Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2015, Article ID 960602, 9 pages http://dx.doi.org/10.1155/2015/960602
Research Article A Fourth-Order Conservative Compact Finite Difference Scheme for the Generalized RLW Equation Shuguang Li, Jue Wang, and Yuesheng Luo School of Science, Harbin Engineering University, Harbin 150001, China Correspondence should be addressed to Shuguang Li;
[email protected] and Jue Wang;
[email protected] Received 31 October 2014; Accepted 29 January 2015 Academic Editor: Sandile Motsa Copyright Š 2015 Shuguang Li et al. This 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. The generalized regularized long-wave (GRLW) equation is studied by finite difference method. A new fourth-order energy conservative compact finite difference scheme was proposed. It is proved by the discrete energy method that the compact scheme is solvable, the convergence and stability of the difference schemes are obtained, and its numerical convergence order is đ(đ2 + â4 ) in the đżâ -norm. Further, the compact schemes are conservative. Numerical experiment results show that the theory is accurate and the method is efficient and reliable.
1. Introduction
where đ˘ = đ˘(đĽ, đĄ) is a real-valued function defined on (đĽđ , đĽđ ) Ă (0, đ], đź, đ˝ > 0, đ ⼠2 is a positive integer, and đ˘0 is a given function with Dirichlet boundary condition. The GRLW equation was first put forward as a model for small-amplitude long waves on the surface of water in a channel by Peregrine [1, 2]. A special case of (1), that is,
phenomena, such as shallow waves and ionic waves. The GRLW equation can also describe that wave motion to the same order of approximation as the KDV equation, so it plays a major role in the study of nonlinear dispersive waves [3]. It is difficult to find the analytical solution for (1), which has been studied by many researchers. The finite difference method for the initial-boundary value problem of the GRLW equation had been studied in [4â8]. Other mathematical theory and numerical methods for GRLW equation were considered in [9â11]. Reference [12] solved the GRLW equation by the Petrov-Galerkin method. Numerical solution of GRLW equation used Sinc-collocation method in [13]. In [14], a time-linearization method that uses a Crank-Nicolson procedure in time and three-point, fourth-order accurate in space, compact difference equations, is presented and used to determine the solutions of the generalized regularized-long wave (GRLW) equation. Recently, there has been growing interest in high-order compact methods for solving partial differential equations [15â17]. In this paper, we consider problem (1)â(3); it has the following conservation law:
đ˘đĄ + đ˘đĽ + đźđ˘đ˘đĽ â đ˝đ˘đĽđĽđĄ = 0,
óľŠ óľŠ2 đ¸ (đĄ) = âđ˘â2đż2 + đ˝ óľŠóľŠóľŠđ˘đĽ óľŠóľŠóľŠđż2 = const.
In this paper, we consider the following generalized regularized long-wave equation: đ˘đĄ + đ˘đĽ + đź (đ˘đ )đĽ â đ˝đ˘đĽđĽđĄ = 0,
(đĽ, đĄ) â (đĽđ , đĽđ ) Ă (0, đ] , (1)
with the boundary conditions đ˘ (đĽđ , đĄ) = đ˘ (đĽđ , đĄ) = 0,
đĄ â (0, đ] ,
(2)
and the initial condition đ˘ (đĽ, 0) = đ˘0 (đĽ) ,
đĽ â (đĽđ , đĽđ ) ,
(3)
(4)
is usually called the regularized long-wave (RLW) equation. The RLW equation is a representation form of nonlinear long wave and can describe a lot of important physical
(5)
Using a customary designation, we will refer to the functional đ¸(đĄ) as the energy integral, although it is not necessarily identifiable with energy in the original physical problem.
2
Mathematical Problems in Engineering
We aim to present a conservative finite difference scheme for problem (1)â(3), which simulates conservation law (5) that the differential equation (1) possesses, and prove convergence and stability of the scheme. This paper is organized as follows. In Section 2, some notations are given and some useful lemmas are proposed. In Section 3, we present a nonlinear compact conservative difference scheme, discuss its discrete conservative law, prove the existence of difference solution by Brouwer fixed point theorem, give some a priori estimates, and then prove by discrete energy method that the difference scheme is uniquely solvable, unconditionally stable and that convergence of the difference solutions with đ(đ2 + â4 ) order is based on some a priori estimates. In Section 4, numerical results are provided to test the theoretical results.
Let Ίâ = {đĽđ = đĽđ + đâ | 0 ⤠đ ⤠đ˝}. It is convenient to let đż2â (Ίâ ) denote the normed vector space as {R0đ˝ , â â
ââ }. The corresponding matrices are defined, respectively, as đ
đ ) , đ˘đ = (đ˘1đ , đ˘2đ , . . . , đ˘đ˝â1 đ
đ
10 1 1 ( đ´1 = ( 12
0 â
â
â
0
1 10 1 â
â
â
0 d d d
2. Notations and Lemmas
4 1
Let â = (đĽđ â đĽđ )/đ˝ and đ = đ/đ be the spatial and temporal step sizes, respectively. Denote đĽđ = đĽđ + đâ, (0 ⤠đ ⤠đ˝), đĄđ = đđ, (0 ⤠đ ⤠đ). Let đ˘đđ denote the approximation of đ˘(đĽđ , đĄđ ), and let
(6)
đżđĽ+ đ˘đđ =
â
đżđĽ đ˘đđ = đ˘đđ
=
đ˘đđ+1 + đ˘đđâ1 2
đżđĽâ đ˘đđ =
,
đ đ đ˘đ+1 â đ˘đâ1
,
2â đżđĄ đ˘đđ
=
đ đ˘đđ â đ˘đâ1
â
,
2đ
,
(7)
2
A2 đ˘đđ = (1 +
â2 + â đ đż đż )đ˘ . 6 đĽ đĽ đ
óľŠóľŠ đ óľŠóľŠ đ đ 1/2 óľŠóľŠđ˘ óľŠóľŠâ = (đ˘ , đ˘ )â .
(đ˘đ , Vđ )â = â â đ˘đđ Vđđ , đ=1
đ
1 â
â
â
0 d d d ) )
0 â
â
â
1
.
