Compact Local Structure-Preserving Algorithms for the Nonlinear Schro¨Dinger Equation with Wave Operator

Hindawi Mathematical Problems in Engineering Volume 2020, Article ID 4345278, 12 pages https://doi.org/10.1155/2020/4345278

Research Article Compact Local Structure-Preserving Algorithms for the Nonlinear Schro¨dinger Equation with Wave Operator

Langyang Huang ,1,2 Zhaowei Tian,1 and Yaoxiong Cai1

1Fujian Province University Key Laboratory of Computational Science, School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China 2Key Laboratory of Applied Mathematics (Putian University), Fujian Province University, Putian 351100, Fujian, China

Correspondence should be addressed to Langyang Huang; [email protected]

Received 7 November 2019; Accepted 6 January 2020; Published 28 January 2020

Academic Editor: Bernardo Spagnolo

Copyright © 2020 Langyang Huang et al. *is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 Schro¨dinger 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.

1. Introduction preserving algorithm is that it can maintain certain invariant quantities and has the ability of long-term simulation. It *e nonlinear Schrodinger¨ equation with wave operator should be pointed out that Wang et al. [22] presented the (NSEW) is a very important model in mathematical physics concept of the local structure-preserving algorithm for PDEs with applications in a wide range, such as plasma physics, and then proposed several algorithms which preserved the water waves, nonlinear optics, and bimolecular dynamics multisymplectic conservation law and local energy and [1, 2]. In this paper, we consider the periodic initial- momentum conservation laws for the Klein–Gordon boundary value problem of the NSEW as equation. For the next few years, the theory of the local

2 structure-preserving algorithm was used successfully for utt − uxx + iαut + β|u| u � 0, x ∈ xL, xR �, 0 < t ≤ T, solving the PDEs (Refs. [23–30] and references therein), and (1) the main advantage of the method is that it can keep the local structures of PDEs independent of boundary conditions.

u(x, 0) � u0(x), ut(x, 0) � u1(x), x ∈ �xL, xR �, (2) 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 u xL, t � � u xR, t �, 0 ≤ t ≤ T, (3) with high-order approximation to equation (1) in the literature. where u(x, t) is a complex function, u0(x) and u1(x) are In this paper, using the high precision of the compact algorithm, known complex functions, α and β are real constants, and we construct two new schemes (i.e., compact local energy- i2 � − 1. Several numerical algorithms have been studied for preserving scheme and compact local momentum-preserving solving the NSEW (Refs. [3–11] and references therein). scheme) with fourth-order accuracy in the space for the NSEW. Recently, structure-preserving algorithms were pro- *e outline of this paper is as follows: In Section 2, some posed to solving the Hamiltonian systems [12–15] and preliminary knowledge is given, such as divided grid point, applied to various PDEs, such as the nonlinear Schrodinger-¨ operator deﬁnitions, and their properties. In Section 3, the local type equation [16–19], wave equation [20], and KdV energy-preserving algorithm is proposed and the local energy equation [21]. *e important feature of the structure- conservation law is proved. In Section 4, the local momentum- 2 Mathematical Problems in Engineering

preserving algorithm is presented and the local momentum into N parts and Nt parts. *us, we introduce some nota- conservation law is analyzed. Numerical experiments are shown tions: h � (xR − xL)/N, τ � T/Nt, xj � xL + jh, j � in Section 5. At last, we make some conclusions in Section 6. 0, 1, ... ,N, and tn � nτ, n � 0, 1, ... ,Nt, where h is the spatial step span and τ is the temporal length. *e numerical 2. Preliminary Knowledge solution and exact value of the function u(x, t) at the node n (xj, tn) are denoted by uj and u(xj, tn), respectively. Firstly, let N and Nt be two positive integers; we then divide Secondly, we deﬁne operators as follows: the space region [xL, xR] and the time interval, respectively,

un+1 − un un − un un+1 + un un + un D un � j j ,D un � j+1 j ,A un � j j ,A un � j+1 j , t j τ x j h t j 2 x j 2 n+2 n+1 n n n n n n n uj + 2uj + uj uj+ + 2uj+ + uj uj+ + 4uj + uj− A2un � ,A2 un � 2 1 , δ un � 1 1, (4) t j 4 x j 4 x j 6 n n n n+1 n n− 1 n n n uj+ + 10uj + uj− uj − 2uj + uj uj+ − 2uj + uj− δ2 un � 1 1,D2un− 1 � ,D2 un � 1 1. x j 12 t j τ2 x j− 1 h2

According to Taylor’s expansion, it is easy to get that In particular, we have the following two equalities for n θ � 1/2 and θ � 1: n zu 4 Dxuj � δx� � + O� h �, zx j 1 (5) θ � ,Dx(f · g)i � Axfi · Dxgi + Dxfi · Axgi, n 2 z2u (10) D2 un � δ2 � � + O� h4 �. x j− 1 x 2 θ � 1,D (f · g) � f · D g + D f · g . zx j x i i+1 x i x i i

*en, Letting f � g � u, we have n − 1 n zu 4 δx Dxuj �� + O� h �, 1 n 2 n n zx j Dt�u � � � Dtu · Atu , ( ) 2 n 6 (11) z2u 1 − 2D2 un �� + O� h4 �. D ��un · un− 1 � � � un · D �A un− 1 �. δx x j− 1 2 t t t zx j 2

By some simple computations, it is not diﬃcult to obtain Furthermore, we get the following: D �− 1fg � � − 1f · D g + D − 1f · g . (i) Commutative law: x δx j δx j+1 x j xδx j j (12)

DxDt � DtDx, , � | � (v , v , ... , v )T, v � v (7) For any u v ∈ Vh �v v 0 1 N 0 N�, AD � AD, we deﬁne the inner product and norms as N

where A represents At or Ax and D represents Dt or (u, v) � h � ujvj, j�0 Dx. �� N (ii) Chain rule: � �2 � n� (13) ‖u‖2 � h � �uj � , n n n j�0 DtF u � � DuF u �Dtu + O(τ), � � (8) � n� ‖u‖∞ � max �uj �. DxF� uj � � DuF� uj �Dxuj + O(h). 0≤j≤N

(iii) Discrete Leibniz rule: For constructing algorithms conveniently, taking u � p + iq in equation (1), where p and q are real-valued Dx(f · g)j ��θfj+1 +(1 − θ)fj� · Dxgj functions, we derive + ( − )g + g · D f , , 2 2 � 1 θ j+1 θ j� x j ∀0 ≤ θ ≤ 1 ⎧⎨ ptt − pxx − αqt + βp + q �p � 0, n n+ n n (14) D (f · g) �� f 1 +( − )f � · D g ⎩ 2 2 t θ 1 θ t qtt − qxx + αpt + βp + q �q � 0. n+1 n n +� (1 − θ)g + θg � · Dtf , ∀0 ≤ θ ≤ 1. Furthermore, letting pt � ξ, qt � η, px � ω, and qx � ], (9) system (14) can be written as Mathematical Problems in Engineering 3

− q − � − p2 + q2 �p, ⎪⎧ ξt α t ωx β n− 1 n− 1 n β n 2 ⎨ DtAtη + αDtAtp − ϕ + ��Atp � 2 2 j j j 4 j ⎪ ηt + αpt − ]x � − βp + q �q, (15) ⎩⎪ n 2 n− 2 n− 2 n n− +�A q � +�A p 1 � +�A q 1 � ��A q + A q 1� � . pt � ξ, qt � η, px � ω, qx � ]. t j t j t j t j t j 0 (28) For system (15), using ωx � φ and ]x � ϕ, we obtain

px � ω, ωx � φ, qx � ], ]x � ϕ, (16) From equations (25)–(28), eliminating φ and ϕ, we have

n− 1 n− 1 − 1 n β n 2 DtAtξ − αDtAtq − Dxδ ω + ��Atp � pt � ξ, qt � η, (17) j j x j 4 j

2 2 n 2 n− 1 2 n− 1 2 n n− 1 ξt − αqt − φ + β�p + q �p � 0, (18) +�Atqj � +�Atpj � +�Atqj � ��Atpj + Atpj � � 0,

2 2 (29) ηt + αpt − ϕ + β�p + q �q � 0. (19) β 2 D A ηn− 1 + αD A pn− 1 − D δ− 1]n + ��A pn � t t j t t j x x j 4 t j 3. Compact Local Energy-Preserving Algorithm n 2 n− 1 2 n− 1 2 n n− 1 +�Atqj � +�Atpj � +�Atqj � ��Atqj + Atqj � � 0. Firstly, we consider the local energy conservation law for (30) system (15). Multiplying the ﬁrst line of equation (15) by pt, and the second line of equation (15) by qt, we derive Furthermore, eliminating ω, ], ξ, and η, we get the fol- 2 2 lowing discrete scheme: ξtpt − αqtpt − ωxpt + β�p + q �ppt � 0, (20) β 2 2 D2pn− 1 − D A qn− 1 − D2 − 2pn + ��A pn � +�A qn � 2 2 t j α t t j xδx j− 1 t j t j ηtqt + αptqt − ]xqt + β�p + q �qqt � 0. (21) 4

n− 1 2 n− 1 2 n n− 1 Summing equations (20) and (21), we obtain + �Atpj � +�Atqj � ��Atpj + Atpj � � 0, 2 2 ξtpt + ηtqt − ωxpt − ]xqt + β�p + q �ppt + qqt � � 0. β 2 (22) D2qn− 1 + αD A pn− 1 − D2 δ− 2qn + ��A pn � t j t t j x x j− 1 4 t j Also by equations (16) and (17), we have n 2 n− 1 2 n− 1 2 n n− 1 2 2 + �Atqj � +�Atpj � +�Atqj � ��Atqj + Atqj � � 0, ξtξ + ηtη + ωωt + ]]t + β�p + q �ppt + qqt � (23) (31) − ωxξ + ωξx + ]xη + ]ηx� � 0. i.e., *rough further processing, we know that system (15) � � � � 2 n− 1 n− 1 2 − 2 n β � n�2 � n− 1�2 admits the local energy conservation law: D u + iαD A u − D δ u + ��A u � +�A u � � t j t t j x x j− 1 4 t j t j 1 β 2 z � �ξ2 + η2 + ω2 + ]2� + �p2 + q2 � � � z (ωξ + ]η). t 2 4 x n n− 1 · �Atuj + Atuj � � 0. (24) (32) In equations (16)–(19), discretizing the space derivatives by using the compact leap-frog rule, the time derivatives by n using the midpoint rule, and the nonlinear term with the Lemma 1. Grid function fj satisﬁes the following identical discrete chain rule in the time direction, we obtain equation: n n n n n n fn D A fn− 1 + fn D A fn− 1 Dxpj � δxωj+1,Dxωj � δxφj ,Dxqj � δx]j+1, j+1 t t j+2 j+1 t t j (25) n n 1 n n+ n− n n n− Dx]j � δxϕj , � �hD �f f 1 � + τD �f 1f � − hD �f f 1 � 2τ x j j+1 t j j+1 x j j+1 n n n n Dtpj � Atξj ,Dtqj � Atηj , (26) n n− 1 + τDt�fj fj+1 ��. (33) n− 1 n− 1 n β n 2 n 2 DtAtξj − αDtAtqj − φj + ��Atpj � +�Atqj � 4 (27) n− 1 2 n− 1 2 n n− 1 +�Atpj � +�Atqj � ��Atpj + Atpj � � 0, Proof. *e left-hand term in equation (33) is equal to 4 Mathematical Problems in Engineering

n n− 1 n n− 1 fj+1DtAtfj+2 + fj+1DtAtfj

1 � �fn fn+1 − fn fn− 1 + fn fn+1 − fn fn− 1� 2τ j+1 j+2 j+1 j+2 j+1 j j+1 j (34) 1 � ��fn fn+1 − fnfn+1� +�fnfn+1 − fn− 1fn � − �fn fn− 1 − fnfn− 1� +�fn fn+1 − fnfn− 1�� 2τ j+1 j+2 j j+1 j j+1 j j+1 j+1 j+2 j j+1 j+1 j j j+1

