A Fourth-Order Conservative Compact Finite Difference Scheme for ...

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 © 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 𝑂(𝜏2 + ℎ4 ) in the 𝐿∞ -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 𝑢 = 𝑢(𝑥, 𝑡) is a real-valued function defined on (𝑥𝑙 , 𝑥𝑟 ) × (0, 𝑇], 𝛼, 𝛽 > 0, 𝑝 ≥ 2 is a positive integer, and 𝑢0 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 [4–8]. Other mathematical theory and numerical methods for GRLW equation were considered in [9–11]. 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 [15–17]. In this paper, we consider problem (1)–(3); it has the following conservation law:

𝑢𝑡 + 𝑢𝑥 + 𝛼𝑢𝑢𝑥 − 𝛽𝑢𝑥𝑥𝑡 = 0,

󵄩 󵄩2 𝐸 (𝑡) = ‖𝑢‖2𝐿2 + 𝛽 󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩𝐿2 = const.

In this paper, we consider the following generalized regularized long-wave equation: 𝑢𝑡 + 𝑢𝑥 + 𝛼 (𝑢𝑝 )𝑥 − 𝛽𝑢𝑥𝑥𝑡 = 0,

(𝑥, 𝑡) ∈ (𝑥𝑙 , 𝑥𝑟 ) × (0, 𝑇] , (1)

with the boundary conditions 𝑢 (𝑥𝑙 , 𝑡) = 𝑢 (𝑥𝑟 , 𝑡) = 0,

𝑡 ∈ (0, 𝑇] ,

(2)

and the initial condition 𝑢 (𝑥, 0) = 𝑢0 (𝑥) ,

𝑥 ∈ (𝑥𝑙 , 𝑥𝑟 ) ,

(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 𝐸(𝑡) 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)–(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 𝑂(𝜏2 + ℎ4 ) order is based on some a priori estimates. In Section 4, numerical results are provided to test the theoretical results.

Let Ωℎ = {𝑥𝑗 = 𝑥𝑙 + 𝑗ℎ | 0 ≤ 𝑗 ≤ 𝐽}. It is convenient to let 𝐿2ℎ (Ωℎ ) denote the normed vector space as {R0𝐽 , ‖ ⋅ ‖ℎ }. The corresponding matrices are defined, respectively, as 𝑇

𝑛 ) , 𝑢𝑛 = (𝑢1𝑛 , 𝑢2𝑛 , . . . , 𝑢𝐽−1 𝑝

𝑝

10 1 1 ( 𝐴1 = ( 12

0 ⋅⋅⋅ 0

1 10 1 ⋅ ⋅ ⋅ 0 d d d

2. Notations and Lemmas

4 1

Let ℎ = (𝑥𝑟 − 𝑥𝑙 )/𝐽 and 𝜏 = 𝑇/𝑁 be the spatial and temporal step sizes, respectively. Denote 𝑥𝑗 = 𝑥𝑙 + 𝑗ℎ, (0 ≤ 𝑗 ≤ 𝐽), 𝑡𝑛 = 𝑛𝜏, (0 ≤ 𝑛 ≤ 𝑁). Let 𝑢𝑗𝑛 denote the approximation of 𝑢(𝑥𝑗 , 𝑡𝑛 ), and let

(6)

𝛿𝑥+ 𝑢𝑗𝑛 =

ℎ

𝛿𝑥 𝑢𝑗𝑛 = 𝑢𝑛𝑗

=

𝑢𝑗𝑛+1 + 𝑢𝑗𝑛−1 2

𝛿𝑥− 𝑢𝑗𝑛 =

,

𝑛 𝑛 𝑢𝑗+1 − 𝑢𝑗−1

,

2ℎ 𝛿𝑡 𝑢𝑗𝑛

=

𝑛 𝑢𝑗𝑛 − 𝑢𝑗−1

ℎ

,

2𝜏

,

(7)

2

A2 𝑢𝑗𝑛 = (1 +

ℎ2 + − 𝑛 𝛿 𝛿 )𝑢 . 6 𝑥 𝑥 𝑗

󵄩󵄩 𝑛 󵄩󵄩 𝑛 𝑛 1/2 󵄩󵄩𝑢 󵄩󵄩ℎ = (𝑢 , 𝑢 )ℎ .

(𝑢𝑛 , V𝑛 )ℎ = ℎ ∑ 𝑢𝑗𝑛 V𝑗𝑛 , 𝑗=1

𝑗

1 ⋅⋅⋅ 0 d d d ) )

0 ⋅⋅⋅ 1

.

4 1 1 4)𝐽−1

For a simple notation, the discrete function 𝜑 is defined by 𝑝𝛼 [(𝑢)𝑝−1 𝐻2 𝛿𝑥 V + 𝐻2 𝛿𝑥 (𝑢𝑝−1 V)] , 𝑝+1

(𝛿𝑥+ 𝑢, V)ℎ = − (𝑢, 𝛿𝑥− V)ℎ ,

(11)

(𝛿𝑥 𝑢, V)ℎ = − (𝑢, 𝛿𝑥 V)ℎ . (12)

Lemma 2. For any real symmetric positive definite matrices 𝐻 and for 𝑢, V ∈ R0𝐽 , one can get

(𝐻𝛿𝑥 𝑢, V)ℎ = − (𝐻𝑢, 𝛿𝑥 V)ℎ = − (𝑢, 𝐻𝛿𝑥 V)ℎ ,

(13)

where 𝑅 is obtained by Cholesky decomposition of 𝐻, denoted as 𝐻 = 𝑅𝑇 𝑅. Proof. For 𝑢, V ∈ R0𝐽 , we have (𝐻𝛿𝑥+ 𝛿𝑥− 𝑢, V)ℎ = (𝛿𝑥+ 𝛿𝑥− 𝐻𝑢, V)ℎ = − (𝛿𝑥+ 𝐻𝑢, 𝛿𝑥+ V)ℎ

(8)

= −(𝐻𝛿𝑥+ 𝑢, 𝛿𝑥+ V)ℎ = − (𝑅𝑇 𝑅𝛿𝑥+ 𝑢, 𝛿𝑥+ V)ℎ = − (𝑅𝛿𝑥+ 𝑢, 𝑅𝛿𝑥+ V)ℎ ,

The discrete 𝐿∞ -norm ‖ ⋅ ‖∞,ℎ is defined as 󵄨 󵄨 ‖𝑢‖∞,ℎ = max 󵄨󵄨󵄨󵄨𝑢𝑗 󵄨󵄨󵄨󵄨 .

1 4

(𝐻𝛿𝑥+ 𝛿𝑥− 𝑢, V)ℎ = − (𝐻𝛿𝑥+ 𝑢, 𝛿𝑥+ V)ℎ = − (𝑅𝛿𝑥+ 𝑢, 𝑅𝛿𝑥+ V)ℎ ,

We now introduce the discrete 𝐿2 -inner product and the associated norm 𝐽−1

1 10)𝐽−1

Lemma 1 (see [18]). For 𝑢, V ∈ R0𝐽 , one has

𝑢𝑗𝑛+1 − 𝑢𝑗𝑛−1

ℎ + − 𝑛 𝛿 𝛿 )𝑢 , 12 𝑥 𝑥 𝑗

(10)

−1 where 𝐻1 = 𝐴−1 1 , 𝐻2 = 𝐴 2 . Obviously, 𝐴 1 , 𝐴 2 , 𝐻1 , 𝐻2 are symmetric positive definite matrices. To obtain some important results, we introduce the following lemmas.

