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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 308410, 14 pages
doi:10.1155/2012/308410
Research Article

Numerical Analysis of a Linear-Implicit Average Scheme for Generalized Benjamin-Bona-Mahony-Burgers Equation

Hai-tao Che,1,2 Xin-tian Pan,2 Lu-ming Zhang,3 and Yi-ju Wang1

1School of Management Science, Qufu Normal University, Rizhao 276800, China
2School of Mathematics and Information Science, Weifang University, Weifang 261061, China
3Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

Received 14 October 2011; Accepted 21 December 2011

Academic Editor: Yuantong Gu

Copyright © 2012 Hai-tao Che 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.

Abstract

A linear-implicit finite difference scheme is given for the initial-boundary problem of GBBM-Burgers equation, which is convergent and unconditionally stable. The unique solvability of numerical solutions is shown. A priori estimate and second-order convergence of the finite difference approximate solution are discussed using energy method. Numerical results demonstrate that the scheme is efficient and accurate.

1. Introduction

The generalized Benjamin-Bona-Mahony-Burgers (GBBM-Burgers) equation is in the form [1] where are constants, is an integer, and represents the velocity of fluid in the horizontal direction . When , (1.1) is called as the Benjamin-Bona-Mahony-Burgers (BBM-Burgers) equation. In the special case, when , (1.1) is described as the generalized Benjamin-Bona-Mahony equation The Equation (1.2) which is usually called as the generalized regularized long-wave equation proposed by Peregrine [2] and Benjamin et al. [3], so-called generalized Benjamin-Bona-Mahony equation, has been studied by many authors [4–7]. This equation features a balance between the nonlinear dispersive effect but takes no account of dissipation.

In recent years, a vast amount of work and computation has been devoted to the initial value problem for the GBBM-Burgers equation. In [1], Al-Khaled et al. studied the GBBM-Burgers by Decomposition method. In [8], Hayashi et al. investigated large time asymptotics of solutions to the BBM-Burgers equation. In [9], Jiang and Xu investigated the asymptotic behavior of solutions of the initial-boundary value problem for the GBBM-Burgers equations. In [10], Yin et al. studied the large time behavior of traveling wave solutions to the Cauchy problem of the GBBM-Burgers equations. In [11], Mei studied the large time behavior of global solutions to the Cauchy problem of GBBM-Burgers equations. In [12], Kondo and Webler studied the global existence of solutions for multidimensional GBBM-Burgers equations. Kinami et al. discussed the Cauchy problem of the GBBM-Burgers equations by Fourier transform method and energy method [13]. However, there are few studies on finite difference approximations for (1.1) which we consider in this paper.

In a recent work [14], we have made some preliminary computation by proposing a linearized difference scheme for GRLW equation which is unconditionally stable and reduces the computational work, and the numerical results are encouraging. In this paper, we continue our work and propose a linear-implicit difference scheme for generalized BBM-Burgers equation which is unconditionally stable and second-order convergent.

In this paper, we consider the following initial-boundary value problem of the GBBM-Burgers equation An outline of the paper is as follows. In Section 2, we describe a linear-implicit finite difference scheme for the GBBM-Burgers equation and prove the error estimates of 2 order. In Section 3, we show that the scheme is uniquely solvable. In Section 4, convergence and stability of the scheme are proved. In Section 5, numerical results are provided to test the theoretical results.

2. Finite Difference Scheme and Estimate for the Difference Solution

As usual, the following notations will be used: where and are the uniform spatial and temporal step sizes, respectively, Let denote the approximation of , . In this paper, we will denote as a generic constant independent of step sizes and .

We propose a three-level linear-implicit difference scheme for the solution of the problem (1.3)

For convenience, the last term of (2.3) is defined by

Lemma 2.1 (see [15]). For any two mesh functions , one has

Lemma 2.2. For any mesh function , one has

Proof. For , one has

Lemma 2.3 (Discrete Sobolev Inequality [16]). For any discrete function and for any given , there exists a constant , depending only and , such that

Theorem 2.4. Assume , then there is the estimation for the solution of difference scheme (2.3)–(2.6),

Proof. Computing the inner product of (2.3) with (i.e., ), we obtain Now, computing the fourth term of the left-hand side in (2.13), we have According to Lemmas 2.1 and 2.2, and using (2.14), we get We let It follows from (2.15) that Then we have Using (2.18), we obtain Equation (2.19) yields Using Lemma 2.3, the proof of Theorem 2.4 is completed.

Remark 2.5. Theorem 2.4 implies that scheme (2.3)–(2.6) is unconditionally stable.

3. Solvability

Next, we will discuss the solvability of the scheme (2.3) based on the technique of Omrani et al. [17].

Theorem 3.1. The finite difference scheme (2.3) is uniquely solvable.

