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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

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 [718]. 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.

tab1
Table 1: The error at various time step.
tab2
Table 2: The error comparison when .
tab3
Table 3: The verification of the second convergence.
tab4
Table 4: Numerical simulations on conservation invariant .

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

986098.fig.001
Figure 1: When and , the wave graph of at various times.
986098.fig.002
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).

References

  1. J.-M. Zuo, “Solitons and periodic solutions for the Rosenau-KdV and Rosenau-Kawahara equations,” Applied Mathematics and Computation, vol. 215, no. 2, pp. 835–840, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. J. Hu, Y. Xu, and B. Hu, “Conservative linear difference scheme for Rosenau-KdV equation,” Advances in Mathematical Physics, vol. 2013, Article ID 423718, 7 pages, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. A. Esfahani, “Solitary wave solutions for generalized Rosenau-KdV equation,” Communications in Theoretical Physics, vol. 55, no. 3, pp. 396–398, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. P. Razborova, H. Triki, and A. Biswas, “Perturbation of dispersive shallow water waves,” Ocean Engineering, vol. 63, pp. 1–7, 2013. View at Publisher · View at Google Scholar
  5. G. Ebadi, A. Mojaver, H. Triki, A. Yildirim, and A. Biswas, “Topological solitons and other solutions of the Rosenau-KdV equation with power law nonlinearity,” Romanian Journal of Physics, vol. 58, no. 1-2, pp. 3–14, 2013. View at MathSciNet
  6. A. Saha, “Topological 1-soliton solutions for the Generalized Rosenau-Kdv equation,” Fundamental Journal of Mathematical Physics, vol. 2, no. 1, pp. 19–23, 2012.
  7. M. Zheng and J. Zhou, “An average linear difference scheme for the generalized Rosenau-KdV equation,” Journal of Applied Mathematics, vol. 2014, Article ID 202793, 9 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  8. S. Li and L. Vu-Quoc, “Finite difference calculus invariant structure of a class of algorithms for the nonlinear Klein-Gordon equation,” SIAM Journal on Numerical Analysis, vol. 32, no. 6, pp. 1839–1875, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. Q. Chang, E. Jia, and W. Sun, “Difference schemes for solving the generalized nonlinear Schrödinger equation,” Journal of Computational Physics, vol. 148, no. 2, pp. 397–415, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. T.-C. Wang and L.-M. Zhang, “Analysis of some new conservative schemes for nonlinear Schrödinger equation with wave operator,” Applied Mathematics and Computation, vol. 182, no. 2, pp. 1780–1794, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. T. Wang, B. Guo, and L. Zhang, “New conservative difference schemes for a coupled nonlinear Schrödinger system,” Applied Mathematics and Computation, vol. 217, no. 4, pp. 1604–1619, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. L. Zhang, “A finite difference scheme for generalized regularized long-wave equation,” Applied Mathematics and Computation, vol. 168, no. 2, pp. 962–972, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. Z. Fei and L. Vázquez, “Two energy conserving numerical schemes for the sine-Gordon equation,” Applied Mathematics and Computation, vol. 45, no. 1, pp. 17–30, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. Y. S. Wong, Q. Chang, and L. Gong, “An initial-boundary value problem of a nonlinear Klein-Gordon equation,” Applied Mathematics and Computation, vol. 84, no. 1, pp. 77–93, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. Q. S. Chang, B. L. Guo, and H. Jiang, “Finite difference method for generalized Zakharov equations,” Mathematics of Computation, vol. 64, no. 210, pp. 537–553, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. J. S. Hu and K. L. Zheng, “Two conservative difference schemes for the generalized Rosenau equation,” Boundary Value Problems, vol. 2010, Article ID 543503, 18 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. X. Pan and L. Zhang, “On the convergence of a conservative numerical scheme for the usual Rosenau-RLW equation,” Applied Mathematical Modelling. Simulation and Computation for Engineering and Environmental Systems, vol. 36, no. 8, pp. 3371–3378, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. X. Pan and L. Zhang, “Numerical simulation for general Rosenau-RLW equation: an average linearized conservative scheme,” Mathematical Problems in Engineering, vol. 2012, Article ID 517818, 15 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. F. E. Browder, “Existence and uniqueness theorems for solutions of nonlinear boundary value problems,” in Proceedings of the Symposia in Applied Mathematics, vol. 17, pp. 24–49, 1965. View at Zentralblatt MATH · View at MathSciNet
  20. T. Wang and B. Guo, “A robust semi-explicit difference scheme for the Kuramoto-Tsuzuki equation,” Journal of Computational and Applied Mathematics, vol. 233, no. 4, pp. 878–888, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. J. Hu, B. Hu, and Y. Xu, “C-N difference schemes for dissipative symmetric regularized long wave equations with damping term,” Mathematical Problems in Engineering, vol. 2011, Article ID 651642, 16 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet