Advances in Mathematical Physics

Volume 2014 (2014), Article ID 986098, 7 pages

http://dx.doi.org/10.1155/2014/986098

Research Article

Conservative Difference Scheme for Generalized Rosenau-KdV Equation

Yan Luo,¹ Youcai Xu,² and Minfu Feng²

¹School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 610054, China

²School of Mathematics, Sichuan University, Chengdu 610064, China

Received 12 January 2014; Revised 9 April 2014; Accepted 22 April 2014; Published 14 May 2014

Academic Editor: Ricardo Weder

Copyright © 2014 Yan Luo 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 conservative Crank-Nicolson finite difference scheme for the initial-boundary value problem of generalized Rosenau-KdV equation is proposed. The difference scheme shows a discrete analogue of the main conservation law associated to the equation. On the other hand the scheme is implicit and stable with second order convergence. Numerical experiments verify the theoretical results.

1. Introduction

In this paper, we consider the following initial-boundary value problem of the generalized Rosenau-KdV equation:

with an initial condition

and boundary conditions

where is a integer and is a known smooth function. When , (1) is called usual Rosenau-KdV equation:

Zuo [1] discussed the solitary wave solutions and periodic solutions for Rosenau-KdV equation. In [2], a conservative nonlinear finite difference scheme for an initial-boundary value problem of Rosenau-Kdv equation is considered.

In [3, 4] the solitary solution and invariant for generalized Rosenau-KdV equation are given. In [4] the singular 1-soliton solution is derived by the ansatz method, and the adiabatic parameter dynamics of the water waves is obtained by perturbation theory. In [5, 6], the ansatz method is applied to obtain the topological soliton solution of the generalized Rosenau-KdV equation. The method as well as the exp-function method is also applied to extract a few more solutions to this equation. In [7], Zheng and Zhou give an average linear scheme for the generalized Rosenau-KdV equation. In this paper, we propose a conservative Crank-Nicolson finite difference scheme for an initial-boundary value problem of the generalized Rosenau-Kdv equation.

The initial-boundary value problem (1)–(3) possesses the following conservative property [3, 4]:

When , , the initial-boundary value problem (1)–(3) and the Cauchy problem (1) are consistent, so that the boundary conditions (3) are reasonable.

It is known that the conservative scheme is better than the nonconservative ones. The nonconservative scheme may easily show nonlinear blow-up. A lot of numerical experiments show that the conservative scheme can possess some invariant properties of the original differential equation [7–18]. The conservative scheme is more suitable for long-time calculations. In [18] Pan and Zhang said “… in some areas, the ability to preserve some invariant properties of the original differential equation is a criterion to judge the success of a numerical simulation”.

The rest of this paper is organized as follows. In Section 2, we propose a Crank-Nicolson implicit nonlinear finite difference scheme for the generalized Rosenau-KdV equation and discuss the property of its solution. In Section 3, we prove that the finite difference scheme is of second order convergence. Finally, some numerical tests are given in Section 4 to verify our theoretical analysis.

2. Finite Difference Scheme and Its Property

Let and let be the uniform step size in the spatial and temporal direction, respectively.  Denote    ,    ,  , and . Throughout this paper, we denote as a generic positive constant independent of and , which may have different values in different occurrences. We introduce the following notations:

We propose a conservative Crank-Nicolson finite difference scheme for the solution of (1)–(3):

From the boundary conditions (3), we know that (10) is reasonable.

Lemma 1. It follows from summation by parts that, for any two mesh functions ,

Then one has

Furthermore, if , then

To show the existence of the solution for (7)–(10), the following Brouwer fixed point theorem should be introduced. For the proof, see [19].

Lemma 2. Let be a finite dimensional inner product space, let be the associated norm, and let be continuous. Assume, moreover, that there exists an , for all and , . Then there exists such that and .

Then one has the following theorem.

Theorem 3. There exists which satisfies the difference scheme (7)–(10) .

Proof. In order to prove the theorem by the mathematical induction, we assume that which satisfy (7)–(10) exist for . Next prove that there also exists which satisfies (7)–(10).

We define on as follows:

Taking an inner product of (14) with and considering

we have

Hence, it is obvious that for all   with . It follows from Lemma 2 that there exists such that . Let ; then satisfies (7).

The difference scheme (7)–(10) simulates the conservation property of the problem (1)–(3) as follows.

Theorem 4. Suppose that , then the difference scheme (7)–(10) is conservative:

Proof. Taking an inner product of (7) with (i.e., ), according to the boundary condition (10) and Lemma 1, we obtain

where .