,

A1 𝑢𝑗𝑛 = (1 +

,

0 ⋅⋅⋅ 0

(0 ⋅ ⋅ ⋅ 0

𝜑 (𝑢, V) =

As usual, the following notations will be used: 𝑛 𝑢𝑗+1 − 𝑢𝑗𝑛

1( 𝐴2 = ( 6

) )

0 ⋅ ⋅ ⋅ 1 10 1

( 0 ⋅⋅⋅ 0

R0𝐽 = {𝑢𝑗 = (𝑢𝑗 )𝑗∈Z | 𝑢0 = 𝑢𝐽 = 0} .

𝑝

𝑝

𝑛 (𝑢𝑛 ) = diag ((𝑢1𝑛 ) , (𝑢2𝑛 ) , . . . , (𝑢𝐽−1 ) ),

(𝐻𝛿𝑥 𝑢, V)ℎ = (𝛿𝑥 𝐻𝑢, V)ℎ = −(𝐻𝑢, 𝛿𝑥 V)ℎ = − (𝑢, 𝐻𝛿𝑥 V)ℎ . (14) (9)


3

Lemma 3 (see [16]). On the matrices 𝐴 1 , 𝐴 2 . The eigenvalues of the matrices 𝐴 1 and 𝐴 2 are, respectively, as follows: 𝜆 𝐴 1 ,𝑖 =

1 2𝑖𝜋 (5 + cos ), 6 𝐽

𝜆 𝐴 2 ,𝑖 =

1 2𝑖𝜋 (2 + cos ), 3 𝐽 𝑖 = 1, 2, . . . , 𝐽 − 1.

Lemma 7 (see [19]). Let (𝐻, (⋅, ⋅)) be a finite-dimensional inner product space, let ‖ ⋅ ‖ be the associated norm, and let 𝑔 : 𝐻 → 𝐻 be continuous. Assume, moreover, that ∃𝛼 > 0, ∀𝑧 ∈ 𝐻, ‖𝑧‖ = 𝛼, (𝑔(𝑧), 𝑧) > 0. Then, there exists a 𝑧∗ ∈ 𝐻 such that 𝑔(𝑧∗ ) = 0 and ‖𝑧∗ ‖ ≤ 𝛼. Lemma 8 (see [18]). Suppose that the discrete function 𝑤ℎ satisfies recurrence formula

(15) Lemma 4. For 𝑢 ∈

R0𝐽 ,

we can get

󵄩 󵄩2 3 ≤ (𝐻1 𝑢, 𝑢)ℎ = 󵄩󵄩󵄩𝑅1 𝑢󵄩󵄩󵄩ℎ ≤ ‖𝑢‖2ℎ , 2 (16) 󵄩󵄩 󵄩󵄩2 2 2 ‖𝑢‖ℎ ≤ (𝐻2 𝑢, 𝑢)ℎ = 󵄩󵄩𝑅2 𝑢󵄩󵄩ℎ ≤ 3 ‖𝑢‖ℎ , where 𝑅𝑙 are obtained by Cholesky decomposition of 𝐻𝑙 , denoted as 𝐻𝑙 = 𝑅𝑙𝑇 𝑅𝑙 , (𝑙 = 1, 2). ‖𝑢‖2ℎ

Proof. It follows from Lemma 3 that the eigenvalues of 𝐻1 and 𝐻2 satisfy

𝑤𝑛 − 𝑤𝑛−1 ≤ 𝐴𝜏𝑤𝑛 + 𝐵𝜏𝑤𝑛−1 + 𝐶𝑛 𝜏,

(23)

where 𝐴, 𝐵, and 𝐶𝑛 (𝑛 = 1, . . . , 𝑁) are nonnegative constants. Then 𝑁

󵄩󵄩 󵄩󵄩 2(𝐴+𝐵)𝜏 , 󵄩󵄩𝑤ℎ 󵄩󵄩 ≤ (𝑤0 + 𝜏 ∑ 𝐶𝑘 ) 𝑒

(24)

𝑘=1

where 𝜏 is sufficiently small, such that (𝐴 + 𝐵)𝜏 ≤ (𝑁 − 1)/2𝑁 (𝑁 > 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)–(3) and give its numerical analysis.

This gives the spectral radius 𝜌(𝐻1 ) ≤ 3/2, 𝜌(𝐻2 ) ≤ 3, and consequently

3.1. The Nonlinear-Implicit Scheme and Its Conservative Law. Next we consider the compact finite difference scheme for problem (1)–(3) as follows:

3 1 ≤ 𝜆 𝐻1 ,𝑘 ≤ , 2

𝑘 = 1, 2, . . . , 𝐽 − 1,

1 ≤ 𝜆 𝐻2 ,𝑘 ≤ 3,

𝑘 = 1, 2, . . . , 𝐽 − 1.

3 󵄩 󵄩 1 ≤ 󵄩󵄩󵄩𝐻1 󵄩󵄩󵄩 = 𝜌 (𝐻1 ) ≤ , 2 Thus

󵄩 󵄩 1 ≤ 󵄩󵄩󵄩𝐻2 󵄩󵄩󵄩 = 𝜌 (𝐻2 ) ≤ 3. (18)

3 󵄩 󵄩 ‖𝑢‖2ℎ ≤ (𝐻1 𝑢, 𝑢)ℎ = (𝑅1 𝑢, 𝑅1 𝑢)ℎ ≤ 󵄩󵄩󵄩𝐻1 󵄩󵄩󵄩 (𝑢, 𝑢)ℎ ≤ ‖𝑢‖2ℎ , 2 󵄩 󵄩 ‖𝑢‖2ℎ ≤ (𝐻2 𝑢, 𝑢)ℎ = (𝑅2 𝑢, 𝑅2 𝑢)ℎ ≤ 󵄩󵄩󵄩𝐻2 󵄩󵄩󵄩 (𝑢, 𝑢)ℎ ≤ 3 ‖𝑢‖2ℎ . (19)

(20)