4 1 1 4)đ˝â1
For a simple notation, the discrete function đ is defined by đđź [(đ˘)đâ1 đť2 đżđĽ V + đť2 đżđĽ (đ˘đâ1 V)] , đ+1
(đżđĽ+ đ˘, V)â = â (đ˘, đżđĽâ V)â ,
(11)
(đżđĽ đ˘, V)â = â (đ˘, đżđĽ V)â . (12)
Lemma 2. For any real symmetric positive definite matrices đť and for đ˘, V â R0đ˝ , one can get
(đťđżđĽ đ˘, V)â = â (đťđ˘, đżđĽ V)â = â (đ˘, đťđżđĽ V)â ,
(13)
where đ
is obtained by Cholesky decomposition of đť, denoted as đť = đ
đ đ
. Proof. For đ˘, V â R0đ˝ , we have (đťđżđĽ+ đżđĽâ đ˘, V)â = (đżđĽ+ đżđĽâ đťđ˘, V)â = â (đżđĽ+ đťđ˘, đżđĽ+ V)â
(8)
= â(đťđżđĽ+ đ˘, đżđĽ+ V)â = â (đ
đ đ
đżđĽ+ đ˘, đżđĽ+ V)â = â (đ
đżđĽ+ đ˘, đ
đżđĽ+ V)â ,
The discrete đżâ -norm â â
ââ,â is defined as óľ¨ óľ¨ âđ˘ââ,â = max óľ¨óľ¨óľ¨óľ¨đ˘đ óľ¨óľ¨óľ¨óľ¨ .
1 4
(đťđżđĽ+ đżđĽâ đ˘, V)â = â (đťđżđĽ+ đ˘, đżđĽ+ V)â = â (đ
đżđĽ+ đ˘, đ
đżđĽ+ V)â ,
We now introduce the discrete đż2 -inner product and the associated norm đ˝â1
1 10)đ˝â1
Lemma 1 (see [18]). For đ˘, V â R0đ˝ , one has
đ˘đđ+1 â đ˘đđâ1
â + â đ đż đż )đ˘ , 12 đĽ đĽ đ
(10)
â1 where đť1 = đ´â1 1 , đť2 = đ´ 2 . Obviously, đ´ 1 , đ´ 2 , đť1 , đť2 are symmetric positive definite matrices. To obtain some important results, we introduce the following lemmas.
,
A1 đ˘đđ = (1 +
,
0 â
â
â
0
(0 â
â
â
0
đ (đ˘, V) =
As usual, the following notations will be used: đ đ˘đ+1 â đ˘đđ
1( đ´2 = ( 6
) )
0 â
â
â
1 10 1
( 0 â
â
â
0
R0đ˝ = {đ˘đ = (đ˘đ )đâZ | đ˘0 = đ˘đ˝ = 0} .
đ
đ
đ (đ˘đ ) = diag ((đ˘1đ ) , (đ˘2đ ) , . . . , (đ˘đ˝â1 ) ),
(đťđżđĽ đ˘, V)â = (đżđĽ đťđ˘, V)â = â(đťđ˘, đżđĽ V)â = â (đ˘, đťđżđĽ V)â . (14) (9)
Mathematical Problems in Engineering
3
Lemma 3 (see [16]). On the matrices đ´ 1 , đ´ 2 . The eigenvalues of the matrices đ´ 1 and đ´ 2 are, respectively, as follows: đ đ´ 1 ,đ =
1 2đđ (5 + cos ), 6 đ˝
đ đ´ 2 ,đ =
1 2đđ (2 + cos ), 3 đ˝ đ = 1, 2, . . . , đ˝ â 1.
Lemma 7 (see [19]). Let (đť, (â
, â
)) be a finite-dimensional inner product space, let â â
â be the associated norm, and let đ : đť â đť be continuous. Assume, moreover, that âđź > 0, âđ§ â đť, âđ§â = đź, (đ(đ§), đ§) > 0. Then, there exists a đ§â â đť such that đ(đ§â ) = 0 and âđ§â â ⤠đź. Lemma 8 (see [18]). Suppose that the discrete function đ¤â satisfies recurrence formula
(15) Lemma 4. For đ˘ â
R0đ˝ ,
we can get
óľŠ óľŠ2 3 ⤠(đť1 đ˘, đ˘)â = óľŠóľŠóľŠđ
1 đ˘óľŠóľŠóľŠâ ⤠âđ˘â2â , 2 (16) óľŠóľŠ óľŠóľŠ2 2 2 âđ˘ââ ⤠(đť2 đ˘, đ˘)â = óľŠóľŠđ
2 đ˘óľŠóľŠâ ⤠3 âđ˘ââ , where đ
đ are obtained by Cholesky decomposition of đťđ , denoted as đťđ = đ
đđ đ
đ , (đ = 1, 2). âđ˘â2â
Proof. It follows from Lemma 3 that the eigenvalues of đť1 and đť2 satisfy
đ¤đ â đ¤đâ1 ⤠đ´đđ¤đ + đľđđ¤đâ1 + đśđ đ,
(23)
where đ´, đľ, and đśđ (đ = 1, . . . , đ) are nonnegative constants. Then đ
óľŠóľŠ óľŠóľŠ 2(đ´+đľ)đ , óľŠóľŠđ¤â óľŠóľŠ ⤠(đ¤0 + đ â đśđ ) đ
(24)
đ=1
where đ is sufficiently small, such that (đ´ + đľ)đ ⤠(đ â 1)/2đ (đ > 1).
3. A Nonlinear-Implicit Conservative Scheme
(17)
In this section, we propose a nonlinear-implicit conservative scheme for the initial-boundary value problem (1)â(3) and give its numerical analysis.
This gives the spectral radius đ(đť1 ) ⤠3/2, đ(đť2 ) ⤠3, and consequently
3.1. The Nonlinear-Implicit Scheme and Its Conservative Law. Next we consider the compact finite difference scheme for problem (1)â(3) as follows:
3 1 ⤠đ đť1 ,đ ⤠, 2
đ = 1, 2, . . . , đ˝ â 1,
1 ⤠đ đť2 ,đ ⤠3,
đ = 1, 2, . . . , đ˝ â 1.