1 � �hD �fnfn+1 � + τD �fn− 1fn � − hD �fnfn− 1 � + τD �fnfn− 1 ��. 2τ x j j+1 t j j+1 x j j+1 t j j+1

*is completes the proof. □ Theorem 1. Scheme (32) meets the discrete local energy conservation law:

1 2 2 1 en � D � ��A ξn− 1 � +�A ηn− 1 � � + �10�δ− 1ωn− 1δ− 1ωn � j t 2 t j t j 24 x j+1 x j+1

− 1 n− 1 − 1 n − 1 n − 1 n− 1 − 1 n− 1 − 1 n + δx ωj δx ωj+1 + δx ωj δx ωj+1 + 10�δx ]j+1 δx ]j+1�

2 − 1 n− 1 − 1 n − 1 n − 1 n− 1 β n− 1 2 n− 1 2 + δx ]j δx ]j+ + δx ]j δx ]j+ � + ��Atpj � +�Atqj � � � 1 1 4 (35)

h − D �δ− 1ωnD A pn− 1 − �δ− 1ωnδ− 1ωn+1 − δ− 1ωnδ− 1ωn− 1� x x j t t j 24τ x j x j+1 x j x j+1

h + δ− 1]nD A pn− 1 − �δ− 1]nδ− 1]n+1 − δ− 1]nδ− 1]n− 1�� 0. x j t t j 24τ x j x j+1 x j x j+1

That is to say, scheme (32) is a local energy-preserving By the discrete Leibniz rule and equations (26)–(28), the algorithm. ﬁrst and third terms in the left side of (36) are 1 2 D A n− 1D A pn− 1 � D A n− 1A A n− 1 � D ��A n− 1 � �, n− 1 t tξj t t j t tξj t tξj t tξj Proof. Multiplying equation (29) by DtAtpj and equation 2 n− 1 (30) by DtAtqj and then adding them together, we get (37) n− 1 n− 1 − 1 n n− 1 n− 1 n− 1 DtAtξj DtAtpj − Dxδx ωj DtAtpj + DtAtηj DtAtqj n− 1 n− 1 n− 1 n− 1 1 n− 1 2 DtAtηj DtAtqj � DtAtηj AtAtηj � Dt��Atηj � �. − 1 n n− 1 β n 2 n 2 n− 1 2 n− 1 2 2 − D δ ] D A q + 􏰃�A p � +�A q � +�A p � +�A q � � x x j t t j 4 t j t j t j t j (38)

n n− 1 n− 1 n n− 1 n− 1 ×��Atpj + Atpj �DtAtpj +�Atqj + Atqj �DtAtqj �� 0. From Lemma 1, equation (25), and the second term in (36) the left side of equation (36), we obtain Mathematical Problems in Engineering 5

− 1 n n− 1 − 1 n n− 1 − 1 n n− 1 Dxδx ωj DtAtpj � Dx�δx ωj DtAtpj � − δx ωj+1DtAtDxpj

− 1 n n− 1 − 1 n n− 1 � Dx�δx ωj DtAtpj � − δx ωj+1DtAtδxωj+1

− 1 n n− 1 − 1 n 2 − 1 n− 1 � Dx�δx ωj DtAtpj � − δx ωj+1δxDtAtδx ωj+1

1 � D �δ− 1ωnD A pn− 1� − δ− 1ωn �D A δ− 1ωn− 1 + 10D A δ− 1ωn− 1 + D A δ− 1ωn− 1� x x j t t j 12 x j+1 t t x j+2 t t x j+1 t t x j

1 1 (39) � D �δ− 1ωnD A pn− 1� − D �10�δ− 1ωn δ− 1ωn− 1�� − �hD �δ− 1ωnδ− 1ωn+1� x x j t t j 24 t x j+1 x j+1 24τ x x j x j+1

− 1 n− 1 − 1 n − 1 n − 1 n− 1 − 1 n − 1 n− 1 + τDt�δx ωj δx ωj+1� − hDx�δx ωj δx ωj+1 � + τDt�δx ωj δx ωj+1 ��

h � D �δ− 1ωnD A pn− 1 − �δ− 1ωnδ− 1ωn+1 − δ− 1ωnδ− 1ωn− 1�� x x j t t j 24τ x j x j+1 x j x j+1

1 − D � �10δ− 1ωn δ− 1ωn− 1 + δ− 1ωn− 1δ− 1ωn + δ− 1ωnδ− 1ωn− 1��. t 24 x j+1 x j+1 x j x j+1 x j x j+1

Similarly, the fourth term in the left side of equation (36) is equal to

h D δ− 1]nD A qn− 1 � D �δ− 1]nD A qn− 1 − �δ− 1]nδ− 1]n+1 − δ− 1]nδ− 1]n− 1�� x x j t t j x x j t t j 24τ x j x j+1 x j x j+1 (40) 1 − D � �10δ− 1]n δ− 1]n− 1 + δ− 1]n− 1δ− 1]n + δ− 1]nδ− 1]n− 1��. t 24 x j+1 x j+1 x j x j+1 x j x j+1

