Fourier Spectral Method for a Class of Nonlinear Schrödinger Models

Zhang, Lei; Yang, Weihua Ou; Liu, Xuan; Qu, Haidong

doi:https://doi.org/10.1155/2021/9934858

Advances in Mathematical Physics

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 9934858 | https://doi.org/10.1155/2021/9934858

Fourier Spectral Method for a Class of Nonlinear Schrödinger Models

Lei Zhang,¹Weihua Ou Yang,¹Xuan Liu,¹and Haidong Qu¹

Academic Editor: Jorge E. Macias-Diaz

Received15 Mar 2021

Revised29 Apr 2021

Accepted23 May 2021

Published01 Jul 2021

Abstract

In this paper, Fourier spectral method combined with modified fourth order exponential time-differencing Runge-Kutta is proposed to solve the nonlinear Schrödinger equation with a source term. The Fourier spectral method is applied to approximate the spatial direction, and fourth order exponential time-differencing Runge-Kutta method is used to discrete temporal direction. The proof of the conservation law of the mass and the energy for the semidiscrete and full-discrete Fourier spectral scheme is given. The error of the semidiscrete Fourier spectral scheme is analyzed in the proper Sobolev space. Finally, several numerical examples are presented to support our analysis.

1. Introduction

Schrödinger equation, being known as basic assumption of quantum mechanics, is one of the most important equations in quantum mechanics that proposed by Austrian physicist Schrödinger. Known to all, the standard Schrödinger equation is a second-order partial differential equation established by combining the concept of matter wave and wave equation, and it can describe the movement of microparticles. Indeed, each microsystem has a corresponding Schrodinger equation. The specific form of wave function and corresponding energy can be obtained by solving this equation, so as to understand the properties of microsystem. Schrödinger equation shows that in quantum mechanics, particles appear in the form of probability, with uncertainty, and can be ignored at the macroscale.

In recent decades, more and more scholars have been interested in studying Schrödinger equations, and the research on Schrödinger equations can be divided into two categories: integral order (or local) and fractional order (or nonlocal) models. Now, we will make a brief literature review. (1)Schrödinger Equations of Fractional Order. In 2000, firstly, Laskin [1] extended the fractional concept in quantum physics to construct a fractional path integral, and over these paths, the standard nonlinear Schrödinger equation was generalized to the fractional one. Very recently, many scholars studied the numerical solutions of fractional Schrödinger equations. In 2008, Rida et al. [2] applied the Adomian decomposition method to get the analytic and approximate solutions for different types of fractional differential equations. In 2011, the generalized two-dimensional differential transform method was directly applied to solve the coupled Burger’s equations by Liu et al. [3]. In 2012, Wei et al. [4] presented an implicit fully discrete local discontinuous Galerkin finite element method for solving the time-fractional Schrödinger equation with Caputo fractional derivative. In 2013, Secchi et al. [5] obtained the solutions to a class of Schrödinger equations by using the fractional Laplacian. In 2014, the Jacobi spectral collocation method was constructed, and the numerical solutions of the time-fractional Schrödinger equation and the space-fractional Schrödinger equation were got by Bhrawy et al. [6]. In literature [7], Wang et al. proposed a linearly implicit conservative difference scheme for the coupled nonlinear Schrödinger equations with space fractional derivative, and she proved that the scheme can conserve the mass and energy in the discrete level. In 2015, Wang et al. [8] studied the energy conservative Crank-Nicolson difference scheme for nonlinear Riesz space-fractional Schrödinger equations and gave the proof of mass conservation and energy conservation in the discrete sense. In 2016, Ran et al. [9], under some assumptions, an implicit difference scheme was proposed to solve the strongly coupled nonlinear space fractional Schrödinger equations. In the literature [10–12], the semidiscrete scheme was fully constructed by employing finite element method. In 2020, Wang et al. [13] proposed a preconditioning method for the space fractional coupled nonlinear Schrödinger equations. For other literature on the fractional Schrödinger equations, see [14–16](2)Schrödinger Equations of Integral Order. The study of the standard Schrödinger equations has attracted many scholars. In 1995, Edwards et al. [17] presented a numerical treatment to solve the time-independent nonlinear Schrödinger equation with an external potential. In 1999, Chang et al. [18] studied the finite difference schemes for solving the generalized nonlinear Schrödinger equations, and several schemes, including Crank-Nicolson-type schemes, split step Fourier scheme, and pseudo-spectral scheme, were employed for solving three models of the generalized nonlinear Schrödinger equations. In 2003, Bao et al. [19] studied the performance of time-splitting spectral approximations for general nonlinear Schrödinger equations and proved that the discrete scheme was unconditionally stable and conserved in norm. In 2005, Wang [20] got the numerical solutions to nonlinear Schrödinger equations by using the split-step finite difference method. In 2017, Dehghan et al. [21] presented a numerical scheme based on radial basis function for solving the high-dimensional nonlinear Schrödinger equations using the explicit Runge-Kutta method. In 2019, Ahsan et al. [22] proposed a Haar wavelet collocation method for linear and nonlinear Schrödinger equations, and the stability analysis of this method was given. In 2020, Ban et al. [23] proposed a family of algorithms, which were special implementations of Lubich’s Convolution Quadrature, for the numerical solution to the Schrödinger equation. For other literature on the standard Schrödinger equations, see [24–28]