3 óľŠ óľŠ 1 ⤠óľŠóľŠóľŠđť1 óľŠóľŠóľŠ = đ (đť1 ) ⤠, 2 Thus
óľŠ óľŠ 1 ⤠óľŠóľŠóľŠđť2 óľŠóľŠóľŠ = đ (đť2 ) ⤠3. (18)
3 óľŠ óľŠ âđ˘â2â ⤠(đť1 đ˘, đ˘)â = (đ
1 đ˘, đ
1 đ˘)â ⤠óľŠóľŠóľŠđť1 óľŠóľŠóľŠ (đ˘, đ˘)â ⤠âđ˘â2â , 2 óľŠ óľŠ âđ˘â2â ⤠(đť2 đ˘, đ˘)â = (đ
2 đ˘, đ
2 đ˘)â ⤠óľŠóľŠóľŠđť2 óľŠóľŠóľŠ (đ˘, đ˘)â ⤠3 âđ˘â2â . (19)
(20)
Proof. For đ˘ â R0đ˝ , we have đđź (đ˘đâ1 đť2 đżđĽ V + đť2 đżđĽ (đ˘đâ1 V) , V)â đ+1
=
đđź [(đ˘đâ1 đť2 đżđĽ V, V)â + (đť2 đżđĽ (đ˘đâ1 V) , V)â ] đ+1
=
đđź [(đť2 đżđĽ V, đ˘đâ1 V)â â (đ˘đâ1 V, đť2 đżđĽ V)â ] = 0. đ+1
đâ1 đđź đ â1 đ đ [(đ˘đđ ) Aâ1 2 đżđĽ đ˘đ + A2 đżđĽ (đ˘đ ) ] đ+1
+ â đ â đ˝Aâ1 1 đżđĽ đżđĽ (đđĄ đ˘đ ) = 0,
1 ⤠đ ⤠đ˝ â 1, 1 ⤠đ ⤠đ â 1,
đ˘0đ = đ˘đ˝đ = 0,
1 ⤠đ ⤠đ,
đ˘đ0 = đ˘0 (đĽđ ) ,
0 ⤠đ ⤠đ˝,
where weight coefficient đ â [0, 1]. Note that we need another finite difference scheme to calculate đ˘1 , so the following scheme will be used: + â 1 đ˘đ1 â đ˝Aâ1 1 đżđĽ đżđĽ đ˘đ = đ˘0 (đĽđ ) â đ˝
(đ (đ˘, V) , V)â =
+
(25)
Lemma 5. For đ˘, V â R0đ˝ , one has (đ (đ˘, V) , V)â = 0.
đ â1 đ đđĄ đ˘đđ + đAâ1 2 đżđĽ đ˘đ + (1 â đ) A2 đżđĽ đ˘đ
â đđ˘0 (đĽđ ) (21)
Lemma 6 (see [18]). For any discrete function đ˘ â R0đ˝ and for any given đ > 0, there exists a constant đž(đ, đ), depending only on đ and đ, such that óľŠóľŠ + đ óľŠóľŠ óľŠóľŠ đ óľŠóľŠ óľŠóľŠ đ óľŠóľŠ (22) óľŠóľŠđ˘ óľŠóľŠâ,â ⤠đ óľŠóľŠđżđĽ đ˘ óľŠóľŠâ + đž (đ, đ) óľŠóľŠđ˘ óľŠóľŠâ .
đ2 đ˘0 đđ˘ (đĽđ ) â đ 0 (đĽđ ) đđĽ2 đđĽ đđ˘0 (đĽ ) . đđĽ đ
(26) The matrix form of the difference scheme (25) can be written as đđĄ đ˘đ + đđť2 đżđĽ đ˘đ + (1 â đ) đť2 đżđĽ đ˘đ +
đđź đâ1 đ [(đ˘đ ) đť2 đżđĽ đ˘đ + đť2 đżđĽ (đ˘đ ) ] đ+1
â đ˝đť1 đżđĽ+ đżđĽâ (đđĄ đ˘đ ) = 0,
1 ⤠đ ⤠đ â 1,
(27)
4
Mathematical Problems in Engineering đ˘0đ = đ˘đ˝đ = 0,
1 ⤠đ ⤠đ,
(28)
đ˘đ0 = đ˘0 (đĽđ ) ,
0 ⤠đ ⤠đ˝.
(29)
Proof. Assume that there exist đ˘0 , đ˘1 , . . . , đ˘đ which satisfy (25) as đ ⤠đ â 1; now we try to prove that đ˘đ+1 satisfy (25). We define the mapping đ : R0đ˝ â R0đ˝ as follows:
Theorem 9. Suppose that đ˘0 â đť01 (Ί); then the finite difference scheme (25) is conservative for discrete energy; that is,
đ (V) = V â đ˘đ â đ˝ (đť1 đżđĽ+ đżđĽâ V â đť1 đżđĽ+ đżđĽâ đ˘đ )
đ˝ óľŠ 1 óľŠ óľŠ2 óľŠ óľŠ2 óľŠ2 óľŠ óľŠ2 đ¸ = (óľŠóľŠóľŠóľŠđ˘đ+1 óľŠóľŠóľŠóľŠâ + óľŠóľŠóľŠđ˘đ óľŠóľŠóľŠâ ) + (óľŠóľŠóľŠóľŠđ
1 đżđĽ+ đ˘đ+1 óľŠóľŠóľŠóľŠâ + óľŠóľŠóľŠđ
1 đżđĽ+ đ˘đ óľŠóľŠóľŠâ ) 2 2
is obviously continuous. Taking in (35) the inner product with V, from Lemmas 2, 4, and 5, we obtain
đ
+ đđ (đ
2 đżđĽ đ˘đ , đ
2 đ˘đ+1 ) = đ¸đâ1 = â
â
â
= đ¸0 , (30)
+ đđť2 đżđĽ V + đđ (V, V)
(đ (V) , V)â óľŠ óľŠ2 = âVâ2â â (V, đ˘đ )â + đ˝ [óľŠóľŠóľŠđ
1 đżđĽ+ VóľŠóľŠóľŠâ â (đ
1 đżđĽ+ V, đ
1 đżđĽ+ đ˘đ )â ]
where đ
đ are obtained by Cholesky decomposition of đťđ , denoted as đťđ = đ
đđ đ
đ , (đ = 1, 2).
