^{-009}$	3.9023
	${\displaystyle \frac{1}{160}}$	5.372170 × ${10}^{-009}$	3.9169
	${\displaystyle \frac{1}{180}}$	3.382494 × ${10}^{-009}$	3.9277
$1.4$	${\displaystyle \frac{1}{100}}$	6.552095 × ${10}^{-008}$	—
	${\displaystyle \frac{1}{120}}$	3.217388 × ${10}^{-008}$	3.9009
	${\displaystyle \frac{1}{140}}$	1.758679 × ${10}^{-008}$	3.9183
	${\displaystyle \frac{1}{160}}$	1.040516 × ${10}^{-008}$	3.9305
	${\displaystyle \frac{1}{180}}$	6.542308 × ${10}^{-009}$	3.9395
$1.5$	${\displaystyle \frac{1}{100}}$	1.167460 × ${10}^{-007}$	—
	${\displaystyle \frac{1}{120}}$	5.713168 × ${10}^{-008}$	3.9197
	${\displaystyle \frac{1}{140}}$	3.115434 × ${10}^{-008}$	3.9338
	${\displaystyle \frac{1}{160}}$	1.839972 × ${10}^{-008}$	3.9438
	${\displaystyle \frac{1}{180}}$	1.155316 × ${10}^{-008}$	3.9511
$1.6$	${\displaystyle \frac{1}{100}}$	1.954096 × ${10}^{-007}$	—
	${\displaystyle \frac{1}{120}}$	9.533607 × ${10}^{-008}$	3.9364
	${\displaystyle \frac{1}{140}}$	5.187624 × ${10}^{-008}$	3.9477
	${\displaystyle \frac{1}{160}}$	3.058942 × ${10}^{-008}$	3.9557
	${\displaystyle \frac{1}{180}}$	1.918356 × ${10}^{-008}$	3.9615
$1.7$	${\displaystyle \frac{1}{100}}$	3.126458 × ${10}^{-007}$	—
	${\displaystyle \frac{1}{120}}$	1.521159 × ${10}^{-007}$	3.9514
	${\displaystyle \frac{1}{140}}$	8.261266 × ${10}^{-008}$	3.9603
	${\displaystyle \frac{1}{160}}$	4.864358 × ${10}^{-008}$	3.9664
	${\displaystyle \frac{1}{180}}$	3.047202 × ${10}^{-008}$	3.9710
$1.8$	${\displaystyle \frac{1}{100}}$	4.835271 × ${10}^{-007}$	—
	${\displaystyle \frac{1}{120}}$	2.346630 × ${10}^{-007}$	3.9653
	${\displaystyle \frac{1}{140}}$	1.272152 × ${10}^{-007}$	3.9719
	${\displaystyle \frac{1}{160}}$	7.480633 × ${10}^{-008}$	3.9764
	${\displaystyle \frac{1}{180}}$	4.681285 × ${10}^{-008}$	3.9797
$1.9$	${\displaystyle \frac{1}{100}}$	7.283749 × ${10}^{-007}$	—
	${\displaystyle \frac{1}{120}}$	3.526410 × ${10}^{-007}$	3.9785
	${\displaystyle \frac{1}{140}}$	1.908465 × ${10}^{-007}$	3.9830
	${\displaystyle \frac{1}{160}}$	1.120801 × ${10}^{-007}$	3.9860
	${\displaystyle \frac{1}{180}}$	7.006880 × ${10}^{-008}$	3.9881
$2.0$	${\displaystyle \frac{1}{100}}$	1.074528 × ${10}^{-006}$	—
	${\displaystyle \frac{1}{120}}$	5.190008 × ${10}^{-007}$	3.9915
	${\displaystyle \frac{1}{140}}$	2.804060 × ${10}^{-007}$	3.9939
	${\displaystyle \frac{1}{160}}$	1.644688 × ${10}^{-007}$	3.9954
	${\displaystyle \frac{1}{180}}$	1.027196 × ${10}^{-007}$	3.9965

show the computational results under different conditions. These results clearly show a convergence order of 4. When applying this formula to Equation (1), the proposed difference schemes (8) and (9) achieve a convergence order of $\mathcal{O}\left({\tau }^2+h_x^4+h_y^4\right)$.

5. Conclusions

In this paper, we have developed an improved high-order difference scheme for the two-dimensional spatial fractional Schrödinger equation, with a focus on conserving mass and energy. The proposed scheme addresses the challenges posed by the non-local nature of fractional derivatives and the nonlinearity of the equation.