*e last term in the left side of equation (36) is From equations (36)–(41), we complete the proof of β 2 2 2 2 *eorem 1. □ ��A pn � +�A qn � +�A pn− 1 � +�A qn− 1 � � 4 t j t j t j t j 4. Compact Local Momentum- n n− 1 n− 1 · ��Atpj + Atpj �DtAtpj Preserving Algorithm

n n− 1 n− 1 Now, we consider the local momentum conservation law for +�Atqj + Atqj �DtAtqj � system (15). Multiplying the ﬁrst line of equation (15) by px, and the second line of equation (15) by qx, we have 2 2 2 2 2 2 ��A pn � +�A qn � � − ��A pn− 1 � +�A qn− 1 � � β t j t j t j t j p − q p − p + p2 + q2 pp � , ( ) � ξt x α t x ωx x β� � x 0 42 4 n 2 n 2 n− 1 2 n− 1 2 ��Atpj � +�Atqj � � − ��Atpj � +�Atqj � � 2 2 ηtqx + αptqx − ]xqx + β�p + q �qqx � 0. (43)

n 2 n 2 n− 1 2 n− 1 2 ��Atpj � +�Atqj � � − ��Atpj � +�Atqj � � *en adding equations (42) to (43), we get × 2 2 τ ξtpx + ηtqx − αqtpx + αptqx − ωxpx − ]xqx + β�p + q � · pp + qq � � . 2 x x 0 β n− 1 2 n− 1 2 � Dt� ��Atp � +�Atq � � �. ( ) 4 j j 44 (41) Additionally, 6 Mathematical Problems in Engineering

1 z ξp � � ξ p + ξp � ξ p + ξξ � ξ p + z �ξ2 �, t x t x xt t x x t x x 2 1 z ηq � � η q + ηq � η q + ηη � η q + z �η2 �, t x t x xt t x x t x x 2 1 1 1 1 1 1 p q − q p � z � pq − qp � + p q − pq − q p + qp t x t x t 2 x 2 x 2 t x 2 xt 2 t x 2 xt 1 1 1 1 1 1 � zt� pqx − qpx� +� ptqx + qpxt� − � qtpx + pqxt� (45) 2 2 2 2 2 2 1 1 1 1 � z � pq − qp � + z � qp − pq �, t 2 x 2 x x 2 t 2 t 1 ω p + ] q � ω ω + ] ] � z �ω2 + ]2 �, x x x x x x 2 x

β 2 β�p2 + q2 �pp + qq � � z ��p2 + q2 � �. x x x 4

2 n− 1 2 n− 1 2 − 1 n− 1 DtAtAxξj − αDtAtAxqj − DxAt Axδx ωj *us, system (15) possesses the following local momentum conservation law: β 2 n− 1 2 2 n− 1 2 α 1 2 2 2 2 β 2 2 2 + ��At Axpj+ � +�At Axqj+ � (51) z � pq − qp � + �ξ + η + ω + ] � − �p + q � � 2 1 1 x 2 t t 2 4

α α 2 n− 1 2 2 n− 1 2 2 2 n− 1 � z �ξp + ηq + pq − qp �. +�At Axpj � +�At Axqj � ��At Axpj � � 0, t x x 2 x 2 x (46) 2 n− 1 2 n− 1 2 − 1 n− 1 DtAtAxηj + αDtAtAxpj − DxAt Axδx ]j In equations (16)–(19), applying the compact midpoint β 2 n− 1 2 2 n− 1 2 rule to space derivatives, the midpoint rule to time deriv- + ��At Axpj+ � +�At Axqj+ � (52) 2 1 1 atives, and the discrete chain rule to the nonlinear term in 2 n− 1 2 2 n− 1 2 2 2 n− 1 the spatial direction, we obtain +�At Axpj � +�At Axqj � ��At Axqj � � 0. n n n n n n Dxpj � δxAxωj ,Dxωj � δxAxφj ,Dxqj � δxAx]j , From equations (47)–(52), we obtain the following n n Dx]j � δxAxϕj , discrete scheme:

(47) 2 2 n− 1 2 n− 1 2 2 − 2 n− 1 Dt Axpj − αDtAtAxqj − DxAt δx pj n n n n Dtpj � Atξj ,Dtqj � Atηj , (48)

β 2 n− 1 2 2 n− 1 2 2 n− 1 2 n− 1 2 2 n− 1 + ��At Axpj+1 � +�At Axqj+1 � DtAtAxξj − αDtAtAxqj − At Axφj 2 β 2 2 + A2A pn− 1 + A2A qn− 1 ��t x j+1 � �t x j+1 � (49) 2 n− 1 2 2 n− 1 2 2 2 n− 1 2 +�At Axpj � +�At Axqj � ��At Axpj � � 0, 2 n− 1 2 2 n− 1 2 2 2 n− 1 +�At Axpj � +�At Axqj � ��At Axpj � � 0, (53) 2 2 n− 1 2 n− 1 2 2 − 2 n− 1 D A q + αDtAtA p − D A δ q 2 n− 1 2 n− 1 2 2 n− 1 t x j x j x t x j DtAtAxηj + αDtAtAxpj − At Axϕj

