Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2013, Article ID 870291, 8 pages
http://dx.doi.org/10.1155/2013/870291
Research Article

A High-Accuracy Linear Conservative Difference Scheme for Rosenau-RLW Equation

School of Mathematics and Computer Engineering, Xihua University, Chengdu 610039, China

Received 14 July 2013; Revised 15 September 2013; Accepted 16 September 2013

Academic Editor: Chien-Yu Lu

Copyright © 2013 Jinsong Hu and Yulan Wang. 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

We study the initial-boundary value problem for Rosenau-RLW equation. We propose a three-level linear finite difference scheme, which has the theoretical accuracy of . The scheme simulates two conservative properties of original problem well. The existence, uniqueness of difference solution, and a priori estimates in infinite norm are obtained. Furthermore, we analyze the convergence and stability of the scheme by energy method. At last, numerical experiments demonstrate the theoretical results.

1. Introduction

In the study of the dynamics of compact discrete systems, wave-wave and wave-wall interactions cannot be described by the well-known KdV equation. To overcome this shortcoming of KdV equation, Rosenau proposed the following Rosenau equation in [1, 2]: Rosenau equation (1) is usually used to describe the dense discrete system and simulate the long-chain transmission model through an L-C flow in radio and computer fields. The existence and uniqueness of solution to (1) were proved by Park in [3]. Rosenau equation is also regarded as the transformation of the Regularized Long Wave (RLW) equation (see [4]): which is usually used to simulate the long wave in nonlinear emanative medium. RLW equation plays important role in the study of nonlinear diffusion wave because it could model lots of physical phenomena. As RLW equation and KdV equation have the same approximative order when they are used to describe motivations, RLW equation could simulate almost all of the applications of KdV equation [5]. Therefore, there are many works about Rosenau equation (1) and RLW equation (2) (see, e.g., [623]).

Rosenau-RLW equation is the generalization of Rosenau equation (1) and RLW equation (2). The authors of [2427] studied the numerical solution of Rosenau-RLW equation. Motivated by the above works, we consider the initial-boundary value problem of the following Rosenau-RLW equation: where is a smooth function.

As the solitary wave solution of (3) is (see, e.g., [2426]) the initial-boundary problem (3)–(5) is in accordance with the Cauchy problem of (3) when , . It is easy to verify that problem (3)–(5) satisfies the following conservative laws [27]: where and are both constants only depending on initial data.

Li and Vu-Quoc pointed in [28] that 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. It is said in [29] that conservative difference scheme can simulate the conservative law of initial problem well and it could avoid the nonlinear blowup. Therefore, constructing conservative difference scheme is an important and significant job. To our knowledge, the theoretic accuracy of the existing difference scheme for Rosenau-RLW equation (see [24, 25, 27]) is . Particularly, in [27], the authors proposed a three-level linear conservative difference scheme for problem (3)–(5), whose theoretic accuracy is . One does not need iteration when solving the equation numerically using this scheme. Henceforth, it could save some computing time. Using Richardson extrapolation idea, we will propose a three-level linear difference scheme which has the theoretic accuracy of without refined mesh in this paper. Our scheme simulates the two conservative laws (7) and (8) well. And we will study the a priori estimate, existence, and uniqueness of the difference solution. Furthermore, we shall analyze the convergence and stability.

The rest of this paper is organized as follows. We propose the conservative difference scheme in Section 2 and prove the existence and uniqueness of solution to difference scheme by mathematical induction in Section 3. Section 4 is devoted to the prior estimate, convergence, and stability of the difference scheme. In Section 5, we verify our theoretical analysis by numerical experiments.

2. Difference Scheme and Its Conservative Law

Let and be the uniform stepsize in the spatial and temporal directions, respectively. Denote , , and , where . Let be the difference approximation of at ; that is, . Let We define the difference operators, inner product, and norms that will be used in this paper as follows:

Lemma 1. It follows from the Cauchy-Schwarz inequality and summation by parts (see [30]) that, for any ,

In the paper, denotes a general positive constant which may have a different value in a different occurrence.

Consider the following finite difference scheme for problem (3)–(5): The discrete boundary condition (15) is reasonable from the boundary condition of (3)–(5). The following theorem shows how the difference scheme (12)–(15) simulates the conservative law numerically.