In this paper, we study the numerical solution to the nonlinear Schrödinger equation of the following form: with the periodic condition and the initial condition where is the imaginary number of units and ; is a function with a period of ; , , and are the three real numbers.

For this model, two important physical quantities are often considered; these are mass and energy and using the conservation of mass and energy to measure whether the algorithm is effective for this equation. As in [10], we can easily derive the mass and energy conservation where

The rest of this paper is organized as follows. In Section 2, the definitions and properties of the related Sobolev space are introduced; based on this space, we establish a semidiscrete Fourier spectral scheme for model (1), whose solution keeps mass and energy conversation for some function . Then, the error estimation of this semidiscrete scheme is given in the established Sobolev space. Section 3 gives an introduction of modified fourth order exponential time-differencing Runge-Kutta method that is used to discrete the temporal component, and a proof of mass conservation and energy conservation is given in the discrete sense. In Section 4, we carry out some numerical experiments to verify our conclusions. Finally, Section 5 offers some concluding remarks.

2. Preliminaries

In this section, based on the semidiscrete Fourier spectral scheme, we prove the mass and energy conversation for this semidiscrete scheme and give the error estimation of the Fourier spectral approximation. First, we introduce the following Sobolev space and some notations.

Let , and the space is defined by

Define inner product on : and the norm on : where is the conjugate of . We denote the complete space of by , which can be characterized in the frequency space under the norm the seminorm is defined by where .

Suppose

Then, the orthogonal projection operator is defined by where , are the Fourier coefficients of .

According to the definition of inner product, , we define the semidiscrete scheme for model (1):

Usually, when designing a numerical scheme for (14), two important conservation laws often need to be considered: the mass and the total energy of the wave function. In the following lemma, we prove that the semidiscrete Fourier spectral scheme (14) can keep the conservation laws.

Lemma 1. Let , then the semidiscrete scheme (14) is conservative in the sense where are the mass and energy of the wave function, respectively.

Proof. Let in (14), we have Note that , , and are three real numbers; thus, taking the imaginary part of the above equation and noting the periodic boundary condition, we get which indicates Next, we prove the semidiscrete scheme (14) can keep energy conservation. Let in (14), we have Taking the real part of the above equation and noting the periodic boundary condition, we have Computing the real part of , we have Computing the real part of , we have It follows from equations (22), (23), (24), and (25); we get the energy conservation law. This completes the proof.

Next, we analyze the existence and uniqueness of solution for the semidiscrete scheme (14). We need the following lemma.

Lemma 2. Let is the solution for the semidiscrete scheme (14); then, there exists a positive constant , such that

Proof. From Lemma 1, we have (27) and (28) lead to Finally, using Sobolev’s embedding theorem, we obtain which implies both and are bounded. Therefore This completes the proof.

The semidiscrete scheme (14) is a first order nonlinear ordinary differential equation with respect to time on a finite interval; using Lemma 2 and Picard’s theorem, we obtain the following theorem.

Theorem 3. Let the solution , , then the semi-discrete scheme (14) has unique solution in .

Proof. The proof is omit.
In the following theorem, we will give the error estimation of the Fourier spectral method to solve problem (1). For convenience, we still use to denote that there exists a positive constant , which is independent of , , and , such that .

Theorem 4. Let is the solution of (1); is obtained from the semidiscrete scheme (14); then, we have

Proof. Let , with in (14) replaced by , for all , we have Note that , by setting , and taking the imaginary part of (33), we obtain Applying the Schwarz inequality to the above equation, we get We note that Then, the factor is bounded by According to equations (35) and (37) and , we arrive at which in accordance with Gronwall inequality, we have Noting that for all , there exists a positive constant , such that Finally, a combination of the above estimates leads to This completes the proof.

3. The Full-Discrete Fourier Spectral Scheme and Its Conservation

In this section, we first introduce the full-discrete Fourier spectral scheme for (1) and then give the proof of its conservation properties.