β n− 2 n− 2 2 1 2 1 β 2 n− 1 2 2 n− 1 2 + ��At Axpj+1 � +�At Axqj+1 � (50) + ��A A p � +�A A q � 2 2 t x j+1 t x j+1 2 n− 1 2 2 n− 1 2 2 2 n− 1 +�At Axpj � +�At Axqj � ��At Axqj � � 0. 2 2 +�A2A pn− 1 � +�A2A qn− 1 � ��A2A2 qn− 1 � � 0, By equations (47)–(50), we have t x j t x j t x j Mathematical Problems in Engineering 7 i.e., n− 1 n− 1 n− 1 n− 1 Dxfj Axfj+1 + Dxfj Axfj− 1 2 2 n− 1 2 n− 1 2 2 − 2 n− 1 Dt Axuj + iαDtAtAxuj − DxAt δx uj 1 n− 1 n− 1 n− 1 n− 1 n− 1 n− 1 n− 1 n− 1 � Dx�fj fj+1 + fj fj + fj fj− 1 − fj+1 fj− 1 �. � � � � 2 β � 2 n− 1�2 � 2 n− 1�2 2 2 n− 1 + ��A A u � +�A A u � ��A A u � � 0. (55) 2 t x j+1 t x j t x j (54) Proof. *e term in the left-hand side of (55) is equal to n Lemma 2. Grid function fj satisﬁes the following identical equation:

n− 1 n− 1 n− 1 n− 1 Dxfj Axfj+1 + Dxfj Axfj− 1

1 � ��fn− 1 − fn− 1 ��fn− 1 + fn− 1 � +�fn− 1 − fn− 1 ��fn− 1 + fn− 1 �� 2h j+1 j j+2 j+1 j+1 j j j− 1

1 � �fn− 1fn− 1 + fn− 1fn− 1 − fn− 1fn− 1 − fn− 1fn− 1 (56) 2h j+1 j+2 j+1 j+1 j j+2 j j+1

n− 1 n− 1 n− 1 n− 1 n− 1 n− 1 n− 1 n− 1 + fj+1 fj + fj+1 fj− 1 − fj fj − fj fj− 1 �

1 � D �fn− 1fn− 1 + fn− 1fn− 1 + fn− 1fn− 1 − fn− 1fn− 1�. 2 x j j+1 j j j j− 1 j+1 j− 1

*is completes the proof. □ Theorem 2. Scheme (54) possesses the discrete local momentum conservation law as

⎧⎨α 1 2 2 mn � D �D A A pn− 1A2A qn− 1 − D A A qn− 1A2A pn− 1� − ��A2A ξn− 1 � +�A2A ηn− 1 � � j x⎩2 t t x j t x j t t x j t x j 2 t x j t x j

1 2 − �11�A2A δ− 1ωn− 1� + A2A δ− 1ωn− 1A2A δ− 1ωn− 1 + A2A δ− 1ωnA2A δ− 1ωn− 1 24 t x x j t x x j t x x j+1 t x x j t x x j− 1

2 − 1 n− 1 2 − 1 n− 1 2 − 1 n− 1 2 2 − 1 n− 1 2 − 1 n− 1 − At Axδx ωj+1 At Axδx ωj− 1 + 11�At Axδx ]j � + At Axδx ]j At Axδx ]j+1

(57) β 2 + A2A δ− 1]nA2A δ− 1]n− 1 − A2A δ− 1]n− 1A2A δ− 1]n− 1� + ��A2A pn− 1 � t x x j t x x j− 1 t x x j+1 t x x j− 1 4 t x j

2⎬ ⎨ 2 n− 1 2 ⎫ ⎧ 2 n− 1 n− 1 2 n− 1 n− 1 +�At Axqj � � ⎭ + Dt⎩AtAxξj DxAtAxpj + AtAxηj DxAtAxqj

α ⎫⎬ + �A A2 pn− 1D A A qn− 1 − A A2 qn− 1D A A pn− 1� � 0. 2 t x j x t x j t x j x t x j ⎭

2 n− 1 That is to say, scheme (54) is a local momentum-preserving Proof. Multiplying (51) by DxAt Axpj and (52) by 2 n− 1 algorithm. DxAt Axqj and then adding them together, we derive 8 Mathematical Problems in Engineering

2 n− 1 2 n− 1 2 n− 1 2 n− 1 2 − 1 n− 1 2 n− 1 DtAtAxξj DxAt Axpj − αDtAtAxqj DxAt Axpj − DxAt Axδx ωj DxAt Axpj

2 n− 1 2 n− 1 2 n− 1 2 n− 1 2 − 1 n− 1 2 n− 1 + DtAtAxηj DxAt Axqj + αDtAtAxpj DxAt Axqj − DxAt Axδx ]j DxAt Axqj (58) β 2 2 2 2 + ��A2A pn− 1 � +�A2A qn− 1 � +�A2A pn− 1 � +�A2A qn− 1 � � 2 t x j+1 t x j+1 t x j t x j

2 2 n− 1 2 n− 1 2 2 n− 1 2 n− 1 ×�At Axpj DxAt Axpj + At Axqj DxAt Axqj � � 0.

According to discrete Leibniz rules, the ﬁrst term in the left-hand side of (58) can be expressed as

2 n− 1 2 n− 1 2 n− 1 n− 1 2 2 n− 1 n− 1 DtAtAxξj DxAt Axpj � Dt�AtAxξj DxAtAxpj � − At Axξj DtDxAtAxpj

2 n− 1 n− 1 2 2 n− 1 2 n− 1 � Dt�AtAxξj DxAtAxpj � − At Axξj DxAt Axξj (59)

1 2 � D �A A2 ξn− 1D A A pn− 1� − D � �A2A ξn− 1 � �. t t x j x t x j x 2 t x j

Similarly, from the fourth term in the left side of (58), we have

1 2 D A A2 ηn− 1D A2A qn− 1 � D �A A2 ηn− 1D A A qn− 1� − D � �A2A ηn− 1 � �. (60) t t x j x t x j t t x j x t x j x 2 t x j