Proof. It is obvious that and are uniquely determined by (2.4)-(2.5). Now suppose be solved uniquely. Considering the equation of (2.3) for , we have Computing the inner product of (3.1) with , we have where .
In view of difference properties and the boundary conditions (2.6), we obtain It follows from (3.2) and (3.3) that Noting that and following from (3.4), we have That is (3.1) has only a trivial solution. Therefore, the scheme (2.3) determines uniquely. This completes the proof.

Remark 3.2. All results above in this paper are correct for IBV problem of the BBM-Burgers equation with finite or infinite boundary.

4. Convergence and Stability of the Difference Scheme

First, we consider the truncation error of the difference scheme (2.3)–(2.6).

Suppose . Making use of Taylor expansion, we find where and are the truncation errors of the difference scheme (2.3)–(2.6). It can be easily obtained that (see [18, 19])

Lemma 4.1. Assume is smooth enough, then the local truncation error of the finite difference scheme (2.3)–(2.6) is

Lemma 4.2 (see [16]). Suppose that the discrete function satisfies recurrence formula where are nonnegative constants. Then where is small, such that .

Theorem 4.3. Assume and , then the solution of the difference scheme (2.3)–(2.6) converges to the solution of the problem (1.3) with order by the norm.

Proof. Let . Subtracting (2.3)-(2.5) from (4.1)–(4.3), respectively, we have For a simple notation, the last two terms of (4.7) are defined by Computing the inner product of (4.7) with (i.e., ), we get Similarly to the proof of Theorem 2.4, we obtain According to Theorem 2.4, we obtain In addition, there exists obviously that Substituting (4.10)–(4.12) into (4.9), we have Let Then (4.13) can be rewritten as By Lemma 4.2, it can immediately be obtained that To complete the proof, it is enough to find estimate. From (4.7), we obtain Using (4.3) and (4.8), we get It follows from (4.17) and (4.18) that Thus According to Lemma 2.3, there exists that

Similarly, the following theorem can be proved.

Theorem 4.4. Under the conditions of Theorem 4.3, the solution of finite difference scheme (2.3)–(2.6) is stable by the norm.

5. Numerical Experiments

In this section, we will compute several numerical experiments to verify the correction of our theoretical analysis in the above sections.

Example 5.1 (see [20]). Consider the following initial-boundary problem of BBM-Burgers equation: We denote the scheme proposed in [20] as Scheme I and the scheme (2.3) in present paper as Scheme II. In computations, we choose the initial condition [20]. The maximal errors of both schemes are listed in Table 1. We get that a second-order linear scheme is as accurate as Scheme I which is a nonlinear one.

tab1
Table 1: The maximal errors of numerical solutions at with for when .

Example 5.2 (see [13]). Consider the GBBM-Burgers equation with an initial condition and boundary conditions In computations, we choose the initial condition [13]. Without loss of generality, We take and , . Since we do not know the exact solution of (5.4)–(5.6), an error estimate method in [21] is used. A comparison between the numerical solutions on a coarse mesh and those on a refine mesh is made. In order to obtain the error estimates, we consider the solution on mesh as reference solution and obtain error estimates on mesh , and , respectively. We denote the scheme proposed in [13] as Scheme III and make a comparison with the scheme (2.3) in present paper as Scheme II when in Table 2. The corresponding errors in the sense of -norm and -norm are listed in Tables 3, 4, and 5, respectively. These three tables verify the second-order convergence and good stability of the numerical solutions.
Figures 1 and 2 plot the numerical solutions computed by the linearly implicit scheme (2.3) with , , and when at , and 10, respectively. From Figures 1 and 2, it is easy to observe that the height of the numerical approximation to is more and more low with time elapsing due to the effect of dissipative term . Both of them simulates that the continuous energy of the problem (1.3) in Theorem 2.4 decreases in time. Numerical experiments show our scheme is accurate and efficient.

tab2
Table 2: The maximal errors of numerical solutions at with for when .
tab3
Table 3: The errors of numerical solutions at with when .
tab4
Table 4: The errors of numerical solutions at with when .
tab5
Table 5: The errors of numerical solutions at with when .
308410.fig.001
Figure 1: Numerical solution of with when .
308410.fig.002
Figure 2: Numerical solution of with when .

6. Conclusions

In this paper, we have presented a three-level linear-implicit finite difference scheme for the GBBM-Burgers equation, which has a wide range of applications in physics. The convergence and stability as well as second-order error estimate of the finite difference approximate solutions were discussed in detail. Numerical experiments show our scheme is accurate and efficient.

Acknowledgments

This work is supported by the fund of National Natural Science (11171193, 11171180, and 10901096) and the fund of Natural Science of Shandong Province (ZR2009AL019, ZR2011AM016), and the Youth Research Foundation of WFU (no. 2011Z17). The authors thank the referees for their valuable comments.

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