3.1. The Full-Discrete Fourier Spectral Scheme

Due to the periodicity of the solution, it suffices to compute the model in the interval . We choose interpolation points , and , . Then, the semidiscrete scheme (14) can be discretized in space by seeking the approximate solution where is the th discrete Fourier coefficient, for , and for . Thus, for each frequency , we have

Let be the time steps, satisfying then (44) is equivalent to

More detail about this equation can be find in [29, 30]. Now, let and using modified fourth order exponential time-differencing Runge-Kutta (ETDRK4) method to approximate the integral equation (46), we have where with the functions , , and being defined by

Finally, by using inverse Fourier transform, we obtain is the approximate solution of .

For simplicity, the equal step-size is used to discrete time directions. The method presented in this paper allows one to choose different step sizes which are determined by adaptive strategy given in [31]. Next, we proof that the full-discrete scheme (47) satisfies two conservation laws given by Lemma 1.

3.2. The Conservation Laws for ETDRK4

In this subsection, we prove that the scheme given by (47) can keep mass and energy conservation. First, see the following two explanatory notes about ETDRK4.

Note 1. Equation (46) is obtained by integrating from to . Then, from the reasoning process of the method ETDRK4 [29, 30], we get the following form where here, is obtained from the semidiscrete scheme (14). Then, if ignoring the error term , using the method of ETDRK4 to solve (46) is equivalent to get the solution to (50) by the scheme (51).

Note 2. Assume that the function mentioned above satisfies the Lipschitz condition with parameter , that is Actually, should be third-order continuous differentiable with respect to due to using the method of ETDRK4, which ensures that satisfies this assumption easily.

In Lemma 1, we have proved the mass conservation for the semidiscrete scheme (14). Then, if it holds , the full-discrete scheme ETDRK4 is mass conservative, where is the solution to (14) and is obtained from (47) and inverse Fourier transform. Next, we give the proof in the following theorem.

Theorem 5 (mass conservation). Let be obtained from the semidiscrete scheme (14), , and is computed by (47). Define the discrete mass , . Then, ignoring the error comes from machine calculation, we have where , Furthermore,

Proof. Without loss of generality, we only need to prove From Note 1, we have Taking the module of equation (51) minus equation (56), and let , it follows Using Note 2, we get Summing the above inequalities from to , we obtain Let using Gronwall inequality leads to Note that and denote by , the desired result follows, Moreover, we have The above inequalities imply as is big enough and .

Theorem 6 (energy conservation). Let and is obtained from (47). Define the discrete local-energy and the discrete energy here, is a numerical approximation of the fractional Laplacian operator , . Then, we have Furthermore, .

Proof. Without loss of generality, we only need to prove Define bounded linear operator then we have where is a positive constant. By Theorem 5, which implies as is big enough and .

4. Applying

In this section, we present two examples to illustrate our results. As in [10], we define the convergence orders in () norm sense as follows where is given by

All the experiments were done on a computer with the following configuration: Intel(R) Core (TM) i5-8250 CPU @ 1.60 GHz 1.80 GHz and involved using MATLAB 2015b.

4.1. Example 1

We first consider the periodic initial Schrödinger equation (1) with the following conditions: , , , and . The exact solution is

By FFT and ETDRK4, we compute numerical solution of this example. The point-wise errors (in sense) against various are shown in Table 1; then, we plot in semilog scale in Figure 1, which clearly indicates the exponential convergence in space direction, and the fourth order accuracy in time direction is shown in Table 2. The exact solution is shown in Figure 2, and the solution surface is shown in Figure 3, from which we can see that the two pictures are fit very well. Finally, we compute the mass and the energy of the numerical solution at different times. As one can see in Table 3, the scheme has an excellent conservation property and is in well agreement with the results in Theorem 5.

4.2. Example 2

We then consider the periodic initial Schrödinger equation (1) with the following conditions: , , , and . The exact solution to the equation is

We compute the numerical solution for this example until the time with time step on the uniform grid with . We display the errors that derived from temporal direction in Table 4. By Theorem 5, the scheme can keep the mass and energy conservation. Table 5 validates this result very well. It can be seen from Figures 4 and 5 that the approximate solution is in good agreement with the exact solution.

5. Conclusion

In this paper, we propose a method to solve the Schrödinger equations by using the method of Fourier spectral and modified fourth order exponential time-differencing Runge-Kutta, and we give the semidiscrete and full-discrete schemes for the equations. We prove that the schemes can keep the conservation law of mass and energy for full-discrete schemes. Numerical examples verify that our conclusions are correct.

Data Availability

The numerical data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank the referees for their many constructive comments and suggestions to improve the paper. This work was partly supported by the Educational science project of Guangdong Province with Nos. 2020GXJK385 and 2019KTSCX097.

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