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Advances in Mathematical Physics
Volume 2014 (2014), Article ID 986098, 7 pages
Conservative Difference Scheme for Generalized Rosenau-KdV Equation
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.
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.
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  discussed the solitary wave solutions and periodic solutions for Rosenau-KdV equation. In , 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  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 , 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.
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  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:
Lemma 1. It follows from summation by parts that, for any two mesh functions , Then one has Furthermore, if , then
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.
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).
In order to prove the bounded quality of the difference solution, we introduce the following lemma.
Lemma 5 (discrete Sobolev’s inequality ). There exist two constants and such that
In order to prove the convergence of the difference scheme, we need to introduce the lemma as follows:
Lemma 8 (discrete Gronwall inequality ). Suppose and are nonnegative functions and is nondecreasing. If and then
Proof. It follows from (5) that By Holder inequality and Schwarz inequality, we get which implies Using Sobolev inequality, we get
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.
4. Numerical Simulations
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  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.
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 . 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.
The work was supported by NSFC (61170309) and Chinese Universities Specialized Research Fund for the Doctoral Program (20110185110020).
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