Proof. For 𝑢 ∈ R0𝐽 , we have 𝑝𝛼 (𝑢𝑝−1 𝐻2 𝛿𝑥 V + 𝐻2 𝛿𝑥 (𝑢𝑝−1 V) , V)ℎ 𝑝+1

=

𝑝𝛼 [(𝑢𝑝−1 𝐻2 𝛿𝑥 V, V)ℎ + (𝐻2 𝛿𝑥 (𝑢𝑝−1 V) , V)ℎ ] 𝑝+1

=

𝑝𝛼 [(𝐻2 𝛿𝑥 V, 𝑢𝑝−1 V)ℎ − (𝑢𝑝−1 V, 𝐻2 𝛿𝑥 V)ℎ ] = 0. 𝑝+1

𝑝−1 𝑝𝛼 𝑛 −1 𝑛 𝑝 [(𝑢𝑛𝑗 ) A−1 2 𝛿𝑥 𝑢𝑗 + A2 𝛿𝑥 (𝑢𝑗 ) ] 𝑝+1

+ − 𝑛 − 𝛽A−1 1 𝛿𝑥 𝛿𝑥 (𝜕𝑡 𝑢𝑗 ) = 0,

1 ≤ 𝑗 ≤ 𝐽 − 1, 1 ≤ 𝑛 ≤ 𝑁 − 1,

𝑢0𝑛 = 𝑢𝐽𝑛 = 0,

1 ≤ 𝑛 ≤ 𝑁,

𝑢𝑗0 = 𝑢0 (𝑥𝑗 ) ,

0 ≤ 𝑗 ≤ 𝐽,

where weight coefficient 𝜃 ∈ [0, 1]. Note that we need another finite difference scheme to calculate 𝑢1 , so the following scheme will be used: + − 1 𝑢𝑗1 − 𝛽A−1 1 𝛿𝑥 𝛿𝑥 𝑢𝑗 = 𝑢0 (𝑥𝑗 ) − 𝛽

(𝜑 (𝑢, V) , V)ℎ =

+

(25)

Lemma 5. For 𝑢, V ∈ R0𝐽 , one has (𝜑 (𝑢, V) , V)ℎ = 0.

𝑛 −1 𝑛 𝜕𝑡 𝑢𝑗𝑛 + 𝜃A−1 2 𝛿𝑥 𝑢𝑗 + (1 − 𝜃) A2 𝛿𝑥 𝑢𝑗

− 𝜏𝑢0 (𝑥𝑗 ) (21)

Lemma 6 (see [18]). For any discrete function 𝑢 ∈ R0𝐽 and for any given 𝜀 > 0, there exists a constant 𝐾(𝜀, 𝑛), depending only on 𝜀 and 𝑛, such that 󵄩󵄩 + 𝑛 󵄩󵄩 󵄩󵄩 𝑛 󵄩󵄩 󵄩󵄩 𝑛 󵄩󵄩 (22) 󵄩󵄩𝑢 󵄩󵄩∞,ℎ ≤ 𝜀 󵄩󵄩𝛿𝑥 𝑢 󵄩󵄩ℎ + 𝐾 (𝜀, 𝑛) 󵄩󵄩𝑢 󵄩󵄩ℎ .

𝜕2 𝑢0 𝜕𝑢 (𝑥𝑗 ) − 𝜏 0 (𝑥𝑗 ) 𝜕𝑥2 𝜕𝑥 𝜕𝑢0 (𝑥 ) . 𝜕𝑥 𝑗

(26) The matrix form of the difference scheme (25) can be written as 𝜕𝑡 𝑢𝑛 + 𝜃𝐻2 𝛿𝑥 𝑢𝑛 + (1 − 𝜃) 𝐻2 𝛿𝑥 𝑢𝑛 +

𝑝𝛼 𝑝−1 𝑝 [(𝑢𝑛 ) 𝐻2 𝛿𝑥 𝑢𝑛 + 𝐻2 𝛿𝑥 (𝑢𝑛 ) ] 𝑝+1

− 𝛽𝐻1 𝛿𝑥+ 𝛿𝑥− (𝜕𝑡 𝑢𝑛 ) = 0,

1 ≤ 𝑛 ≤ 𝑁 − 1,

(27)

4

Mathematical Problems in Engineering 𝑢0𝑛 = 𝑢𝐽𝑛 = 0,

1 ≤ 𝑛 ≤ 𝑁,

(28)

𝑢𝑗0 = 𝑢0 (𝑥𝑗 ) ,

0 ≤ 𝑗 ≤ 𝐽.

(29)

Proof. Assume that there exist 𝑢0 , 𝑢1 , . . . , 𝑢𝑛 which satisfy (25) as 𝑛 ≤ 𝑁 − 1; now we try to prove that 𝑢𝑛+1 satisfy (25). We define the mapping 𝑔 : R0𝐽 → R0𝐽 as follows:

Theorem 9. Suppose that 𝑢0 ∈ 𝐻01 (Ω); then the finite difference scheme (25) is conservative for discrete energy; that is,

𝑔 (V) = V − 𝑢𝑛 − 𝛽 (𝐻1 𝛿𝑥+ 𝛿𝑥− V − 𝐻1 𝛿𝑥+ 𝛿𝑥− 𝑢𝑛 )

𝛽 󵄩 1 󵄩 󵄩2 󵄩 󵄩2 󵄩2 󵄩 󵄩2 𝐸 = (󵄩󵄩󵄩󵄩𝑢𝑛+1 󵄩󵄩󵄩󵄩ℎ + 󵄩󵄩󵄩𝑢𝑛 󵄩󵄩󵄩ℎ ) + (󵄩󵄩󵄩󵄩𝑅1 𝛿𝑥+ 𝑢𝑛+1 󵄩󵄩󵄩󵄩ℎ + 󵄩󵄩󵄩𝑅1 𝛿𝑥+ 𝑢𝑛 󵄩󵄩󵄩ℎ ) 2 2

is obviously continuous. Taking in (35) the inner product with V, from Lemmas 2, 4, and 5, we obtain

𝑛

+ 𝜏𝜃 (𝑅2 𝛿𝑥 𝑢𝑛 , 𝑅2 𝑢𝑛+1 ) = 𝐸𝑛−1 = ⋅ ⋅ ⋅ = 𝐸0 , (30)

+ 𝜏𝐻2 𝛿𝑥 V + 𝜏𝜑 (V, V)

(𝑔 (V) , V)ℎ 󵄩 󵄩2 = ‖V‖2ℎ − (V, 𝑢𝑛 )ℎ + 𝛽 [󵄩󵄩󵄩𝑅1 𝛿𝑥+ V󵄩󵄩󵄩ℎ − (𝑅1 𝛿𝑥+ V, 𝑅1 𝛿𝑥+ 𝑢𝑛 )ℎ ]

where 𝑅𝑙 are obtained by Cholesky decomposition of 𝐻𝑙 , denoted as 𝐻𝑙 = 𝑅𝑙𝑇 𝑅𝑙 , (𝑙 = 1, 2).