From

we have

Then (17) is gotten from (21).

In order to prove the bounded quality of the difference solution, we introduce the following lemma.

Lemma 5 (discrete Sobolev’s inequality [2]). There exist two constants and such that

Theorem 6. Suppose ; then the solution of (7)–(10) satisfies

which yield

Proof. It follows from (17) that

According to (12) and Schwarz inequality, we get

Using Lemma 5, we have

Remark 7. Theorem 6 implies that the solution of difference scheme (7)–(10) is stable in the sense of norm .

3. Convergence

In order to prove the convergence of the difference scheme, we need to introduce the lemma as follows:

Lemma 8 (discrete Gronwall inequality [2]). Suppose and are nonnegative functions and is nondecreasing. If and

then

Theorem 9. Suppose that ; then the solution of (1)–(3) satisfies

Proof. It follows from (5) that

By Holder inequality and Schwarz inequality, we get

which implies

Using Sobolev inequality, we get

Let be the solution of problem (1)–(3), ; then the truncation error of the difference scheme (7)–(10) is

Making use of Taylor expansion, we know that holds if .

Theorem 10. Suppose ; then the solution of the difference scheme (7)–(10) converges to the solution of the problem (1)–(3) with order in norm .

Proof. Subtracting (7) from (35) and letting , we have

Computing the inner product of (36) with , we obtain

Similar to the proof of (19), we have

This indicates

where

Noting that

and with

we have

Similar to the proof of Theorem 6, we have

This yields

Let . We claim that

which yields

If is sufficiently small which satisfies , then

Summing up (48) from to , we have

Since

and , we obtain

By Lemma 8, we get

which implies

From (44), we have

By Lemma 5 we get

Finally, we can similarly prove the results as follows.

Theorem 11. The solution of (7)–(10) is unique.

4. Numerical Simulations

The difference scheme (7)–(10) is a nonlinear system about that can be easily solved by the Newton iterative algorithm.

Let , and . According to [3, 4], when , the soliton solution is as follows:

and the initial condition is

When , the soliton solution is as follows:

and the initial condition is

In Table 1 we give the error at various time step. We denote the C-N scheme in this paper as scheme I and the difference scheme in [7] as scheme II. In Table 2 we give the error comparison between scheme I and scheme II. It is easy to see that the calculation results of scheme I are slightly better than scheme II. Using the method in [20, 21], we verified the second convergence of the difference scheme in Table 3. Numerical simulations on the conservation invariant are given in Table 4.

Table 1: The error at various time step.

Table 2: The error comparison when .

Scheme I

Scheme II

Table 3: The verification of the second convergence.

—
3.98355
3.99527
—
3.97764
3.99497

—
3.97483
3.99443
—
3.97073
3.99275

—
3.97233
3.99319
—
3.96382
3.99112

—
3.96837
3.99261
—
3.95742
3.98917

Table 4: Numerical simulations on conservation invariant .

1.68252899330
3.11067490241
1.68254308255
3.11070293879
1.68254661109
3.11070996431

1.68252899329
3.11067490241
1.68254308255
3.11070293879
1.68254661108
3.11070996430

1.68252899328
3.11067490240
1.68254308255
3.11070293879
1.68254661095
3.11070996426

1.68252899327
3.11067490240
1.68254308255
3.11070293879
1.68254661102
3.11070996417

1.68252899325
3.11067490240
1.68254308254
3.11070293879
1.68254661095
3.11070996435

The wave graph comparison of at various times is given in Figures 1 and 2 when and .

Figure 1: When and , the wave graph of at various times.

Figure 2: When and , the wave graph of at various times.

5. Conclusions

In this paper, we propose a conservative Crank-Nicolson finite difference scheme for the initial-boundary value problem of the generalized Rosenau-KdV equation. The two-level finite difference scheme is of second order convergence and unconditionally stable, which can start by itself. From Table 2 we conclude that the C-N scheme is more efficient than scheme II in [7]. From Table 3 we conclude that the C-N scheme is of second order convergence obviously. Numerical simulations on the conservation invariant are given in Table 4. Figures 1 and 2 show that the height of the wave graph at different time is almost identical. Table 4 and Figures 1 and 2 imply that the finite difference scheme is conservative and efficient.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The work was supported by NSFC (61170309) and Chinese Universities Specialized Research Fund for the Doctoral Program (20110185110020).

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