CompactLocal Structure-PreservingAlgorithms for theNonlinear Schrödinger Equation with Wave Operator

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). *e convergence rates of both schemes are O(h4 + τ2). *e 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.


Introduction
e 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 initialboundary value problem of the NSEW as where u(x, t) is a complex function, u 0 (x) and u 1 (x) are known complex functions, α and β are real constants, and i 2 � − 1. Several numerical algorithms have been studied for solving the NSEW (Refs. [3][4][5][6][7][8][9][10][11] and references therein).
Recently, structure-preserving algorithms were proposed to solving the Hamiltonian systems [12][13][14][15] and applied to various PDEs, such as the nonlinear Schrödingertype equation [16][17][18][19], wave equation [20], and KdV equation [21]. e 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][24][25][26][27][28][29][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 energypreserving scheme and compact local momentum-preserving scheme) with fourth-order accuracy in the space for the NSEW. e 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.

Preliminary Knowledge
Firstly, let N and N t be two positive integers; we then divide the space region [x L , x R ] and the time interval, respectively, into N parts and N t parts. us, we introduce some notations: x j � x L + jh, j � 0, 1, . . . , N, and t n � nτ, n � 0, 1, . . . , N t , where h is the spatial step span and τ is the temporal length. e numerical solution and exact value of the function u(x, t) at the node (x j , t n ) are denoted by u n j and u(x j , t n ), respectively. Secondly, we define operators as follows: According to Taylor's expansion, it is easy to get that en, By some simple computations, it is not difficult to obtain the following: (i) Commutative law: where A represents A t or A x and D represents D t or D x . (ii) Chain rule: (iii) Discrete Leibniz rule: In particular, we have the following two equalities for θ � 1/2 and θ � 1: Letting f � g � u, we have Furthermore, we get we define the inner product and norms as For constructing algorithms conveniently, taking u � p + iq in equation (1), where p and q are real-valued functions, we derive Furthermore, letting p t � ξ, q t � η, p x � ω, and q x � ], system (14) can be written as 2 Mathematical Problems in Engineering For system (15), using ω x � φ and ] x � ϕ, we obtain

Compact Local Energy-Preserving Algorithm
Firstly, we consider the local energy conservation law for system (15). Multiplying the first line of equation (15) by p t , and the second line of equation (15) by q t , we derive Summing equations (20) and (21), we obtain Also by equations (16) and (17), we have rough 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 ω, ], ξ, and η, we get the following discrete scheme: i.e., Lemma 1. Grid function f n j satisfies the following identical equation: Proof. e left-hand term in equation (33) is equal to Mathematical Problems in Engineering is 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 D t A t p n− 1 j and equation (30) by D t A t q n− 1 j and then adding them together, we get By the discrete Leibniz rule and equations (26)-(28), the first and third terms in the left side of (36) are From Lemma 1, equation (25), and the second term in the left side of equation (36), we obtain Similarly, the fourth term in the left side of equation (36) is equal to e last term in the left side of equation (36) is

Compact Local Momentum-Preserving Algorithm
Now, we consider the local momentum conservation law for system (15). Multiplying the first line of equation (15) by p x , and the second line of equation (15) by q x , we have en adding equations (42) to (43), we get Additionally,

Mathematical Problems in Engineering
(45) us, 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: 6 Mathematical Problems in Engineering i.e., Lemma 2. Grid function f n j satisfies the following identical equation: Proof. e term in the left-hand side of (55) is equal to is completes the proof.

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 α � β � 1, equation (1) has exact solution u(x, t) � e i(x+t) . In numerical calculations, we let x ∈ [− π, π].
In order to verify the convergence rates of the proposed schemes (32) and (54), we define ε � u N t j − u(x j , t N t ) and Tables 1-4 show the L ∞ and L 2 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 O(τ 2 + h 4 ).
Next, we investigate the local conservation properties of the schemes (32) and (54). To compute the discrete local energy and local momentum at t n � nτ, we define e n local � max 0≤j≤N |e n j | and m n local � max 0≤j≤N |m n j |, where e n j and m n j can be calculated by (35) and (57), respectively. In our experiments, we take T �100, h � π/128, and τ � 0.02. Figures 1 and 2 show the numerical results for the local Mathematical Problems in Engineering   Table 2: Spatial errors and convergence rates of scheme (32) at T �1 with τ � 0.0005.
h      energy and local momentum errors using schemes (32) and (54), respectively. e 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 eorem 1 and eorem 2 in this paper.

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

Conflicts of Interest
e authors declare that they have no conflicts of interest.