≥

𝛽 󵄩 1 󵄩2 󵄩 󵄩2 󵄩2 󵄩 (‖V‖2ℎ − 󵄩󵄩󵄩𝑢𝑛 󵄩󵄩󵄩ℎ ) + (󵄩󵄩󵄩𝑅1 𝛿𝑥+ V󵄩󵄩󵄩ℎ − 󵄩󵄩󵄩𝑅1 𝛿𝑥+ 𝑢𝑛 󵄩󵄩󵄩ℎ ) 2 2

Proof. Taking an inner product of (27) with 𝑢𝑛+1 + 𝑢𝑛−1 , from Lemma 5, we obtain

≥

3𝛽 󵄩󵄩 + 𝑛 󵄩󵄩2 1 󵄩 󵄩2 (‖V‖2ℎ − 󵄩󵄩󵄩𝑢𝑛 󵄩󵄩󵄩ℎ ) − 󵄩𝛿 𝑢 󵄩 . 2 4 󵄩 𝑥 󵄩ℎ

1 󵄩󵄩 𝑛+1 󵄩󵄩2 󵄩󵄩 𝑛−1 󵄩󵄩2 (󵄩󵄩𝑢 󵄩󵄩󵄩ℎ − 󵄩󵄩󵄩𝑢 󵄩󵄩󵄩ℎ ) + 𝜃(𝐻2 𝛿𝑥 𝑢𝑛 , 2𝑢𝑛 )ℎ 2𝜏 󵄩 +

𝛽 󵄩󵄩 󵄩2 󵄩 󵄩2 (󵄩󵄩𝑅 𝛿+ 𝑢𝑛+1 󵄩󵄩󵄩󵄩ℎ − 󵄩󵄩󵄩󵄩𝑅1 𝛿𝑥+ 𝑢𝑛−1 󵄩󵄩󵄩󵄩ℎ ) = 0. 2𝜏 󵄩 1 𝑥

(31)

(𝐻2 𝛿𝑥 𝑢𝑛 , 𝑢𝑛+1 + 𝑢𝑛−1 )ℎ = (𝑅2 𝛿𝑥 𝑢𝑛 , 𝑅2 𝑢𝑛+1 )ℎ − (𝑅2 𝛿𝑥 𝑢𝑛−1 , 𝑅2 𝑢𝑛 )ℎ ,

(32)

from (31)-(32), we obtain

Thus for ‖V‖2ℎ = ‖𝑢𝑛 ‖2ℎ + (3𝛽/2)‖𝛿𝑥+ 𝑢𝑛 ‖2ℎ + 1, there exists (𝑔(V), V)ℎ > 0. The existence of 𝑢𝑛 follows from Lemma 7 and consequently the existence of 𝑢𝑛+1 = 2V − 𝑢𝑛−1 is obtained. This completes the proof.

Lemma 11. Suppose that 𝑢0 ∈ 𝐻01 (Ω); then there exists the estimation for the solution of problem (1)–(3): 󵄩󵄩 󵄩󵄩 ‖𝑢‖𝐿∞ ≤ 𝐾0 . ‖𝑢‖𝐿2 ≤ 𝐶, (37) 󵄩󵄩𝑢𝑥 󵄩󵄩𝐿2 ≤ 𝐶, Proof. It follows from (5) that

1 󵄩󵄩 𝑛+1 󵄩󵄩2 󵄩󵄩 𝑛−1 󵄩󵄩2 (󵄩󵄩𝑢 󵄩󵄩󵄩ℎ − 󵄩󵄩󵄩𝑢 󵄩󵄩󵄩ℎ ) 2𝜏 󵄩 + 𝜃 [(𝑅2 𝛿𝑥 𝑢𝑛 , 𝑅2 𝑢𝑛+1 )ℎ − (𝑅2 𝛿𝑥 𝑢𝑛−1 , 𝑅2 𝑢𝑛 )ℎ ]

‖𝑢‖𝐿2 ≤ 𝐶, (33)

‖𝑢‖𝐿∞ ≤ 𝐾0 .

(38)

(39)

Lemma 12. Suppose that 𝑢0 ∈ 𝐻01 (Ω); then there exists the estimation for the solution of the difference scheme (25):

Let 1 󵄩󵄩 𝑛+1 󵄩󵄩2 󵄩󵄩 𝑛 󵄩󵄩2 (󵄩󵄩𝑢 󵄩󵄩󵄩ℎ + 󵄩󵄩𝑢 󵄩󵄩ℎ ) 2 󵄩 𝛽 󵄩 󵄩2 󵄩 󵄩2 + (󵄩󵄩󵄩󵄩𝑅1 𝛿𝑥+ 𝑢𝑛+1 󵄩󵄩󵄩󵄩ℎ + 󵄩󵄩󵄩𝑅1 𝛿𝑥+ 𝑢𝑛 󵄩󵄩󵄩ℎ ) 2

󵄩󵄩 󵄩󵄩 󵄩󵄩𝑢𝑥 󵄩󵄩𝐿2 ≤ 𝐶.

Hence, it follows from the Sobolev inequality that

𝛽 󵄩󵄩 󵄩2 󵄩 󵄩2 (󵄩󵄩󵄩𝑅1 𝛿𝑥+ 𝑢𝑛+1 󵄩󵄩󵄩󵄩ℎ − 󵄩󵄩󵄩󵄩𝑅1 𝛿𝑥+ 𝑢𝑛−1 󵄩󵄩󵄩󵄩ℎ ) = 0. 2𝜏

𝐸𝑛 =

(36)

Next we will give some a priori estimates of difference solutions.

Noting that

+

(35)

󵄩󵄩 𝑛 󵄩󵄩 󵄩󵄩𝑢 󵄩󵄩ℎ ≤ 𝐾1 , (34)

+ 𝜏𝜃 (𝑅2 𝛿𝑥 𝑢𝑛 , 𝑅2 𝑢𝑛+1 ) . Then, from (33), we get 𝐸𝑛 = 𝐸𝑛−1 . This completes the proof.

3.2. Existence and Prior Estimates of Difference Solution Theorem 10. Suppose that 𝑢0 ∈ 𝐻01 (Ω); then the finite difference scheme (25) is solvable.

󵄩󵄩 + 𝑛 󵄩󵄩 󵄩󵄩𝛿𝑥 𝑢 󵄩󵄩ℎ ≤ 𝐾2 ,

󵄩󵄩 𝑛 󵄩󵄩 󵄩󵄩𝑢 󵄩󵄩∞,ℎ ≤ 𝐾3 ,

(40)

= max{√6𝐸0 /(3 − 5𝜃𝜏), √2𝐸0 }, 𝐾2 = where 𝐾4 0 0 max{√6𝐸 /(3𝛽 − 5𝜃𝜏), √2𝐸 /𝛽}, 𝐾3 = 𝜀𝐾2 + 𝐾(𝜀, 𝑛)𝐾1 . Proof. From Theorem 9, we obtain 𝛽 󵄩 1 󵄩󵄩 𝑛+1 󵄩󵄩2 󵄩󵄩 𝑛 󵄩󵄩2 󵄩2 󵄩 󵄩2 (󵄩󵄩𝑢 󵄩󵄩󵄩ℎ + 󵄩󵄩𝑢 󵄩󵄩ℎ ) + (󵄩󵄩󵄩󵄩𝑅1 𝛿𝑥+ 𝑢𝑛+1 󵄩󵄩󵄩󵄩ℎ + 󵄩󵄩󵄩𝑅1 𝛿𝑥+ 𝑢𝑛 󵄩󵄩󵄩ℎ ) 2 󵄩 2 󵄨 󵄨 ≤ 𝐸0 + 𝜏𝜃 󵄨󵄨󵄨󵄨(𝑅2 𝛿𝑥 𝑢𝑛 , 𝑅2 𝑢𝑛+1 )ℎ 󵄨󵄨󵄨󵄨 (41) ≤ 𝐸0 +

