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.
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: subject to the boundary condition and the initial conditions where and are positive real constants, , and . and are given real functions. The fractional Laplacian operator can be defined as a pseudo-differential operator with the symbol : where is the Fourier transform and is the Fourier transform of .
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 interval , it is converted into periodic boundary conditions in this paper and studied on and below. Let
Denote , , , and . Thus, (1)–(3) become
where .
In fact, the nonlinear fractional Schrödinger equation ((6)–(8)) has two conserved quantities: where with
The outline of the remainder of this paper is as follows. In Section 2, 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
Let be the set of all complex-valued and -periodic -functions on . Denote as the inner product on the space with the norm (abbreviated as ):
For as a nonnegative real number, let be the closure of . Note that . For any function , the following equations [17] can be developed easily: where the Fourier coefficients are arranged as
For the Fourier transform of fractional Laplacian , we have
In order to discretize the equation in the temporal direction, the time step is defined by . Denote difference operator where is a positive integer (). Therefore, the Crank-Nicolson method was used to discretize equation (6) in the time axis.
where
2.1. Crank-Nicolson Fourier Galerkin Method
For positive even number , the basis function space can be constructed as where the norm and seminorm of are characterized by
Let
The orthogonal operators are defined as follows:
Lemma 1 [18, 19]. Suppose that for all ; it holds that Denote 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: where , .
2.2. Crank-Nicolson Fourier Collocation Method
For positive even number , consider the points , , as collocation nodes. The discrete Fourier coefficients [18] of a function on with respect to the collocation points are the following form:
Using the inversion formula, we have
Define the interpolation operator [18] at the collocation points:
According to (27),
Lemma 2 [18, 19]. For any , , the estimate in the sense of the Sobolev norm.
Combining Lemma 2 and the triangle inequality, Corollary 3 is drawn.
Corollary 3. For any , , there exists a constant independent of and , such that 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: Applying the Fourier transformation to (24), we get the following form: where .
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: where
Proof. We derive the full discrete Fourier Galerkin method:
Let in equation (37); it holds that
Taking the imaginary part of equation (38), due to
Therefore,
thus,
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.
Let ; according to equation (37), we also get
Taking the real part of (42), due to
therefore, using (43)–(46), we obtain
thus,
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 that is a positive constant in this article, which might be different in every formula.
Lemma 5 [20]. For any discrete function , it holds that
Lemma 6. For , there exists a positive constant , such that
Proof. Using Theorem 4, it yields thus, Because of , it satisfies Sum the inequalities of Lemma 5 from 0 to yields Adding (53) and (54), we can obtain the following items: For is sufficiently small (), this implies According to the discrete Gronwall’s inequality, there is Therefore, ☐
Theorem 7. If , assume that is the exact solution of (6)–(8), and is the numerical solution of (24). It possesses the following conclusion:
Proof. Let , , , and ; then, . From triangle inequality and Lemma 1, it yields According to the orthogonality of the projection operator , we get The authors derive the full discrete Fourier Galerkin method: Subtracting equation (62) from equation (61), due to thus, According to the orthogonality of operator , i.e., . Therefore, Let in (64), and taking the real part, due to therefore, using (66)-(68), this implies where Thus, according to Lemma 6, we can get Note Lemma 1; it gives that Then, Thus, (69) becomes Because of and from Lemma 5, it gives that Then, combining (74) and (76) leads to Summing above inequalities (77) from 1 to yields Hence, using the discrete Gronwall’s inequality gives thus, Substituting (80) into (60) can yield which immediately gives conclusion.☐
Similar to the proof of Theorem 7, we can obtain the following theorem.
Theorem 8. Let ; assume that is the exact solution of (6)–(8), and is the numerical solution of (32). It possesses the following conclusion:
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: Let , , and . Figures 1 and 2 present the numerical solutions for and . We can find that the order of will affect the shape of the solution.
There is no exact solution of (83) known for . Therefore, numerical solution calculated by the method (24) with and is taken as the reference solution. Let be the numerical solution, and calculate the error at in the sense of the discrete norm:
The convergence rates in the direction of time and space are calculated as
Let . 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 and .
Figures 3–6 show the ability of the numerical method (24) to maintain the discrete mass and discrete energy when and . It can be seen from the figure that the numerical method (24) maintains the discrete mass and discrete energy well.
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., . 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.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work is supported by the National Natural Science Foundation of China under grant no. 11901329, the Natural Science Foundation of Shandong Province under grant nos. ZR2019BA022 and ZR2020QA008, and the Teaching Reform Projects of Qingdao University of Technology (F2019-044 and F2020-079).