âĽ
đ˝ óľŠ 1 óľŠ2 óľŠ óľŠ2 óľŠ2 óľŠ (âVâ2â â óľŠóľŠóľŠđ˘đ óľŠóľŠóľŠâ ) + (óľŠóľŠóľŠđ
1 đżđĽ+ VóľŠóľŠóľŠâ â óľŠóľŠóľŠđ
1 đżđĽ+ đ˘đ óľŠóľŠóľŠâ ) 2 2
Proof. Taking an inner product of (27) with đ˘đ+1 + đ˘đâ1 , from Lemma 5, we obtain
âĽ
3đ˝ óľŠóľŠ + đ óľŠóľŠ2 1 óľŠ óľŠ2 (âVâ2â â óľŠóľŠóľŠđ˘đ óľŠóľŠóľŠâ ) â óľŠđż đ˘ óľŠ . 2 4 óľŠ đĽ óľŠâ
1 óľŠóľŠ đ+1 óľŠóľŠ2 óľŠóľŠ đâ1 óľŠóľŠ2 (óľŠóľŠđ˘ óľŠóľŠóľŠâ â óľŠóľŠóľŠđ˘ óľŠóľŠóľŠâ ) + đ(đť2 đżđĽ đ˘đ , 2đ˘đ )â 2đ óľŠ +
đ˝ óľŠóľŠ óľŠ2 óľŠ óľŠ2 (óľŠóľŠđ
đż+ đ˘đ+1 óľŠóľŠóľŠóľŠâ â óľŠóľŠóľŠóľŠđ
1 đżđĽ+ đ˘đâ1 óľŠóľŠóľŠóľŠâ ) = 0. 2đ óľŠ 1 đĽ
(31)
(đť2 đżđĽ đ˘đ , đ˘đ+1 + đ˘đâ1 )â = (đ
2 đżđĽ đ˘đ , đ
2 đ˘đ+1 )â â (đ
2 đżđĽ đ˘đâ1 , đ
2 đ˘đ )â ,
(32)
from (31)-(32), we obtain
Thus for âVâ2â = âđ˘đ â2â + (3đ˝/2)âđżđĽ+ đ˘đ â2â + 1, there exists (đ(V), V)â > 0. The existence of đ˘đ follows from Lemma 7 and consequently the existence of đ˘đ+1 = 2V â đ˘đâ1 is obtained. This completes the proof.
Lemma 11. Suppose that đ˘0 â đť01 (Ί); then there exists the estimation for the solution of problem (1)â(3): óľŠóľŠ óľŠóľŠ âđ˘âđżâ ⤠đž0 . âđ˘âđż2 ⤠đś, (37) óľŠóľŠđ˘đĽ óľŠóľŠđż2 ⤠đś, Proof. It follows from (5) that
1 óľŠóľŠ đ+1 óľŠóľŠ2 óľŠóľŠ đâ1 óľŠóľŠ2 (óľŠóľŠđ˘ óľŠóľŠóľŠâ â óľŠóľŠóľŠđ˘ óľŠóľŠóľŠâ ) 2đ óľŠ + đ [(đ
2 đżđĽ đ˘đ , đ
2 đ˘đ+1 )â â (đ
2 đżđĽ đ˘đâ1 , đ
2 đ˘đ )â ]
âđ˘âđż2 ⤠đś, (33)
âđ˘âđżâ ⤠đž0 .
(38)
(39)
Lemma 12. Suppose that đ˘0 â đť01 (Ί); then there exists the estimation for the solution of the difference scheme (25):
Let 1 óľŠóľŠ đ+1 óľŠóľŠ2 óľŠóľŠ đ óľŠóľŠ2 (óľŠóľŠđ˘ óľŠóľŠóľŠâ + óľŠóľŠđ˘ óľŠóľŠâ ) 2 óľŠ đ˝ óľŠ óľŠ2 óľŠ óľŠ2 + (óľŠóľŠóľŠóľŠđ
1 đżđĽ+ đ˘đ+1 óľŠóľŠóľŠóľŠâ + óľŠóľŠóľŠđ
1 đżđĽ+ đ˘đ óľŠóľŠóľŠâ ) 2
óľŠóľŠ óľŠóľŠ óľŠóľŠđ˘đĽ óľŠóľŠđż2 ⤠đś.
Hence, it follows from the Sobolev inequality that
đ˝ óľŠóľŠ óľŠ2 óľŠ óľŠ2 (óľŠóľŠóľŠđ
1 đżđĽ+ đ˘đ+1 óľŠóľŠóľŠóľŠâ â óľŠóľŠóľŠóľŠđ
1 đżđĽ+ đ˘đâ1 óľŠóľŠóľŠóľŠâ ) = 0. 2đ
đ¸đ =
(36)
Next we will give some a priori estimates of difference solutions.
Noting that
+
(35)
óľŠóľŠ đ óľŠóľŠ óľŠóľŠđ˘ óľŠóľŠâ ⤠đž1 , (34)
+ đđ (đ
2 đżđĽ đ˘đ , đ
2 đ˘đ+1 ) . Then, from (33), we get đ¸đ = đ¸đâ1 . This completes the proof.
3.2. Existence and Prior Estimates of Difference Solution Theorem 10. Suppose that đ˘0 â đť01 (Ί); then the finite difference scheme (25) is solvable.
óľŠóľŠ + đ óľŠóľŠ óľŠóľŠđżđĽ đ˘ óľŠóľŠâ ⤠đž2 ,
óľŠóľŠ đ óľŠóľŠ óľŠóľŠđ˘ óľŠóľŠâ,â ⤠đž3 ,
(40)
= max{â6đ¸0 /(3 â 5đđ), â2đ¸0 }, đž2 = where đž4 0 0 max{â6đ¸ /(3đ˝ â 5đđ), â2đ¸ /đ˝}, đž3 = đđž2 + đž(đ, đ)đž1 . Proof. From Theorem 9, we obtain đ˝ óľŠ 1 óľŠóľŠ đ+1 óľŠóľŠ2 óľŠóľŠ đ óľŠóľŠ2 óľŠ2 óľŠ óľŠ2 (óľŠóľŠđ˘ óľŠóľŠóľŠâ + óľŠóľŠđ˘ óľŠóľŠâ ) + (óľŠóľŠóľŠóľŠđ
1 đżđĽ+ đ˘đ+1 óľŠóľŠóľŠóľŠâ + óľŠóľŠóľŠđ
1 đżđĽ+ đ˘đ óľŠóľŠóľŠâ ) 2 óľŠ 2 óľ¨ óľ¨ â¤ đ¸0 + đđ óľ¨óľ¨óľ¨óľ¨(đ
2 đżđĽ đ˘đ , đ
2 đ˘đ+1 )â óľ¨óľ¨óľ¨óľ¨ (41) ⤠đ¸0 +
đđ óľŠóľŠ óľŠ2 óľŠ2 óľŠ (óľŠóľŠđ
2 đżđĽ đ˘đ óľŠóľŠóľŠâ + óľŠóľŠóľŠóľŠđ
2 đ˘đ+1 óľŠóľŠóľŠóľŠâ ) ; 2
Mathematical Problems in Engineering
5
then from Lemma 4, we have đ˝ óľŠ 1 óľŠóľŠ đ+1 óľŠóľŠ2 óľŠóľŠ đ óľŠóľŠ2 óľŠ2 óľŠ óľŠ2 (óľŠóľŠóľŠđ˘ óľŠóľŠóľŠâ + óľŠóľŠđ˘ óľŠóľŠâ ) + (óľŠóľŠóľŠóľŠđżđĽ+ đ˘đ+1 óľŠóľŠóľŠóľŠâ + óľŠóľŠóľŠđżđĽ+ đ˘đ óľŠóľŠóľŠâ ) 2 2 3đđ óľŠóľŠ + đ óľŠóľŠ2 óľŠóľŠ đ+1 óľŠóľŠ2 (óľŠđż đ˘ óľŠ + óľŠóľŠđ˘ óľŠóľŠóľŠâ ) . 2 óľŠ đĽ óľŠâ óľŠ
0.9 0.8
(42)
0.7 0.6 un
⤠đ¸0 + That is
1 3đđ óľŠóľŠ đ+1 óľŠóľŠ2 1 óľŠóľŠ đ óľŠóľŠ2 đ˝ óľŠóľŠ + đ+1 óľŠóľŠ2 ( â ) óľŠóľŠđ˘ óľŠóľŠóľŠâ + óľŠóľŠđ˘ óľŠóľŠâ + óľŠóľŠóľŠđżđĽ đ˘ óľŠóľŠóľŠ 2 2 óľŠ 2 2 đ˝ 3đđ óľŠóľŠ + đ óľŠóľŠ2 +( â ) óľŠđż đ˘ óľŠ â¤ đ¸0 ; 2 2 óľŠ đĽ óľŠ
0.3 0.2
(43)
0.1
where
0 â50
0 x
50
t=0 t=5 t = 10
Figure 1: Numerical solution đ˘đ of scheme with đ = 2 and đ = â = 0.1.