𝜏𝜃 󵄩󵄩 󵄩2 󵄩2 󵄩 (󵄩󵄩𝑅2 𝛿𝑥 𝑢𝑛 󵄩󵄩󵄩ℎ + 󵄩󵄩󵄩󵄩𝑅2 𝑢𝑛+1 󵄩󵄩󵄩󵄩ℎ ) ; 2


5

then from Lemma 4, we have 𝛽 󵄩 1 󵄩󵄩 𝑛+1 󵄩󵄩2 󵄩󵄩 𝑛 󵄩󵄩2 󵄩2 󵄩 󵄩2 (󵄩󵄩󵄩𝑢 󵄩󵄩󵄩ℎ + 󵄩󵄩𝑢 󵄩󵄩ℎ ) + (󵄩󵄩󵄩󵄩𝛿𝑥+ 𝑢𝑛+1 󵄩󵄩󵄩󵄩ℎ + 󵄩󵄩󵄩𝛿𝑥+ 𝑢𝑛 󵄩󵄩󵄩ℎ ) 2 2 3𝜏𝜃 󵄩󵄩 + 𝑛 󵄩󵄩2 󵄩󵄩 𝑛+1 󵄩󵄩2 (󵄩𝛿 𝑢 󵄩 + 󵄩󵄩𝑢 󵄩󵄩󵄩ℎ ) . 2 󵄩 𝑥 󵄩ℎ 󵄩

0.9 0.8

(42)

0.7 0.6 un

≤ 𝐸0 + That is

1 3𝜏𝜃 󵄩󵄩 𝑛+1 󵄩󵄩2 1 󵄩󵄩 𝑛 󵄩󵄩2 𝛽 󵄩󵄩 + 𝑛+1 󵄩󵄩2 ( − ) 󵄩󵄩𝑢 󵄩󵄩󵄩ℎ + 󵄩󵄩𝑢 󵄩󵄩ℎ + 󵄩󵄩󵄩𝛿𝑥 𝑢 󵄩󵄩󵄩 2 2 󵄩 2 2 𝛽 3𝜏𝜃 󵄩󵄩 + 𝑛 󵄩󵄩2 +( − ) 󵄩𝛿 𝑢 󵄩 ≤ 𝐸0 ; 2 2 󵄩 𝑥 󵄩

0.3 0.2

(43)

0.1

where

0 −50

0 x

50

t=0 t=5 t = 10

Figure 1: Numerical solution 𝑢𝑛 of scheme with 𝑝 = 2 and 𝜏 = ℎ = 0.1.

(45)

Energy

10

{ 2𝐸0 2𝐸0 } ,√ . 𝐾2 = max {√ 𝛽 − 3𝜃𝜏 𝛽 } } {

9.8 9.6 9.4 9.2

(46)

En

It follows from Lemma 6 that 󵄩󵄩 𝑛 󵄩󵄩 󵄩󵄩𝑢 󵄩󵄩∞,ℎ ≤ 𝐾3 ,

0.5 0.4

let 𝜏 be small, such that min{1/2 − 3𝜏𝜃/2, 𝛽/2 − 3𝜏𝜃/2} > 0; then we can get 󵄩󵄩󵄩𝑢𝑛 󵄩󵄩󵄩 ≤ 𝐾1 , 󵄩󵄩󵄩𝛿+ 𝑢𝑛 󵄩󵄩󵄩 ≤ 𝐾2 , (44) 󵄩 󵄩ℎ 󵄩 𝑥 󵄩ℎ { 2𝐸0 √ 0 } 𝐾1 = max {√ , 2𝐸 } , 1 − 3𝜃𝜏 } {

Numerical solution

1

9 8.8

where 𝐾3 = 𝜀𝐾2 + 𝐾(𝜀, 𝑛)𝐾1 . 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 V𝑗𝑛 = 𝑢(𝑥𝑗 , 𝑡𝑛 ), which is the solution of problem (1)–(3). Then we have

8.2 8

0

2

𝑝𝛼 𝑝−1 𝑝 [(V𝑛 ) 𝐻2 𝛿𝑥 V𝑛 + 𝐻2 𝛿𝑥 (V𝑛 ) ] ; 𝑝+1

(47)

according to Taylor’s expansion, 𝑟𝑗𝑛 = 𝑂(𝜏2 + ℎ4 ) can be easily obtained. Next, we consider convergence and stability of the finite difference scheme (25). Theorem 13. Suppose that 𝑢0 ∈ 𝐻01 (Ω) and 𝑢 ∈ 𝐶(5,3) ; then the solution of the conservative difference scheme (25) converges to the solution of problem (1)–(3) with the order 𝑂(𝜏2 + ℎ4 ) by 𝐿∞ norm. Proof. Subtracting (27) from (47), and letting 𝑒𝑛 = V𝑛 − 𝑢𝑛 , we have 𝑟𝑛 = 𝜕𝑡 𝑒𝑛 + 𝜃𝐻2 𝛿𝑥 𝑒𝑛 + (1 − 𝜃) 𝐻2 𝛿𝑥 𝑒𝑛 − 𝛽𝐻1 𝛿𝑥+ 𝛿𝑥− 𝜕𝑡 𝑒𝑛 + 𝜑 (V𝑛 , V𝑛 ) − 𝜑 (𝑢𝑛 , 𝑢𝑛 ) ;

(48)

6

8

10

t

𝑟𝑛 = 𝜕𝑡 V𝑛 + 𝜃𝐻2 𝛿𝑥 V𝑛 + (1 − 𝜃) 𝐻2 𝛿𝑥 V𝑛 − 𝛽𝐻1 𝛿𝑥+ 𝛿𝑥− 𝜕𝑡 V𝑛 +

4

E(t) En

Figure 2: Discrete energy 𝐸𝑛 of scheme with 𝑝 = 2, 𝑇 = 10, and 𝜏 = ℎ = 0.1.

taking an inner product of (48) with 𝑒𝑛+1 + 𝑒𝑛−1 , we obtain (𝑟𝑛 , 𝑒𝑛+1 + 𝑒𝑛−1 )ℎ =

1 󵄩󵄩 𝑛+1 󵄩󵄩2 󵄩󵄩 𝑛−1 󵄩󵄩2 (󵄩󵄩𝑒 󵄩󵄩󵄩ℎ − 󵄩󵄩󵄩𝑒 󵄩󵄩󵄩ℎ ) 2𝜏 󵄩 +

𝛽 󵄩󵄩 󵄩2 󵄩 󵄩2 (󵄩󵄩󵄩𝑅1 𝛿𝑥+ 𝑒𝑛+1 󵄩󵄩󵄩󵄩ℎ − 󵄩󵄩󵄩󵄩𝑅1 𝛿𝑥+ 𝑒𝑥𝑛−1 󵄩󵄩󵄩󵄩ℎ ) 2𝜏

+ 𝜃 (𝑅2 𝛿𝑥 𝑒𝑛 , 𝑒𝑛+1 + 𝑒𝑛−1 )ℎ + (𝜑 (V𝑛 , V𝑛 ) − 𝜑 (𝑢𝑛 , 𝑢𝑛 ) , 𝑒𝑛+1 + 𝑒𝑛−1 )ℎ ;

(49)

6

Mathematical Problems in Engineering Table 1: Errors computed by the proposed compact scheme with 𝜎 = 4/3, 𝑝 = 2 and ℎ = 𝜏 = 0.1.