Theorem 2. Suppose that and . Then the scheme (12)–(15) is conservative for discrete energy; that is, , where

Proof. Multiplying in the two sides of (12) and taking summation for , we could obtain from (15) and summation by parts [30] that On the other hand, From the definition of , we know that (16) could be deduced from (19)-(20).
Taking an inner product of (12) and (i.e., ), we could obtain from boundary condition (15) and summation by parts [30] that where On the other hand, Taking (23)–(25) into (21), we have From the definition of , we know that (17) could be obtained by deducing the above equality from to .

3. Solvability

Theorem 3. The solution of difference scheme (12)–(15) is unique.

Proof. We will use the mathematical induction to prove our theorem. We first note that and are determined uniquely by (13) and (14).
Suppose that are the unique solution to scheme (12)–(15). Next we prove that there exists unique which satisfies (12)–(15).
Consider
Taking an inner product of (27) and , we could obtain from boundary condition (15) and summation by parts [30] that Note that Furthermore, from Lemma 1, one can easily obtain that Henceforth we could have from (28)–(30).
That is, (27) only admits zero solution. So, there exits unique that satisfies (12).

4. Convergence and Stability of the Difference Scheme

Suppose that is the solution to problem (3)–(5). Let . Then the truncation error of the difference scheme (12)–(15) is Making use of the Taylor expansion theorem, we know that as , .

Lemma 4. Assume that , . Then the solution to problem (3)–(5) satisfies

Proof. From (8), we know that Then from the Sobolev inequality.

Theorem 5. Suppose that , . Then the solution to difference scheme (12)–(15) satisfies thus,

Proof. From Theorem 2 and (30) we can get that is, Then the discrete Sobolev inequality (see [30]) shows that

Theorem 6. Suppose that , . Then the solution to difference scheme (12)–(15) converges to the solution of problem (3)–(5) in the sense of norm , and the convergent rate is .

Proof. Subtracting (12)–(15) from (32) and letting , we have Taking the inner product of (43) and , and using boundary condition (44), we obtain Noticing that we can obtain that from the Cauchy-Schwarz inequality and (30), and (44).
Then taking the inner product of (41) and , and combining with boundary condition (44) again, one can get
Similar to (23), we have By using Lemma 4, Theorem 5, Lemma 1, and the Cauchy-Schwarz inequality, we can get Taking (49)-(50) into (48), we obtain Set Summarizing (51) from to , we get From (42) and (47), we know that
Furthermore, Therefore, The discrete Gronwall inequality (see [30]) implies that Then we can obtain from the discrete Sobolev inequality.

We could prove the following theorems in a similar way of Theorem 6.

Theorem 7. Suppose that and . The solution to difference scheme (12)–(15) is stable in the sense of norm .

5. Numerical Simulations

As the difference scheme (12)–(15) is a linear system about , it does not need iteration when we solve it numerically.

Let , , , and For some different values of and , we list errors at several times in Table 1 and verify the accuracy of the difference scheme in Table 2. The numerical simulation of two conservative quantities (7) and (8) is listed in Table 3.

tab1
Table 1: The errors estimates of numerical solution with various and .
tab2
Table 2: The numerical verification of theoretical accuracy .
tab3
Table 3: The numerical simulation for conservative quantities (7) and (8).

The stability and convergence of the scheme are verified by these numerical experiments. And it shows that our proposed algorithm is effective and reliable.

Acknowledgments

The authors are very grateful to the anonymous referees for their careful reading and useful suggestions, which greatly improved the presentation of the paper. This work is supported in part by Scientific Research Foundation of Sichuan Provincial Education Department (no. 11ZB009), the Key Scientific Research Foundation of Xihua University (no. Z0912611), and the fund of Key Disciplinary of Computer Software and Theory, Sichuan, China, Grant no. SZD0802-09-1.