(45)
Energy
10
{ 2đ¸0 2đ¸0 } ,â . đž2 = max {â đ˝ â 3đđ đ˝ } } {
9.8 9.6 9.4 9.2
(46)
En
It follows from Lemma 6 that óľŠóľŠ đ óľŠóľŠ óľŠóľŠđ˘ óľŠóľŠâ,â ⤠đž3 ,
0.5 0.4
let đ be small, such that min{1/2 â 3đđ/2, đ˝/2 â 3đđ/2} > 0; then we can get óľŠóľŠóľŠđ˘đ óľŠóľŠóľŠ ⤠đž1 , óľŠóľŠóľŠđż+ đ˘đ óľŠóľŠóľŠ ⤠đž2 , (44) óľŠ óľŠâ óľŠ đĽ óľŠâ { 2đ¸0 â 0 } đž1 = max {â , 2đ¸ } , 1 â 3đđ } {
Numerical solution
1
9 8.8
where đž3 = đđž2 + đž(đ, đ)đž1 . This completes the proof.
8.6 8.4
3.3. Convergence and Stability of Difference Solution. First, we consider the truncation error of the finite difference scheme (25). Suppose that Vđđ = đ˘(đĽđ , đĄđ ), which is the solution of problem (1)â(3). Then we have
8.2 8
0
2
đđź đâ1 đ [(Vđ ) đť2 đżđĽ Vđ + đť2 đżđĽ (Vđ ) ] ; đ+1
(47)
according to Taylorâs expansion, đđđ = đ(đ2 + â4 ) can be easily obtained. Next, we consider convergence and stability of the finite difference scheme (25). Theorem 13. Suppose that đ˘0 â đť01 (Ί) and đ˘ â đś(5,3) ; then the solution of the conservative difference scheme (25) converges to the solution of problem (1)â(3) with the order đ(đ2 + â4 ) by đżâ norm. Proof. Subtracting (27) from (47), and letting đđ = Vđ â đ˘đ , we have đđ = đđĄ đđ + đđť2 đżđĽ đđ + (1 â đ) đť2 đżđĽ đđ â đ˝đť1 đżđĽ+ đżđĽâ đđĄ đđ + đ (Vđ , Vđ ) â đ (đ˘đ , đ˘đ ) ;
(48)
6
8
10
t
đđ = đđĄ Vđ + đđť2 đżđĽ Vđ + (1 â đ) đť2 đżđĽ Vđ â đ˝đť1 đżđĽ+ đżđĽâ đđĄ Vđ +
4
E(t) En
Figure 2: Discrete energy đ¸đ of scheme with đ = 2, đ = 10, and đ = â = 0.1.
taking an inner product of (48) with đđ+1 + đđâ1 , we obtain (đđ , đđ+1 + đđâ1 )â =
1 óľŠóľŠ đ+1 óľŠóľŠ2 óľŠóľŠ đâ1 óľŠóľŠ2 (óľŠóľŠđ óľŠóľŠóľŠâ â óľŠóľŠóľŠđ óľŠóľŠóľŠâ ) 2đ óľŠ +
đ˝ óľŠóľŠ óľŠ2 óľŠ óľŠ2 (óľŠóľŠóľŠđ
1 đżđĽ+ đđ+1 óľŠóľŠóľŠóľŠâ â óľŠóľŠóľŠóľŠđ
1 đżđĽ+ đđĽđâ1 óľŠóľŠóľŠóľŠâ ) 2đ
+ đ (đ
2 đżđĽ đđ , đđ+1 + đđâ1 )â + (đ (Vđ , Vđ ) â đ (đ˘đ , đ˘đ ) , đđ+1 + đđâ1 )â ;
(49)
6
Mathematical Problems in Engineering Table 1: Errors computed by the proposed compact scheme with đ = 4/3, đ = 2 and â = đ = 0.1.