𝑡

𝜃=0 2.6213 × 10−5 5.2311 × 10−5 7.8509 × 10−5 1.0476 × 10−4 1.3090 × 10−4

0.2 0.4 0.6 0.8 1.0

󵄩󵄩 𝑛 󵄩󵄩 󵄩󵄩𝑒 󵄩󵄩∞,ℎ 𝜃 = 0.5 1.5794 × 10−5 3.1520 × 10−5 4.7095 × 10−5 6.2474 × 10−5 7.7647 × 10−5

𝜃 = 0.25 1.2528 × 10−5 2.4917 × 10−5 3.7267 × 10−5 4.9503 × 10−5 6.1653 × 10−5

𝜃 = 0.75 3.2838 × 10−5 6.5530 × 10−5 9.7948 × 10−5 1.3006 × 10−4 1.6222 × 10−4

𝜃=1 5.0401 × 10−5 1.0012 × 10−4 1.4917 × 10−4 1.9758 × 10−4 2.4638 × 10−4

Table 2: Errors computed by the proposed compact scheme with 𝜎 = 4/3, 𝑝 = 4 and ℎ = 𝜏 = 0.1. 𝑡

𝜃=0 1.2498 × 10−4 2.3629 × 10−4 3.3108 × 10−4 4.1205 × 10−4 4.9896 × 10−4

0.2 0.4 0.6 0.8 1.0

󵄩󵄩 𝑛 󵄩󵄩 󵄩󵄩𝑒 󵄩󵄩∞,ℎ 𝜃 = 0.5 2.3183 × 10−4 4.5445 × 10−4 6.6455 × 10−4 8.6651 × 10−4 1.0667 × 10−3

𝜃 = 0.25 1.7684 × 10−4 3.4550 × 10−4 5.0029 × 10−4 6.4180 × 10−4 7.7376 × 10−4

×10−4 5

from Lemma 4 and Cauchy-Schwarz inequality, we obtain

(50)

2

en

1

𝑛

𝑛+1

(𝜑 (V , V ) − 𝜑 (𝑢 , 𝑢 ) , 𝑒

𝑛−1

+𝑒

0 −1

according to Lemmas 2 and 4, we have 𝑛

Error

3

󵄩2 1 󵄩 󵄩2 󵄩2 1 󵄩 󵄩 ≤ 3 󵄩󵄩󵄩𝛿𝑥 𝑒𝑥𝑛 󵄩󵄩󵄩ℎ + 󵄩󵄩󵄩󵄩𝑒𝑛+1 󵄩󵄩󵄩󵄩ℎ + 󵄩󵄩󵄩󵄩𝑒𝑛−1 󵄩󵄩󵄩󵄩ℎ ; 2 2

𝑛

𝜃=1 3.4464 × 10−4 6.7531 × 10−4 9.9436 × 10−4 1.3159 × 10−3 1.6550 × 10−3

4

(𝑅2 𝛿𝑥 𝑒𝑛 , 𝑒𝑛+1 + 𝑒𝑛−1 )ℎ

𝑛

𝜃 = 0.75 2.8729 × 10−4 5.6499 × 10−4 8.3064 × 10−4 1.0913 × 10−3 1.3595 × 10−3

−2 −3

)ℎ

−4

𝑝−1 2𝑝𝛼 { 𝐽−1 𝑛 𝑛 𝑝−1 𝑛 𝑛 = A−1 ℎ ∑ [(V𝑛𝑗 ) A−1 2 𝛿𝑥 V𝑗 − (𝑢𝑗 ) 2 𝛿𝑥 𝑢𝑗 ] 𝑒𝑗 { 𝑝+1 { 𝑗=1 𝐽−1 𝑛 𝑝 −1 𝑛 𝑝 𝑛} + ℎ ∑ [A−1 2 𝛿𝑥 (V𝑗 ) − A2 𝛿𝑥 (𝑢𝑗 ) ] 𝑒𝑗 } 𝑗=0 } 𝐽−1 󵄨 𝑛 󵄨󵄨 󵄨󵄨 𝑛 󵄨󵄨 󵄨󵄨 𝑛 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ≤ 𝐾4 ℎ ∑ (󵄨󵄨󵄨󵄨A−1 2 𝛿𝑥 𝑒𝑗 󵄨󵄨 + 󵄨󵄨𝑒𝑗 󵄨󵄨) 󵄨󵄨𝑒𝑗 󵄨󵄨 𝑗=0

−5 −50

0 x

50

t=5 t = 10

Figure 3: Absolute error 𝑒𝑁 of scheme with 𝑝 = 2 and 𝜏 = ℎ = 0.1.

Substituting (50) and (51) into (49), we obtain

𝐽−1 󵄨 󵄨󵄨 𝑛 󵄨󵄨 󵄨 + 𝐾4 ℎ ∑ 󵄨󵄨󵄨󵄨𝑒𝑛𝑗 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨A−1 2 𝛿𝑥 𝑒𝑗 󵄨󵄨

𝛽 󵄩󵄩 1 󵄩󵄩 𝑛+1 󵄩󵄩2 󵄩󵄩 𝑛−1 󵄩󵄩2 󵄩2 󵄩 󵄩2 (󵄩󵄩󵄩𝑒 󵄩󵄩󵄩ℎ − 󵄩󵄩󵄩𝑒 󵄩󵄩󵄩ℎ ) + (󵄩󵄩󵄩𝑅1 𝛿𝑥+ 𝑒𝑛+1 󵄩󵄩󵄩󵄩ℎ − 󵄩󵄩󵄩󵄩𝑅1 𝛿𝑥+ 𝑒𝑥𝑛−1 󵄩󵄩󵄩󵄩ℎ ) 2𝜏 2𝜏

𝑗=0

󵄩2 󵄩 󵄩 󵄩2 ≤ 𝐾4 (9 󵄩󵄩󵄩𝛿𝑥 𝑒𝑛 󵄩󵄩󵄩ℎ + 2 󵄩󵄩󵄩𝑒𝑛 󵄩󵄩󵄩ℎ ) 9󵄩 󵄩2 9 󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩2 ≤ 𝐾4 ( 󵄩󵄩󵄩󵄩𝛿𝑥 𝑒𝑛+1 󵄩󵄩󵄩󵄩ℎ + 󵄩󵄩󵄩󵄩𝛿𝑥 𝑒𝑛−1 󵄩󵄩󵄩󵄩ℎ + 󵄩󵄩󵄩󵄩𝑒𝑛+1 󵄩󵄩󵄩󵄩ℎ + 󵄩󵄩󵄩󵄩𝑒𝑛−1 󵄩󵄩󵄩󵄩ℎ ) , 2 2 (51) 𝑝−1

𝑝−2

where 𝐾4 = (𝑝𝛼/2(𝑝 + 1)) max{𝐾0 , 2(𝑝 − 1)𝐾0 𝐾3 , 2(𝑝 − 𝑝−1 1)𝐾3 }.