*e second and ﬁfth terms in the left side of (58) are equal to

2 n− 1 2 n− 1 2 n− 1 2 n− 1 αDtAtAxpj DxAt Axqj − αDtAtAxqj DxAt Axpj

α α α � D � A A2 pn− 1D A A qn− 1 − A A2 qn− 1D A A pn− 1� + D A A2 pn− 1D A2A qn− 1 t 2 t x j x t x j 2 t x j x t x j 2 t t x j x t x j

α α α − A2A2 pn− 1D D A A qn− 1 − D A A2 qn− 1D A2A pn− 1 + A2A2 qn− 1D D A A pn− 1 2 t x j x t t x j 2 t t x j x t x j 2 t x j x t t x j (61)

α α α � D � A A2 pn− 1D A A qn− 1 − A A2 qn− 1D A A pn− 1� + D � D A A pn− 1A2A qn− 1 t 2 t x j x t x j 2 t x j x t x j x 2 t t x j t x j

α − D A A qn− 1A2A pn− 1�. 2 t t x j t x j Mathematical Problems in Engineering 9

From the third term in the left side of (58) and Lemma 2, we can obtain 2 − 1 n− 1 2 n− 1 2 − 1 n− 1 2 2 n− 1 DxAt Axδx ωj DxAt Axpj � DxAt Axδx ωj At Axδxωj

2 − 1 n− 1 2 2 2 − 1 n− 1 � DxAt Axδx ωj δx�At Axδx ωj � 1 � �D A2A δ− 1ωn− 1A2A2 δ− 1ωn− 1 + 10D A2A δ− 1ωn− 1A2A2 δ− 1ωn− 1 12 x t x x j t x x j+1 x t x x j t x x j 2 − 1 n− 1 2 2 − 1 n− 1 + DxAt Axδx ωj At Axδx ωj− 1 � (62) 1 2 � D �10�A2A δ− 1ωn− 1� + A2A δ− 1ωn− 1A2A δ− 1ωn− 1 + A2A δ− 1ωn− 1A2A δ− 1ωn− 1 24 x t x x j t x x j t x x j+1 t x x j t x x j 2 − 1 n− 1 2 − 1 n− 1 2 − 1 n− 1 2 − 1 n− 1 + At Axδx ωj At Axδx ωj− 1 − At Axδx ωj+1 At Axδx ωj− 1 �

1 2 � D �11�A2A δ− 1ωn− 1� + A2A δ− 1ωn− 1A2A δ− 1ωn− 1 24 x t x x j t x x j t x x j+1 2 − 1 n− 1 2 − 1 n− 1 2 − 1 n− 1 2 − 1 n− 1 + At Axδx ωj At Axδx ωj− 1 − At Axδx ωj+1 At Axδx ωj− 1 �. Similarly, the sixth term in the left side of (58) is equal to 2 − 1 n− 1 2 n− 1 DxAt Axδx ]j DxAt Axqj

1 2 � D �11�A2A δ− 1]n− 1� + A2A δ− 1]n− 1A2A δ− 1]n− 1 (63) 24 x t x x j t x x j t x x j+1

2 − 1 n− 1 2 − 1 n− 1 2 − 1 n− 1 2 − 1 n− 1 + At Axδx ]j At Axδx ]j− 1 − At Axδx ]j+1 At Axδx ]j− 1 �. From the last term in the left-hand side of (58), we have β 2 2 2 2 ��A2A pn− 1 � +�A2A qn− 1 � +�A2A pn− 1 � +�A2A qn− 1 � � 2 t x j+1 t x j+1 t x j t x j

2 2 n− 1 2 n− 1 2 2 n− 1 2 n− 1 × �At Axpj DxAt Axpj + At Axqj DxAt Axqj �

2 2 2 2 2 2 ��A2A pn− 1 � +�A2A qn− 1 � � − ��A2A pn− 1 � +�A2A qn− 1 � � β t x j+1 t x j+1 t x j t x j � 2 2 n− 1 2 2 n− 1 2 n− 1 2 n− 1 2 (64) ��At Axpj+1 � +�At Axqj+1 �� − ��At Axpj � +�At Axqj � �

2 n− 1 2 2 n− 1 2 n− 1 2 n− 1 2 ��At Axpj+1 � +�At Axqj+1 �� − ��At Axpj � +�At Axqj � � × 2τ

β 2 2 2 � D � ��A2A pn− 1 � +�A2A qn− 1 � � �. x 4 t x j t x j

*rough equations (58)–(64), we complete the proof of Re(Lρ) � ‖Re(ε)‖ρ, Im(Lρ) � ‖Im(ε)‖ρ, where ρ � ∞ or *eorem 2. □ ρ � 2. Tables 1–4 show the L∞ and L2 errors of the numerical solutions with respect to the exact ones for both schemes. In 5. Numerical Experiments these tables, we can confirm that the two schemes have the accuracy of O(τ2 + h4). In this section, numerical experiments are designed to show Next, we investigate the local conservation properties of the accuracies and conservation properties of the schemes the schemes (32) and (54). To compute the discrete local � � which we have obtained above. Taking α β 1, equation energy and local momentum at t � nτ, we define en � u(x, t) � i(x+t) n local (1) has exact solution e . In numerical calcu- max |en| and mn � max |mn|, where en and mn lations, we let x ∈ [− π, π]. 0≤j≤N j local 0≤j≤N j j j In order to verify the convergence rates of the proposed can be calculated by (35) and (57), respectively. In our N � h � � . schemes (32) and (54), we define ε � u t − u(x , t ) and experiments, we take T 100, π/128, and τ 0 02. j j Nt Figures 1 and 2 show the numerical results for the local 10 Mathematical Problems in Engineering