đĄ
đ=0 2.6213 Ă 10â5 5.2311 Ă 10â5 7.8509 Ă 10â5 1.0476 Ă 10â4 1.3090 Ă 10â4
0.2 0.4 0.6 0.8 1.0
óľŠóľŠ đ óľŠóľŠ óľŠóľŠđ óľŠóľŠâ,â đ = 0.5 1.5794 Ă 10â5 3.1520 Ă 10â5 4.7095 Ă 10â5 6.2474 Ă 10â5 7.7647 Ă 10â5
đ = 0.25 1.2528 Ă 10â5 2.4917 Ă 10â5 3.7267 Ă 10â5 4.9503 Ă 10â5 6.1653 Ă 10â5
đ = 0.75 3.2838 Ă 10â5 6.5530 Ă 10â5 9.7948 Ă 10â5 1.3006 Ă 10â4 1.6222 Ă 10â4
đ=1 5.0401 Ă 10â5 1.0012 Ă 10â4 1.4917 Ă 10â4 1.9758 Ă 10â4 2.4638 Ă 10â4
Table 2: Errors computed by the proposed compact scheme with đ = 4/3, đ = 4 and â = đ = 0.1. đĄ
đ=0 1.2498 Ă 10â4 2.3629 Ă 10â4 3.3108 Ă 10â4 4.1205 Ă 10â4 4.9896 Ă 10â4
0.2 0.4 0.6 0.8 1.0
óľŠóľŠ đ óľŠóľŠ óľŠóľŠđ óľŠóľŠâ,â đ = 0.5 2.3183 Ă 10â4 4.5445 Ă 10â4 6.6455 Ă 10â4 8.6651 Ă 10â4 1.0667 Ă 10â3
đ = 0.25 1.7684 Ă 10â4 3.4550 Ă 10â4 5.0029 Ă 10â4 6.4180 Ă 10â4 7.7376 Ă 10â4
Ă10â4 5
from Lemma 4 and Cauchy-Schwarz inequality, we obtain
(50)
2
en
1
đ
đ+1
(đ (V , V ) â đ (đ˘ , đ˘ ) , đ
đâ1
+đ
0 â1
according to Lemmas 2 and 4, we have đ
Error
3
óľŠ2 1 óľŠ óľŠ2 óľŠ2 1 óľŠ óľŠ â¤ 3 óľŠóľŠóľŠđżđĽ đđĽđ óľŠóľŠóľŠâ + óľŠóľŠóľŠóľŠđđ+1 óľŠóľŠóľŠóľŠâ + óľŠóľŠóľŠóľŠđđâ1 óľŠóľŠóľŠóľŠâ ; 2 2
đ
đ=1 3.4464 Ă 10â4 6.7531 Ă 10â4 9.9436 Ă 10â4 1.3159 Ă 10â3 1.6550 Ă 10â3
4
(đ
2 đżđĽ đđ , đđ+1 + đđâ1 )â
đ
đ = 0.75 2.8729 Ă 10â4 5.6499 Ă 10â4 8.3064 Ă 10â4 1.0913 Ă 10â3 1.3595 Ă 10â3
â2 â3
)â
â4
đâ1 2đđź { đ˝â1 đ đ đâ1 đ đ = Aâ1 â â [(Vđđ ) Aâ1 2 đżđĽ Vđ â (đ˘đ ) 2 đżđĽ đ˘đ ] đđ { đ+1 { đ=1 đ˝â1 đ đ â1 đ đ đ} + â â [Aâ1 2 đżđĽ (Vđ ) â A2 đżđĽ (đ˘đ ) ] đđ } đ=0 } đ˝â1 óľ¨ đ óľ¨óľ¨ óľ¨óľ¨ đ óľ¨óľ¨ óľ¨óľ¨ đ óľ¨óľ¨ óľ¨ óľ¨ óľ¨ óľ¨ óľ¨ â¤ đž4 â â (óľ¨óľ¨óľ¨óľ¨Aâ1 2 đżđĽ đđ óľ¨óľ¨ + óľ¨óľ¨đđ óľ¨óľ¨) óľ¨óľ¨đđ óľ¨óľ¨ đ=0
â5 â50
0 x
50
t=5 t = 10
Figure 3: Absolute error đđ of scheme with đ = 2 and đ = â = 0.1.
Substituting (50) and (51) into (49), we obtain
đ˝â1 óľ¨ óľ¨óľ¨ đ óľ¨óľ¨ óľ¨ + đž4 â â óľ¨óľ¨óľ¨óľ¨đđđ óľ¨óľ¨óľ¨óľ¨ óľ¨óľ¨óľ¨óľ¨Aâ1 2 đżđĽ đđ óľ¨óľ¨
đ˝ óľŠóľŠ 1 óľŠóľŠ đ+1 óľŠóľŠ2 óľŠóľŠ đâ1 óľŠóľŠ2 óľŠ2 óľŠ óľŠ2 (óľŠóľŠóľŠđ óľŠóľŠóľŠâ â óľŠóľŠóľŠđ óľŠóľŠóľŠâ ) + (óľŠóľŠóľŠđ
1 đżđĽ+ đđ+1 óľŠóľŠóľŠóľŠâ â óľŠóľŠóľŠóľŠđ
1 đżđĽ+ đđĽđâ1 óľŠóľŠóľŠóľŠâ ) 2đ 2đ
đ=0
óľŠ2 óľŠ óľŠ óľŠ2 ⤠đž4 (9 óľŠóľŠóľŠđżđĽ đđ óľŠóľŠóľŠâ + 2 óľŠóľŠóľŠđđ óľŠóľŠóľŠâ ) 9óľŠ óľŠ2 9 óľŠ óľŠ2 óľŠ óľŠ2 óľŠ óľŠ2 ⤠đž4 ( óľŠóľŠóľŠóľŠđżđĽ đđ+1 óľŠóľŠóľŠóľŠâ + óľŠóľŠóľŠóľŠđżđĽ đđâ1 óľŠóľŠóľŠóľŠâ + óľŠóľŠóľŠóľŠđđ+1 óľŠóľŠóľŠóľŠâ + óľŠóľŠóľŠóľŠđđâ1 óľŠóľŠóľŠóľŠâ ) , 2 2 (51) đâ1
đâ2
where đž4 = (đđź/2(đ + 1)) max{đž0 , 2(đ â 1)đž0 đž3 , 2(đ â đâ1 1)đž3 }.