󵄩2 󵄩 󵄩2 󵄩 󵄩2 1 󵄩 ≤ 󵄩󵄩󵄩𝑟𝑛 󵄩󵄩󵄩ℎ + (󵄩󵄩󵄩󵄩𝑒𝑛+1 󵄩󵄩󵄩󵄩ℎ + 󵄩󵄩󵄩󵄩𝑒𝑛−1 󵄩󵄩󵄩󵄩ℎ ) 2 󵄩2 1 󵄩 󵄩2 󵄩 󵄩2 1 󵄩 + 𝜃 (3 󵄩󵄩󵄩𝛿𝑥 𝑒𝑥𝑛 󵄩󵄩󵄩ℎ + 󵄩󵄩󵄩󵄩𝑒𝑛+1 󵄩󵄩󵄩󵄩ℎ + 󵄩󵄩󵄩󵄩𝑒𝑛−1 󵄩󵄩󵄩󵄩ℎ ) 2 2 9󵄩 󵄩2 9 󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩2 + 𝐾4 ( 󵄩󵄩󵄩󵄩𝛿𝑥 𝑒𝑛+1 󵄩󵄩󵄩󵄩ℎ + 󵄩󵄩󵄩󵄩𝛿𝑥 𝑒𝑛−1 󵄩󵄩󵄩󵄩ℎ + 󵄩󵄩󵄩󵄩𝑒𝑛+1 󵄩󵄩󵄩󵄩ℎ + 󵄩󵄩󵄩󵄩𝑒𝑛−1 󵄩󵄩󵄩󵄩ℎ ) , 2 2 (52)


7

Table 3: Comparison of ‖𝑒‖∞,ℎ by the compact scheme for ℎ = 𝜏 = 0.1 with the Zhang [4] scheme for ℎ = 0.05, 𝜏 = 0.1. 𝑝=2

𝑡

𝑝=4

Compact Scheme 1.2528 × 10−5 2.4917 × 10−5 3.7267 × 10−5 4.9503 × 10−5 6.1653 × 10−5

0.2 0.4 0.6 0.8 1.0

Zhang [4] 7.756 × 10−6 1.575 × 10−5 2.357 × 10−5 3.129 × 10−5 3.963 × 10−5

Compact Scheme 1.2498 × 10−4 2.3629 × 10−4 3.3108 × 10−4 4.1205 × 10−4 4.9896 × 10−4

Zhang [4] 0.0049 0.0098 0.0148 0.0198 0.0239

Table 4: Comparison of ‖𝑒‖∞,ℎ by the compact scheme for 𝜎 = 4/3, 𝜃 = 0.25, 𝑝 = 2, 𝜏 = 0.1 and ℎ = 0.05. 𝑡

Compact Scheme 1.2536 × 10−5 2.4916 × 10−5 3.7265 × 10−5 4.9526 × 10−5 6.1667 × 10−5

0.2 0.4 0.6 0.8 1.0

󵄩󵄩 𝑛 󵄩󵄩 󵄩󵄩𝑒 󵄩󵄩∞,ℎ 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 ‖𝑒‖∞,ℎ by the compact scheme for 𝜎 = 1.03, 𝜃 = 0.25, 𝑝 = 2, 𝜏 = ℎ = 0.1.

Compact Scheme

󵄩󵄩 𝑛 󵄩󵄩 󵄩󵄩𝑒 󵄩󵄩∞,ℎ Bakodah and Banaja [6]

Kutluay and Esen [8]

7.0938 × 10−5 2.1660 × 10−4 5.1012 × 10−4

1.4805 × 10−4 2.9961 × 10−4 4.5367 × 10−4

1.23 × 10−4 1.66 × 10−4 1.79 × 10−4

4 8 12

Numerical solution

1.4 1.2 1 0.8 0.6 0.4 0.2

that is, 𝛽 󵄩 1 󵄩󵄩 𝑛+1 󵄩󵄩2 󵄩󵄩 𝑛−1 󵄩󵄩2 󵄩2 󵄩 󵄩2 (󵄩󵄩𝑒 󵄩󵄩󵄩ℎ − 󵄩󵄩󵄩𝑒 󵄩󵄩󵄩ℎ ) + (󵄩󵄩󵄩󵄩𝛿𝑥+ 𝑒𝑛+1 󵄩󵄩󵄩󵄩ℎ − 󵄩󵄩󵄩󵄩𝛿𝑥+ 𝑒𝑥𝑛−1 󵄩󵄩󵄩󵄩ℎ ) 2 󵄩 2 󵄩2 󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩2 󵄩 ≤ 𝜏𝐾5 (󵄩󵄩󵄩󵄩𝑒𝑛+1 󵄩󵄩󵄩󵄩ℎ + 󵄩󵄩󵄩𝑒𝑛 󵄩󵄩󵄩ℎ + 󵄩󵄩󵄩󵄩𝑒𝑛−1 󵄩󵄩󵄩󵄩ℎ + 󵄩󵄩󵄩󵄩𝛿𝑥+ 𝑒𝑛+1 󵄩󵄩󵄩󵄩ℎ

0 −50

(53)

𝛽 󵄩 1 󵄩󵄩 𝑛+1 󵄩󵄩2 󵄩󵄩 𝑛 󵄩󵄩2 󵄩2 󵄩 󵄩2 (󵄩󵄩𝑒 󵄩󵄩󵄩ℎ + 󵄩󵄩𝑒 󵄩󵄩ℎ ) + (󵄩󵄩󵄩󵄩𝛿𝑥+ 𝑒𝑛+1 󵄩󵄩󵄩󵄩ℎ + 󵄩󵄩󵄩𝛿𝑥+ 𝑒𝑛 󵄩󵄩󵄩ℎ ) ; 2 󵄩 2 (54)

then (53) can be rewritten as (55)

1≤𝑛≤𝑁

𝐵0 ≤ [𝑂(𝜏2 + ℎ4 )] ;

it follows from (56) that 󵄩󵄩 𝑛 󵄩󵄩 2 4 󵄩󵄩𝑒 󵄩󵄩ℎ ≤ 𝑂 (𝜏 + ℎ ) ,

󵄩󵄩 + 𝑛 󵄩󵄩 2 4 󵄩󵄩𝛿𝑥 𝑒 󵄩󵄩ℎ ≤ 𝑂 (𝜏 + ℎ ) ,

(58)

and then, from Lemma 6, we obtain (59)

This completes the proof. (56)

Thus we can choose a fourth-order method to compute 𝑢1 such that 2

t=0 t=5 t = 10

‖𝑒‖∞,ℎ ≤ 𝑂 (𝜏2 + ℎ4 ) .

where 𝐾6 = max{2𝐾5 , 2𝐾5 /𝛽}. From Lemma 8, we have 󵄩 󵄩2 𝐵𝑁 ≤ (𝐵0 + 𝑇 sup 󵄩󵄩󵄩𝑟𝑛 󵄩󵄩󵄩ℎ ) 𝑒4𝐾6 𝑇 .

