Abstract
Combining the compact method with the structure-preserving algorithm, we propose a compact local energy-preserving scheme and a compact local momentum-preserving scheme for the nonlinear Schrödinger equation with wave operator (NSEW). The convergence rates of both schemes are . The discrete local conservative properties of the presented schemes are derived theoretically. Numerical experiments are carried out to demonstrate the convergence order and local conservation laws of the developed algorithms.
1. Introduction
The nonlinear Schrödinger equation with wave operator (NSEW) is a very important model in mathematical physics with applications in a wide range, such as plasma physics, water waves, nonlinear optics, and bimolecular dynamics [1, 2]. In this paper, we consider the periodic initial-boundary value problem of the NSEW aswhere is a complex function, and are known complex functions, α and β are real constants, and . Several numerical algorithms have been studied for solving the NSEW (Refs. [3–11] and references therein).
Recently, structure-preserving algorithms were proposed to solving the Hamiltonian systems [12–15] and applied to various PDEs, such as the nonlinear Schrödinger-type equation [16–19], wave equation [20], and KdV equation [21]. The important feature of the structure-preserving algorithm is that it can maintain certain invariant quantities and has the ability of long-term simulation. It should be pointed out that Wang et al. [22] presented the concept of the local structure-preserving algorithm for PDEs and then proposed several algorithms which preserved the multisymplectic conservation law and local energy and momentum conservation laws for the Klein–Gordon equation. For the next few years, the theory of the local structure-preserving algorithm was used successfully for solving the PDEs (Refs. [23–30] and references therein), and the main advantage of the method is that it can keep the local structures of PDEs independent of boundary conditions.
As is known to all, high accuracy [31] and conservation algorithm are two important aspects of the numerical solutions. However, there are few local structure-preserving algorithms with high-order approximation to equation (1) in the literature. In this paper, using the high precision of the compact algorithm, we construct two new schemes (i.e., compact local energy-preserving scheme and compact local momentum-preserving scheme) with fourth-order accuracy in the space for the NSEW.
The outline of this paper is as follows: In Section 2, some preliminary knowledge is given, such as divided grid point, operator definitions, and their properties. In Section 3, the local energy-preserving algorithm is proposed and the local energy conservation law is proved. In Section 4, the local momentum-preserving algorithm is presented and the local momentum conservation law is analyzed. Numerical experiments are shown in Section 5. At last, we make some conclusions in Section 6.
2. Preliminary Knowledge
Firstly, let N and be two positive integers; we then divide the space region and the time interval, respectively, into N parts and parts. Thus, we introduce some notations: , , and , where h is the spatial step span and τ is the temporal length. The numerical solution and exact value of the function at the node are denoted by and , respectively.
Secondly, we define operators as follows:
According to Taylor’s expansion, it is easy to get that
Then,
By some simple computations, it is not difficult to obtain the following:(i)Commutative law: where A represents or and D represents or .(ii)Chain rule:(iii)Discrete Leibniz rule:
In particular, we have the following two equalities for and :
Letting , we have
Furthermore, we get
For any , we define the inner product and norms as
For constructing algorithms conveniently, taking in equation (1), where p and q are real-valued functions, we derive
Furthermore, letting , , , and , system (14) can be written as
For system (15), using and , we obtain
3. Compact Local Energy-Preserving Algorithm
Firstly, we consider the local energy conservation law for system (15). Multiplying the first line of equation (15) by , and the second line of equation (15) by , we derive
Summing equations (20) and (21), we obtain
Also by equations (16) and (17), we have
Through further processing, we know that system (15) admits the local energy conservation law:
In equations (16)–(19), discretizing the space derivatives by using the compact leap-frog rule, the time derivatives by using the midpoint rule, and the nonlinear term with the discrete chain rule in the time direction, we obtain
From equations (25)–(28), eliminating φ and ϕ, we have
Furthermore, eliminating , and η, we get the following discrete scheme:i.e.,
Lemma 1. Grid function satisfies the following identical equation:
Proof. The left-hand term in equation (33) is equal toThis completes the proof.
Theorem 1. Scheme (32) meets the discrete local energy conservation law:That is to say, scheme (32) is a local energy-preserving algorithm.
Proof. Multiplying equation (29) by and equation (30) by and then adding them together, we getBy the discrete Leibniz rule and equations (26)–(28), the first and third terms in the left side of (36) areFrom Lemma 1, equation (25), and the second term in the left side of equation (36), we obtainSimilarly, the fourth term in the left side of equation (36) is equal toThe last term in the left side of equation (36) isFrom equations (36)–(41), we complete the proof of Theorem 1.
4. Compact Local Momentum-Preserving Algorithm
Now, we consider the local momentum conservation law for system (15). Multiplying the first line of equation (15) by , and the second line of equation (15) by , we have
Then adding equations (42) to (43), we get
Additionally,
Thus, system (15) possesses the following local momentum conservation law:
In equations (16)–(19), applying the compact midpoint rule to space derivatives, the midpoint rule to time derivatives, and the discrete chain rule to the nonlinear term in the spatial direction, we obtain
By equations (47)–(50), we have
From equations (47)–(52), we obtain the following discrete scheme:i.e.,
Lemma 2. Grid function satisfies the following identical equation:
Proof. The term in the left-hand side of (55) is equal toThis completes the proof.
Theorem 2. Scheme (54) possesses the discrete local momentum conservation law asThat is to say, scheme (54) is a local momentum-preserving algorithm.
Proof. Multiplying (51) by and (52) by and then adding them together, we deriveAccording to discrete Leibniz rules, the first term in the left-hand side of (58) can be expressed asSimilarly, from the fourth term in the left side of (58), we haveThe second and fifth terms in the left side of (58) are equal toFrom the third term in the left side of (58) and Lemma 2, we can obtainSimilarly, the sixth term in the left side of (58) is equal toFrom the last term in the left-hand side of (58), we haveThrough equations (58)–(64), we complete the proof of Theorem 2.
5. Numerical Experiments
In this section, numerical experiments are designed to show the accuracies and conservation properties of the schemes which we have obtained above. Taking , equation (1) has exact solution . In numerical calculations, we let .
In order to verify the convergence rates of the proposed schemes (32) and (54), we define and , where or . Tables 1–4 show the and errors of the numerical solutions with respect to the exact ones for both schemes. In these tables, we can confirm that the two schemes have the accuracy of .
Next, we investigate the local conservation properties of the schemes (32) and (54). To compute the discrete local energy and local momentum at , we define and , where and can be calculated by (35) and (57), respectively. In our experiments, we take T = 100, , and . Figures 1 and 2 show the numerical results for the local energy and local momentum errors using schemes (32) and (54), respectively. The figures indicate that the developed schemes can preserve the local energy and local momentum of the system very well over long-time simulations, which is consistent with the theoretical results of Theorem 1 and Theorem 2 in this paper.
6. Conclusions
In this paper, two new compact local structure-preserving algorithms are constructed for solving the NSEW. Local conservation laws of the proposed schemes are derived theoretically. Numerical results are shown to verify the accuracy, validity, and long-time numerical behavior of the schemes obtained in this work. Hence, the compact local structure-preserving method can be used for many Hamiltonian systems.
Data Availability
The data of the numerical experiment used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was partially supported by the Key Laboratory of Applied Mathematics of Fujian Province University (Putian University) (no. SX201802), the Natural Science Foundation of China (nos. 11701196 and 11701197), the Promotion Program for Young and Middle-Aged Teachers in Science and Technology Research of Huaqiao University (no. ZQN-YX502), and the Fundamental Research Funds for the Central Universities (no. ZQN-702).