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

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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 + βjuj 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; ; � j � (v ; v ; ... ; v )T; v � v (7) For any u v ∈ Vh nv v 0 1 N 0 No, 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 X ujvj; j�0 Dx. v�������� t N (ii) Chain rule: � �2 � n� (13) kuk2 � h X �uj � ; n n n j�0 DtF u � � DuF u �Dtu + O(τ); � � (8) � n� kuk∞ � 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 8< 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; >8 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.
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