Table 1: Temporal errors and convergence rates of scheme (32) at T �1 with h � π/64. Re(L ) Order Im(L ) Order τ ρ ρ ρ � ∞ ρ � 2 ρ � ∞ ρ � 2 ρ � ∞ ρ � 2 ρ � ∞ ρ � 2 1/40 5.21E − 05 9.24E − 05 — — 5.21E − 05 9.24E − 05 — — 1/80 1.31E − 05 2.33E − 05 1.99 1.99 1.31E − 05 2.33E − 05 1.99 1.99 1/160 3.31E − 06 5.86E − 06 1.99 1.99 3.31E − 06 5.86E − 06 1.99 1.99

Table 2: Spatial errors and convergence rates of scheme (32) at T �1 with τ � 0.0005. Re(L ) Order Im(L ) Order h ρ ρ ρ � ∞ ρ � 2 ρ � ∞ ρ � 2 ρ � ∞ ρ � 2 ρ � ∞ ρ � 2 π/4 5.50E − 04 9.75E − 04 — — 5.50E − 04 9.75E − 04 — — π/8 3.38E − 05 5.99E − 05 4.02 4.02 3.38E − 05 5.99E − 05 4.02 4.02 π/16 2.12E − 06 3.76E − 06 3.99 3.99 2.12E − 06 3.76E − 06 3.99 3.99

Table 3: Temporal errors and convergence rates of scheme (54) at T �1 with h � π/64. Re(L ) Order Im(L ) Order τ ρ ρ ρ � ∞ ρ � 2 ρ � ∞ ρ � 2 ρ � ∞ ρ � 2 ρ � ∞ ρ � 2 1/40 1.56E − 04 2.77E − 04 — — 1.56E − 04 2.77E − 04 — — 1/80 3.93E − 05 6.96E − 05 1.99 1.99 3.93E − 05 6.96E − 05 1.99 1.99 1/160 9.78E − 06 1.73E − 05 2.01 2.01 9.78E − 06 1.73E − 05 2.01 2.01

Table 4: Spatial errors and convergence rates of scheme (54) at T �1 with τ � 0.0005. Re(L ) Order Im(L ) Order h ρ ρ ρ � ∞ ρ � 2 ρ � ∞ ρ � 2 ρ � ∞ ρ � 2 ρ � ∞ ρ � 2 π/4 8.02E − 03 1.42E − 02 — — 8.02E − 03 1.42E − 02 — — π/8 4.80E − 04 8.46E − 04 4.06 4.07 4.79E − 04 8.46E − 04 4.07 4.07 π/16 2.96E − 05 5.22E − 05 4.02 4.02 2.96E − 05 5.22E − 05 4.02 4.02