References

  1. P. Rosenau, “A quasi-continuous description of a nonlinear transmission line,” Physica Scripta, vol. 34, pp. 827–829, 1986. View at Google Scholar
  2. P. Rosenau, “Dynamics of dense discrete systems,” Progress of Theoretical Physics, vol. 79, pp. 1028–1042, 1988. View at Google Scholar
  3. M. A. Park, “On the Rosenau equation,” Matemática Aplicada e Computacional, vol. 9, no. 2, pp. 145–152, 1990. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. D. H. Peregrine, “Calculations of the development of an unduiar bore,” Journal of Fluid Mechanics, vol. 25, pp. 321–330, 1966. View at Google Scholar
  5. D. J. Kortewag and G. Devries, “On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves,” Philosophical Magazine, vol. 39, pp. 422–443, 1985. View at Google Scholar
  6. S. K. Chung and S. N. Ha, “Finite element Galerkin solutions for the Rosenau equation,” Applicable Analysis, vol. 54, no. 1-2, pp. 39–56, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. 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
  8. K. Omrani, F. Abidi, T. Achouri, and N. Khiari, “A new conservative finite difference scheme for the Rosenau equation,” Applied Mathematics and Computation, vol. 201, no. 1-2, pp. 35–43, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. S. K. Chung, “Finite difference approximate solutions for the Rosenau equation,” Applicable Analysis, vol. 69, no. 1-2, pp. 149–156, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. S. K. Chung and A. K. Pani, “Numerical methods for the Rosenau equation,” Applicable Analysis, vol. 77, no. 3-4, pp. 351–369, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. S. A. V. Manickam, A. K. Pani, and S. K. Chung, “A second-order splitting combined with orthogonal cubic spline collocation method for the Rosenau equation,” Numerical Methods for Partial Differential Equations, vol. 14, no. 6, pp. 695–716, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. J. Hu and K. Zheng, “Two conservative difference schemes for the generalized Rosenau equation,” Boundary Value Problems, vol. 2010, Article ID 543503, 18 pages, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. Y. D. Kim and H. Y. Lee, “The convergence of finite element Galerkin solution for the Roseneau equation,” The Korean Journal of Computational & Applied Mathematics, vol. 5, no. 1, pp. 171–180, 1998. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. Q. S. Chang, G. B. Wang, and B. L. Guo, “Conservative scheme for a model of nonlinear dispersive waves and its solitary waves induced by boundary motion,” Journal of Computational Physics, vol. 93, no. 2, pp. 360–375, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. D. Bhardwaj and R. Shankar, “A computational method for regularized long wave equation,” Computers & Mathematics with Applications, vol. 40, no. 12, pp. 1397–1404, 2000, Advances in partial differential equations, III. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. L. M. Zhang and Q. S. Chang, “A new finite difference method for regularized long wave equation,” Chinese Journal on Numerical Methods and Computer Applications, vol. 23, pp. 58–66, 2000. View at Google Scholar · View at MathSciNet
  17. A. A. Soliman and M. H. Hussien, “Collocation solution for RLW equation with septic spline,” Applied Mathematics and Computation, vol. 161, no. 2, pp. 623–636, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. J. I. Ramos, “Explicit finite difference methods for the EW and RLW equations,” Applied Mathematics and Computation, vol. 179, no. 2, pp. 622–638, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. H. Gu and N. Chen, “Least-squares mixed finite element methods for the RLW equations,” Numerical Methods for Partial Differential Equations, vol. 24, no. 3, pp. 749–758, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. Siraj-ul-Islam, S. Haq, and A. Ali, “A meshfree method for the numerical solution of the RLW equation,” Journal of Computational and Applied Mathematics, vol. 223, no. 2, pp. 997–1012, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. L. Mei and Y. Chen, “Numerical solutions of RLW equation using Galerkin method with extrapolation techniques,” Computer Physics Communications, vol. 183, no. 8, pp. 1609–1616, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  24. J.-M. Zuo, Y.-M. Zhang, T.-D. Zhang, and F. Chang, “A new conservative difference scheme for the general Rosenau-RLW equation,” Boundary Value Problems, vol. 2010, Article ID 516260, 13 pages, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. 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
  26. R. C. Mittal and R. K. Jain, “Numerical solution of general Rosenau-RLW equation using quintic B-splines collocation method,” Communications in Numerical Analysis, vol. 2012, Article ID 00129, 16 pages, 2012. View at Google Scholar · View at MathSciNet
  27. X. Pan and L. Zhang, “On the convergence of a conservative numerical scheme for the usual Rosenau-RLW equation,” Applied Mathematical Modelling, vol. 36, no. 8, pp. 3371–3378, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. 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
  29. Z. Fei, V. M. Pérez-García, and L. Vázquez, “Numerical simulation of nonlinear Schrödinger systems: a new conservative scheme,” Applied Mathematics and Computation, vol. 71, no. 2-3, pp. 165–177, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. Y. L. Zhou, Applications of Discrete Functional Analysis to the Finite Difference Method, International Academic Publishers, Beijing, China, 1990. View at MathSciNet