óľŠ2 óľŠ óľŠ2 óľŠ óľŠ2 1 óľŠ â¤ óľŠóľŠóľŠđđ óľŠóľŠóľŠâ + (óľŠóľŠóľŠóľŠđđ+1 óľŠóľŠóľŠóľŠâ + óľŠóľŠóľŠóľŠđđâ1 óľŠóľŠóľŠóľŠâ ) 2 óľŠ2 1 óľŠ óľŠ2 óľŠ óľŠ2 1 óľŠ + đ (3 óľŠóľŠóľŠđżđĽ đđĽđ óľŠóľŠóľŠâ + óľŠóľŠóľŠóľŠđđ+1 óľŠóľŠóľŠóľŠâ + óľŠóľŠóľŠóľŠđđâ1 óľŠóľŠóľŠóľŠâ ) 2 2 9óľŠ óľŠ2 9 óľŠ óľŠ2 óľŠ óľŠ2 óľŠ óľŠ2 + đž4 ( óľŠóľŠóľŠóľŠđżđĽ đđ+1 óľŠóľŠóľŠóľŠâ + óľŠóľŠóľŠóľŠđżđĽ đđâ1 óľŠóľŠóľŠóľŠâ + óľŠóľŠóľŠóľŠđđ+1 óľŠóľŠóľŠóľŠâ + óľŠóľŠóľŠóľŠđđâ1 óľŠóľŠóľŠóľŠâ ) , 2 2 (52)
Mathematical Problems in Engineering
7
Table 3: Comparison of âđââ,â by the compact scheme for â = đ = 0.1 with the Zhang [4] scheme for â = 0.05, đ = 0.1. đ=2
đĄ
đ=4
Compact Scheme 1.2528 Ă 10â5 2.4917 Ă 10â5 3.7267 Ă 10â5 4.9503 Ă 10â5 6.1653 Ă 10â5
0.2 0.4 0.6 0.8 1.0
Zhang [4] 7.756 Ă 10â6 1.575 Ă 10â5 2.357 Ă 10â5 3.129 Ă 10â5 3.963 Ă 10â5
Compact Scheme 1.2498 Ă 10â4 2.3629 Ă 10â4 3.3108 Ă 10â4 4.1205 Ă 10â4 4.9896 Ă 10â4
Zhang [4] 0.0049 0.0098 0.0148 0.0198 0.0239
Table 4: Comparison of âđââ,â by the compact scheme for đ = 4/3, đ = 0.25, đ = 2, đ = 0.1 and â = 0.05. đĄ
Compact Scheme 1.2536 Ă 10â5 2.4916 Ă 10â5 3.7265 Ă 10â5 4.9526 Ă 10â5 6.1667 Ă 10â5
0.2 0.4 0.6 0.8 1.0
óľŠóľŠ đ óľŠóľŠ óľŠóľŠđ óľŠóľŠâ,â C-N scheme 0.00070 0.03331 0.06337 0.08433 0.11287
Shao et al. [7] 0.00056 0.00085 0.00112 0.00141 0.00169
Table 5: Comparison of âđââ,â by the compact scheme for đ = 1.03, đ = 0.25, đ = 2, đ = â = 0.1.
Compact Scheme
óľŠóľŠ đ óľŠóľŠ óľŠóľŠđ óľŠóľŠâ,â Bakodah and Banaja [6]
Kutluay and Esen [8]
7.0938 Ă 10â5 2.1660 Ă 10â4 5.1012 Ă 10â4
1.4805 Ă 10â4 2.9961 Ă 10â4 4.5367 Ă 10â4
1.23 Ă 10â4 1.66 Ă 10â4 1.79 Ă 10â4
4 8 12
Numerical solution
1.4 1.2 1 0.8 0.6 0.4 0.2
that is, đ˝ óľŠ 1 óľŠóľŠ đ+1 óľŠóľŠ2 óľŠóľŠ đâ1 óľŠóľŠ2 óľŠ2 óľŠ óľŠ2 (óľŠóľŠđ óľŠóľŠóľŠâ â óľŠóľŠóľŠđ óľŠóľŠóľŠâ ) + (óľŠóľŠóľŠóľŠđżđĽ+ đđ+1 óľŠóľŠóľŠóľŠâ â óľŠóľŠóľŠóľŠđżđĽ+ đđĽđâ1 óľŠóľŠóľŠóľŠâ ) 2 óľŠ 2 óľŠ2 óľŠ óľŠ2 óľŠ óľŠ2 óľŠ óľŠ2 óľŠ â¤ đđž5 (óľŠóľŠóľŠóľŠđđ+1 óľŠóľŠóľŠóľŠâ + óľŠóľŠóľŠđđ óľŠóľŠóľŠâ + óľŠóľŠóľŠóľŠđđâ1 óľŠóľŠóľŠóľŠâ + óľŠóľŠóľŠóľŠđżđĽ+ đđ+1 óľŠóľŠóľŠóľŠâ
0 â50
(53)
đ˝ óľŠ 1 óľŠóľŠ đ+1 óľŠóľŠ2 óľŠóľŠ đ óľŠóľŠ2 óľŠ2 óľŠ óľŠ2 (óľŠóľŠđ óľŠóľŠóľŠâ + óľŠóľŠđ óľŠóľŠâ ) + (óľŠóľŠóľŠóľŠđżđĽ+ đđ+1 óľŠóľŠóľŠóľŠâ + óľŠóľŠóľŠđżđĽ+ đđ óľŠóľŠóľŠâ ) ; 2 óľŠ 2 (54)
then (53) can be rewritten as (55)
1â¤đâ¤đ
đľ0 ⤠[đ(đ2 + â4 )] ;
it follows from (56) that óľŠóľŠ đ óľŠóľŠ 2 4 óľŠóľŠđ óľŠóľŠâ ⤠đ (đ + â ) ,
óľŠóľŠ + đ óľŠóľŠ 2 4 óľŠóľŠđżđĽ đ óľŠóľŠâ ⤠đ (đ + â ) ,
(58)
and then, from Lemma 6, we obtain (59)
This completes the proof. (56)
Thus we can choose a fourth-order method to compute đ˘1 such that 2
t=0 t=5 t = 10
âđââ,â ⤠đ (đ2 + â4 ) .
where đž6 = max{2đž5 , 2đž5 /đ˝}. From Lemma 8, we have óľŠ óľŠ2 đľđ ⤠(đľ0 + đ sup óľŠóľŠóľŠđđ óľŠóľŠóľŠâ ) đ4đž6 đ .
50
Figure 4: Numerical solution đ˘đ of scheme with đ = 4 and đ = â = 0.1.
where đž5 = max{(1 + đ)/2 + đž4 , (9/2)đž4 , 3đ}. Let
óľŠ óľŠ2 đľđ â đľđâ1 ⤠đ óľŠóľŠóľŠđđ óľŠóľŠóľŠâ + đđž6 (đľđ + đľđâ1 ) ,
0 x
óľŠ2 óľŠ2 óľŠ óľŠ óľŠ óľŠ2 + óľŠóľŠóľŠđżđĽ+ đđ óľŠóľŠóľŠâ + óľŠóľŠóľŠóľŠđżđĽ+ đđâ1 óľŠóľŠóľŠóľŠâ ) + đ óľŠóľŠóľŠđđ óľŠóľŠóľŠâ ,
đľđ =
Dogan [11] 0.00053 0.00113 0.00175 0.00237 0.00299
un
đĄ
Raslan [10] 0.00190 0.00283 0.00403 0.00481 0.00563
(57)
Below, we can similarly prove stability of the difference solution. Theorem 14. Under the conditions of Theorem 13, the solution of conservative finite difference scheme (25) is stable by đżâ norm.
8
Mathematical Problems in Engineering Energy
10 9.8 9.6 9.4 9.2 En
9 8.8 8.6 8.4 8.2 8
0
2
4
6
8
10
t
E(t) En
Conflict of Interests
Figure 5: Discrete energy đ¸đ of scheme with đ = 4, đ = 10, and đ = â = 0.1. Ă10â3 1.5
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
Error
This research was supported by the National Natural Science Foundation of China (Grant no. 11401183) and Fundamental Research Funds for the Central Universities.
1
en = n â u n
take â = 0.1 and đ = 0.1, respectively. The errors âđđ ââ,â are listed in Tables 1-2, respectively. In Table 3, 4, and 5, the comparison of âđââ,â by the compact scheme for â = đ = 0.1 with the Zhang [4] scheme for â = 0.05, đ = 0.1 when đ = 4/3, đ = 0.25 is shown. From Table 3, we can see that our compact scheme is acceptable. Numerical results show that numerical precision depends on the choice of parameter đ. From Tables 1-2, âđđ ââ,â ⤠đ(đ2 + â4 ) is validated. We take different đ, â, and đ values and compute the errors for the solution of problem (1)â (3). Numerical results are almost identical with the above experiment result. Hence, our schemes are efficient and reliable. In Figures 1, 2, 3, 4, 5, and 6, we show the numerical solution and conservative discrete energy in each case.
0.5
References
0 â0.5 â1 â1.5 â50
0 x
50
t=5 t = 10
Figure 6: Absolute error đđ of scheme with đ = 4 and đ = â = 0.1.
4. Numerical Experiments In this section, two examples are presented to illustrate the effectiveness of the finite difference scheme (25) in [â50, 50]. The single solitary wave solution of (1) is đ˘ (đĽ, đĄ) = đ´ â
sech2/(đâ1) [đ (đĽ + đĽ0 â đđĄ)] ,
(60)
where (đ + 1) (đ â 1) ] đ´=[ 2đź
1/(đâ1)
,
đ=
đâ1 đâ1 â , (61) 2đ˝ đ
and đ, đĽ0 are arbitrary constants and đ ⼠2. Let đĽ0 = 0 in (60), đź = 1/2, đ˝ = 1, and đ˘0 (đĽ) = đ´ â
sech2/(đâ1) (đđĽ) and consider two cases: đ = 2 and đ = 4. We
[1] D. H. Peregrine, âCalculations of the development of an unduiar bore,â Journal of Fluid Mechanics, vol. 25, pp. 321â330, 1966. [2] D. H. Peregrine, âLong waves on a beach,â Journal of Fluid Mechanics, vol. 27, no. 4, pp. 815â827, 1967. [3] J. L. Bona, W. G. Pritchard, and L. R. Scott, âNumerical schemes for a model for nonlinear dispersive waves,â Journal of Computational Physics, vol. 60, no. 2, pp. 167â186, 1985. [4] L. Zhang, âA finite difference scheme for generalized regularized long-wave equation,â Applied Mathematics and Computation, vol. 168, no. 2, pp. 962â972, 2005. [5] Z. Ren, W. Wang, and D. Yu, âA new conservative finite difference method for the nonlinear regularized long wave equation,â Applied Mathematical Sciences, vol. 5, no. 41â44, pp. 2091â2096, 2011. [6] H. O. Bakodah and M. A. Banaja, âThe method of lines solution of the regularized long-wave equation using Runge-Kutta time discretization method,â Mathematical Problems in Engineering, vol. 2013, Article ID 804317, 8 pages, 2013. [7] X. Shao, G. Xue, and C. Li, âA conservative weighted finite difference scheme for regularized long wave equation,â Applied Mathematics and Computation, vol. 219, no. 17, pp. 9202â9209, 2013. [8] S. Kutluay and A. Esen, âA finite difference solution of the regularized long-wave equation,â Mathematical Problems in Engineering, vol. 2006, Article ID 85743, 14 pages, 2006. [9] A. Esen and S. Kutluay, âApplication of a lumped Galerkin method to the regularized long wave equation,â Applied Mathematics and Computation, vol. 174, no. 2, pp. 833â845, 2006. [10] K. R. Raslan, âA computational method for the regularized long wave (RLW) equation,â Applied Mathematics and Computation, vol. 167, no. 2, pp. 1101â1118, 2005.
Mathematical Problems in Engineering [11] A. Dogan, âNumerical solution of RLW equation using linear finite elements within Galerkinâs method,â Applied Mathematical Modelling, vol. 26, no. 7, pp. 771â783, 2002. [12] T. Roshan, âA Petrov-Galerkin method for solving the generalized regularized long wave (GRLW) equation,â Computers & Mathematics with Applications, vol. 63, no. 5, pp. 943â956, 2012. [13] R. Mokhtari and M. Mohammadi, âNumerical solution of GRLW equation using sinc-collocation method,â Computer Physics Communications, vol. 181, no. 7, pp. 1266â1274, 2010. [14] C. M. Garcia-Lopez and J. I. Ramos, âEffects of convection on a modified GRLW equation,â Applied Mathematics and Computation, vol. 219, no. 8, pp. 4118â4132, 2012. [15] X. Li, L. Zhang, and S. Wang, âA compact finite difference scheme for the nonlinear Schr¨odinger equation with wave operator,â Applied Mathematics and Computation, vol. 219, no. 6, pp. 3187â3197, 2012. [16] T. Wang, B. Guo, and Q. Xu, âFourth-order compact and energy conservative difference schemes for the nonlinear Schr¨odinger equation in two dimensions,â Journal of Computational Physics, vol. 243, pp. 382â399, 2013. [17] T. Wang and B. Guo, âUnconditional convergence of two conservative compact difference schemes for nonlinear Schroinger equation in one dimension,â Scientia Sinica Mathematica, vol. 41, no. 3, pp. 207â233, 2011 (Chinese). [18] Y. Zhou, Applications of Discrete Functional Analysis to the Finite Difference Method, International Academic, Beijing, China, 1990. [19] T. Wang, L. Zhang, and F. Chen, âConservative schemes for the symmetric regularized long wave equations,â Applied Mathematics and Computation, vol. 190, no. 2, pp. 1063â1080, 2007.
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Differential Equations Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at http://www.hindawi.com International Journal of
Advances in
Combinatorics Hindawi Publishing Corporation http://www.hindawi.com
Mathematical Physics Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
International Journal of Mathematics and Mathematical Sciences
Mathematical Problems in Engineering
Journal of
Mathematics Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Discrete Dynamics in Nature and Society
Journal of
Function Spaces Hindawi Publishing Corporation http://www.hindawi.com
Abstract and Applied Analysis
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
International Journal of
Journal of
Stochastic Analysis
Optimization
Hindawi Publishing Corporation http://www.hindawi.com
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Volume 2014