×10–12 ×10–12 1.4 2

1.3 1.8 1.2 1.6 1.1

1 1.4

Local energy 0.9 1.2 Local momentum 0.8 1 0.7

0.6 0.8 0 20 40 60 80 100 0 20 40 60 80 100 t t Figure Figure 1: Local energy elocal of scheme (32). 2: Local momentum mlocal of scheme (54). Mathematical Problems in Engineering 11 energy and local momentum errors using schemes (32) and [7] X. Li, L. M. Zhang, and T. Zhang, “A new numerical scheme (54), respectively. *e figures indicate that the developed for the nonlinear Schrodinger¨ equation with wave operator,” schemes can preserve the local energy and local momentum Journal of Applied Mathematics and Computing, vol. 54, no. 1- of the system very well over long-time simulations, which is 2, pp. 109–125, 2017. consistent with the theoretical results of *eorem 1 and [8] L. Guo and Y. Xu, “Energy conserving local discontinuous *eorem 2 in this paper. Galerkin methods for the nonlinear Schrödingerequation with wave operator,” Journal of Scientific Computing, vol. 65, no. 2, pp. 622–647, 2015. 6. Conclusions [9] B. Q. Ji and L. M. Zhang, “An exponential wave integrator Fourier pseudospectral method for the nonlinear Schrödinger In this paper, two new compact local structure-preserving equation with wave operator,” Journal of Applied Mathematics algorithms are constructed for solving the NSEW. Local and Computing, vol. 58, no. 1-2, pp. 273–288, 2018. conservation laws of the proposed schemes are derived [10] J. Wang, “A note on multisymplectic Fourier pseudospectral theoretically. Numerical results are shown to verify the discretization for the nonlinear Schrodinger¨ equation,” Ap- accuracy, validity, and long-time numerical behavior of the plied Mathematics and Computation, vol. 191, no. 1, pp. 31–41, schemes obtained in this work. Hence, the compact local 2007. structure-preserving method can be used for many Ham- [11] L. Wang, L. Kong, L. Zhang, W. Zhou, and X. Zheng, “Multi- iltonian systems. symplectic preserving integrator for the Schrodinger¨ equation with wave operator,” Applied Mathematical Modelling, vol. 39, Data Availability no. 22, pp. 6817–6829, 2015. [12] K. Feng and M. Z. Qin, Symplectic Geometric Algorithms for *e data of the numerical experiment used to support the Hamiltonian Systems, Springer and Zhejiang Science and findings of this study are included within the article. Technology Publishing House, Heidelberg, Germany, 2010. [13] B. Zhu, Z. Hu, Y. Tang, and R. Zhang, “Symmetric and symplectic methods for gyrocenter dynamics in time-inde- Conflicts of Interest pendent magnetic fields,” International Journal of Modeling, *e authors declare that they have no conflicts of interest. Simulation, and Scientific Computing, vol. 07, no. 2, p. 1650008, 2016. [14] T. J. Bridge and S. Reich, “Multi-symplectic integrator: nu- Acknowledgments merical schemes for Hamiltonian PDEs that conserve sym- plecticity,” Physics Letters A, vol. 284, no. 4-5, pp. 184–193, *is work was partially supported by the Key Laboratory of 2001. Applied Mathematics of Fujian Province University (Putian [15] Y. S. Wang and J. L. Hong, “Multi-symplectic algorithms for University) (no. SX201802), the Natural Science Foundation Hamiltonian partial differential equations,” Communication of China (nos. 11701196 and 11701197), the Promotion on Applied Mathematics and Computation, vol. 27, no. 2, Program for Young and Middle-Aged Teachers in Science pp. 163–230, 2013. and Technology Research of Huaqiao University (no. ZQN- [16] L. Y. Huang, Z. F. Weng, and C. Y. Lin, “Compact splitting YX502), and the Fundamental Research Funds for the symplectic scheme for the fourth-order dispersive Central Universities (no. ZQN-702). Schrodinger¨ equation with Cubic-Quintic nonlinear term,” International Journal of Modeling, Simulation, and Scientific References Computing, vol. 10, no. 2, p. 1950007, 2019. [17] J. L. Hong and L. H. Kong, “Novel multi-symplectic inte- [1] L. Berge´ and T. Colin, “A singular perturbation problem for grators for nonlinear fourth-order Schrodinger¨ equation with an envelope equation in plasma physics,” Physica D: Non- trapped term,” Communications in Computational Physics, linear Phenomena, vol. 84, no. 3-4, pp. 437–459, 1995. vol. 7, no. 3, pp. 613–630, 2010. [2] K. Matsunchi, “Nonlinear interactions of counter-travelling [18] Y. F. Tang, L. Vazquez,´ F. Zhang, and V. M. Perez-Garc´ ´ıa, waves,” Journal of the Physical Society of Japan, vol. 48, no. 5, “Symplectic methods for the nonlinear Schrödingerequa- pp. 1746–1754, 1980. tion,” Computers & Mathematics with Applications, vol. 32, [3] L. M. Zhang and Q. S. Chang, “A conservative numerical no. 5, pp. 73–83, 1996. scheme for a class of nonlinear Schrodinger¨ equation with [19] L. H. Kong, Y. Cao, L. Wang, and L. Wang, “Split-step wave operator,” Applied Mathematics and Computation, multisymplectic integrator for the fourth-order Schrödinger vol. 145, no. 2-3, pp. 603–612, 2003. equation with cubic nonlinear term,” Chinese Journal of [4] Z. Q. Liang and B. N. Lu, “*e quasi-spectrum method for Computational Physics, vol. 28, no. 5, pp. 730–736, 2011. non-linear Schrodinger¨ equation with wave operator,” Numer [20] H. C. Li, J. Q. Sun, and M. Z. Qin, “New explicit multi- Math: A Journal of Chinese Universities, vol. 21, no. 3, symplectic scheme for nonlinear wave equation,” Applied pp. 202–211, 1999. Mathematics and Mechanics, vol. 35, no. 3, pp. 369–380, 2014. [5] T. C. Wang and L. M. Zhang, “Analysis of some new con- [21] U. M. Ascher and R. I. McLachlan, “On symplectic and servative schemes for nonlinear Schrödingerequation with multisymplectic schemes for the KdV equation,” Journal of wave operator,” Applied Mathematics and Computation, Scientific Computing, vol. 25, no. 1, pp. 83–104, 2005. vol. 182, no. 2, pp. 1780–1794, 2006. [22] Y. Wang, B. Wang, and M. Qin, “Local structure-preserving [6] X. Li, L. Zhang, and S. Wang, “A compact finite difference algorithms for partial differential equations,” Science in China scheme for the nonlinear Schrödingerequation with wave Series A: Mathematics, vol. 51, no. 11, pp. 2115–2136, 2008. operator,” Applied Mathematics and Computation, vol. 219, [23] J. Cai and J. Shen, “Two classes of linearly implicit local no. 6, pp. 3187–3197, 2012. energy-preserving approach for general multi-symplectic 12 Mathematical Problems in Engineering

Hamiltonian PDEs,” Journal of Computational Physics, vol. 401, p. 108975, 2020. [24] J. Cai, C. Bai, and H. Zhang, “Decoupled local/global energy- preserving schemes for the N-coupled nonlinear Schrödinger equations,” Journal of Computational Physics, vol. 374, pp. 281–299, 2018. [25] J. Wang and Y. Wang, “Local structure-preserving algorithms for the KdV equation,” Journal of Computational Mathe- matics, vol. 35, no. 3, pp. 289–318, 2017. [26] J. Cai, Y. Wang, and H. Liang, “Local energy-preserving and momentum-preserving algorithms for coupled nonlinear Schrodinger¨ system,” Journal of Computational Physics, vol. 239, pp. 30–50, 2013. [27] J. Cai and Y. Wang, “Local structure-preserving algorithms for the "good" Boussinesq equation,” Journal of Computa- tional Physics, vol. 239, pp. 72–89, 2013. [28] Y. Yang, Y. Wang, and Y. Song, “A new local energy-preserving algorithm for the BBM equation,” Applied Mathe- matics and Computation, vol. 324, pp. 119–130, 2018. [29] J. Cai, Y. Wang, and C. Jiang, “Local structure-preserving algorithms for general multi-symplectic Hamiltonian PDEs,” Computer Physics Communications, vol. 235, pp. 210–220, 2019. [30] Z. Mu, Y. Gong, W. Cai, and Y. Wang, “Efficient local energy dissipation preserving algorithms for the Cahn-Hilliard equation,” Journal of Computational Physics, vol. 374, pp. 654–667, 2018. [31] J. Cai and H. Zhang, “Efficient schemes for the damped nonlinear Schrödingerequation in high dimensions,” Applied Mathematics Letters, vol. 102, p. 106158, 2020.