50

Figure 4: Numerical solution 𝑢𝑛 of scheme with 𝑝 = 4 and 𝜏 = ℎ = 0.1.

where 𝐾5 = max{(1 + 𝜃)/2 + 𝐾4 , (9/2)𝐾4 , 3𝜃}. Let

󵄩 󵄩2 𝐵𝑛 − 𝐵𝑛−1 ≤ 𝜏 󵄩󵄩󵄩𝑟𝑛 󵄩󵄩󵄩ℎ + 𝜏𝐾6 (𝐵𝑛 + 𝐵𝑛−1 ) ,

0 x

󵄩2 󵄩2 󵄩 󵄩 󵄩 󵄩2 + 󵄩󵄩󵄩𝛿𝑥+ 𝑒𝑛 󵄩󵄩󵄩ℎ + 󵄩󵄩󵄩󵄩𝛿𝑥+ 𝑒𝑛−1 󵄩󵄩󵄩󵄩ℎ ) + 𝜏 󵄩󵄩󵄩𝑟𝑛 󵄩󵄩󵄩ℎ ,

𝐵𝑛 =

Dogan [11] 0.00053 0.00113 0.00175 0.00237 0.00299

un

𝑡

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 𝐿∞ 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 𝐸𝑛 of scheme with 𝑝 = 4, 𝑇 = 10, and 𝜏 = ℎ = 0.1. ×10−3 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 − u n

take ℎ = 0.1 and 𝜏 = 0.1, respectively. The errors ‖𝑒𝑛 ‖∞,ℎ are listed in Tables 1-2, respectively. In Table 3, 4, and 5, the comparison of ‖𝑒‖∞,ℎ by the compact scheme for ℎ = 𝜏 = 0.1 with the Zhang [4] scheme for ℎ = 0.05, 𝜏 = 0.1 when 𝜎 = 4/3, 𝜃 = 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 𝜃. From Tables 1-2, ‖𝑒𝑛 ‖∞,ℎ ≤ 𝑂(𝜏2 + ℎ4 ) is validated. We take different 𝑝, ℎ, and 𝜃 values and compute the errors for the solution of problem (1)– (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 −0.5 −1 −1.5 −50

0 x

50

t=5 t = 10

Figure 6: Absolute error 𝑒𝑁 of scheme with 𝑝 = 4 and 𝜏 = ℎ = 0.1.

4. Numerical Experiments In this section, two examples are presented to illustrate the effectiveness of the finite difference scheme (25) in [−50, 50]. The single solitary wave solution of (1) is 𝑢 (𝑥, 𝑡) = 𝐴 ⋅ sech2/(𝑝−1) [𝑘 (𝑥 + 𝑥0 − 𝜎𝑡)] ,

(60)

where (𝑝 + 1) (𝜎 − 1) ] 𝐴=[ 2𝛼

1/(𝑝−1)

,

𝑘=

𝑝−1 𝜎−1 √ , (61) 2𝛽 𝜎

and 𝜎, 𝑥0 are arbitrary constants and 𝑝 ≥ 2. Let 𝑥0 = 0 in (60), 𝛼 = 1/2, 𝛽 = 1, and 𝑢0 (𝑥) = 𝐴 ⋅ sech2/(𝑝−1) (𝑘𝑥) and consider two cases: 𝑝 = 2 and 𝑝 = 4. We

[1] D. H. Peregrine, “Calculations of the development of an unduiar bore,” Journal of Fluid Mechanics, vol. 25, pp. 321–330, 1966. [2] D. H. Peregrine, “Long waves on a beach,” Journal of Fluid Mechanics, vol. 27, no. 4, pp. 815–827, 1967. [3] J. L. Bona, W. G. Pritchard, and L. R. Scott, “Numerical schemes for a model for nonlinear dispersive waves,” Journal of Computational Physics, vol. 60, no. 2, pp. 167–186, 1985. [4] L. Zhang, “A finite difference scheme for generalized regularized long-wave equation,” Applied Mathematics and Computation, vol. 168, no. 2, pp. 962–972, 2005. [5] Z. Ren, W. Wang, and D. Yu, “A new conservative finite difference method for the nonlinear regularized long wave equation,” Applied Mathematical Sciences, vol. 5, no. 41–44, pp. 2091–2096, 2011. [6] H. O. Bakodah and M. A. Banaja, “The method of lines solution of the regularized long-wave equation using Runge-Kutta time discretization method,” Mathematical Problems in Engineering, vol. 2013, Article ID 804317, 8 pages, 2013. [7] X. Shao, G. Xue, and C. Li, “A conservative weighted finite difference scheme for regularized long wave equation,” Applied Mathematics and Computation, vol. 219, no. 17, pp. 9202–9209, 2013. [8] S. Kutluay and A. Esen, “A finite difference solution of the regularized long-wave equation,” Mathematical Problems in Engineering, vol. 2006, Article ID 85743, 14 pages, 2006. [9] A. Esen and S. Kutluay, “Application of a lumped Galerkin method to the regularized long wave equation,” Applied Mathematics and Computation, vol. 174, no. 2, pp. 833–845, 2006. [10] K. R. Raslan, “A computational method for the regularized long wave (RLW) equation,” Applied Mathematics and Computation, vol. 167, no. 2, pp. 1101–1118, 2005.

Mathematical Problems in Engineering [11] A. Dogan, “Numerical solution of RLW equation using linear finite elements within Galerkin’s method,” Applied Mathematical Modelling, vol. 26, no. 7, pp. 771–783, 2002. [12] T. Roshan, “A Petrov-Galerkin method for solving the generalized regularized long wave (GRLW) equation,” Computers & Mathematics with Applications, vol. 63, no. 5, pp. 943–956, 2012. [13] R. Mokhtari and M. Mohammadi, “Numerical solution of GRLW equation using sinc-collocation method,” Computer Physics Communications, vol. 181, no. 7, pp. 1266–1274, 2010. [14] C. M. Garcia-Lopez and J. I. Ramos, “Effects of convection on a modified GRLW equation,” Applied Mathematics and Computation, vol. 219, no. 8, pp. 4118–4132, 2012. [15] X. Li, L. Zhang, and S. Wang, “A compact finite difference scheme for the nonlinear Schr¨odinger equation with wave operator,” Applied Mathematics and Computation, vol. 219, no. 6, pp. 3187–3197, 2012. [16] T. Wang, B. Guo, and Q. Xu, “Fourth-order compact and energy conservative difference schemes for the nonlinear Schr¨odinger equation in two dimensions,” Journal of Computational Physics, vol. 243, pp. 382–399, 2013. [17] T. Wang and B. Guo, “Unconditional convergence of two conservative compact difference schemes for nonlinear Schroinger equation in one dimension,” Scientia Sinica Mathematica, vol. 41, no. 3, pp. 207–233, 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, “Conservative schemes for the symmetric regularized long wave equations,” Applied Mathematics and Computation, vol. 190, no. 2, pp. 1063–1080, 2007.

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