 Research Article
 Open Access
A New Conservative Difference Scheme for the General RosenauRLW Equation
 JinMing Zuo^{1}Email author,
 YaoMing Zhang^{1},
 TianDe Zhang^{2} and
 Feng Chang^{2}
 Received: 28 May 2010
 Accepted: 14 October 2010
 Published: 20 October 2010
Abstract
A new conservative finite difference scheme is presented for an initialboundary value problem of the general RosenauRLW equation. Existence of its difference solutions are proved by Brouwer fixed point theorem. It is proved by the discrete energy method that the scheme is uniquely solvable, unconditionally stable, and secondorder convergent. Numerical examples show the efficiency of the scheme.
Keywords
 Difference Scheme
 Fixed Point Theorem
 Truncation Error
 Discrete Function
 Discrete Energy
1. Introduction
It is known the conservative scheme is better than the nonconservative ones. Zhang et al. [1] point out that the nonconservative scheme may easily show nonlinear blow up. In [2] Li and VuQuoc 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". In [3–11], some conservative finite difference schemes were used for a system of the generalized nonlinear Schrödinger equations, Regularized long wave (RLW) equations, SineGordon equation, KleinGordon equation, Zakharov equations, Rosenau equation, respectively. Numerical results of all the schemes are very good. Hence, we propose a new conservative difference scheme for the general RosenauRLW equation, which simulates conservative laws (1.4) and (1.5) at the same time. The outline of the paper is as follows. In Section 2, a nonlinear difference scheme is proposed and corresponding convergence and stability of the scheme are proved. In Section 3, some numerical experiments are shown.
2. A NonlinearImplicit Conservative Scheme
In this section, we propose a nonlinearimplicit conservative scheme for the initialboundary value problem (1.1)–(1.3) and give its numerical analysis.
2.1. The NonlinearImplicit Scheme and Its Conservative Law
and in the paper, denotes a general positive constant, which may have different values in different occurrences.
Lemma 2.1 (see [12]).
Theorem 2.2.
Proof.
Then (2.8) is gotten from (2.10).
Then (2.9) is gotten from (2.15). This completes the proof of Theorem 2.2.
2.2. Existence and Prior Estimates of Difference Solution
To show the existence of the approximations for scheme (2.3)–(2.5), we introduce the following Brouwer fixed point theorem [13].
Lemma 2.3.
Let be a finitedimensional inner product space, be the associated norm, and be continuous. Assume, moreover, that there exist , for all , , . Then, there exists a such that and .
Let , , then have the following.
Theorem 2.4.
There exists which satisfies scheme (2.3)–(2.5).
Proof.
It follows from the original problem (1.1)–(1.3) that satisfies scheme (2.3)–(2.5). Assume there exists which satisfy scheme (2.3)–(2.5), as , now we try to prove that , satisfy scheme (2.3)–(2.5).
Hence, for all , there exists . It follows from Lemma 2.3 that exists which satisfies . Let , then it can be proved that is the solution of scheme (2.3)–(2.5). This completes the proof of Theorem 2.4.
Next we will give some priori estimates of difference solutions. First the following two lemmas [14] are introduced:
Lemma 2.5 (discrete Sobolev's estimate).
where are two constants independent of and step length .
Lemma 2.6 (discrete Gronwall's inequality).
where is sufficiently small, such that .
Theorem 2.7.
hold.
Proof.
This completes the proof of Theorem 2.7.
Remark 2.8.
Theorem 2.7 implies that scheme (2.3)–(2.5) is unconditionally stable.
2.3. Convergence and Uniqueness of Difference Solution
then from Taylor's expansion, we obtain the following.
Theorem 2.9.
as ,
Theorem 2.10.
Suppose that the conditions of Theorem 2.9 are satisfied, then the solution of scheme (2.3)–(2.5) converges to the solution of problem (1.1)–(1.3) with order in the norm.
Proof.
It follows from Lemma 2.5, we have . This completes the proof of Theorem 2.10.
Theorem 2.11.
Scheme (2.3)–(2.5) is uniquely solvable.
Proof.
This completes the proof of Theorem 2.11.
Remark 2.12.
All results above in this paper are correct for initialboundary value problem of the general RosenauRLW equation with finite or infinite boundary.
3. Numerical Experiments
where is wave velocity.
It follows from (3.4) that the initialboundary value problem (3.1)–(3.3) is consistent to the boundary value problem (3.3) for . In the following examples, we always choose , .
The errors of numerical solutions at with for .






0.4 

 
0.2 

 3.953 296  3.930 200 
0.1 

 3.987 130  3.980 050 
0.05 

 3.997 412  3.994 759 
0.025 

 4.221 051  4.151 348 
The errors of numerical solutions at with for .






0.4 

 
0.2 

 3.961 294  3.934 592 
0.1 

 3.996 350  3.988 283 
0.05 

 4.003 273  4.000 165 
0.025 

 4.290 286  4.242 688 
The errors of numerical solutions at with for .






0.4 

 
0.2 

 3.885 945  3.851 797 
0.1 

 3.975 084  3.965 980 
0.05 

 4.000 294  3.995 992 
0.025 

 4.391 818  4.381 199 
Discrete mass and discrete energy with at various for .

 

10  1.897 658 262 960 01  0.533 175 231 580 85 
20  1.897 658 268 873 21  0.533 175 231 872 51 
30  1.897 658 262 993 93  0.533 175 231 177 25 
40  1.897 658 265 568 93  0.533 175 231 478 09 
50  1.897 658 260 975 87  0.533 175 231 776 18 
60  1.897 658 265 384 88  0.533 175 231 074 05 
Discrete mass and discrete energy with at various for .

 

10  2.672 608 675 265 30  1.113 462 678 852 70 
20  2.672 608 676 236 58  1.113 462 678 465 22 
30  2.672 608 674 147 13  1.113 462 678 083 94 
40  2.672 608 672 639 88  1.113 462 678 711 58 
50  2.672 608 672 874 71  1.113 462 678 330 06 
60  2.672 608 679 729 44  1.113 462 678 958 71 
Discrete mass and discrete energy with at various for .

 

10  3.988 663 320 390 89  1.917 613 014 656 71 
20  3.988 663 260 854 26  1.917 613 014 739 89 
30  3.988 663 167 685 49  1.917 613 014 820 83 
40  3.988 663 194 506 97  1.917 613 014 927 44 
50  3.988 663 973 359 89  1.917 613 014 009 71 
60  3.988 663 621 972 59  1.917 613 014 679 17 
Declarations
Acknowledgments
The authors would like to express their sincere thanks to the referees for their valuable suggestions and comments. This paper is supported by the National Natural Science Foundation of China (nos. 10871117 and 10571110).
Authors’ Affiliations
References
 Zhang F, PérezGarcía VM, Vázquez L: Numerical simulation of nonlinear Schrödinger systems: a new conservative scheme. Applied Mathematics and Computation 1995,71(23):165177. 10.1016/00963003(94)00152TMathSciNetView ArticleMATHGoogle Scholar
 Li S, VuQuoc L: Finite difference calculus invariant structure of a class of algorithms for the nonlinear KleinGordon equation. SIAM Journal on Numerical Analysis 1995,32(6):18391875. 10.1137/0732083MathSciNetView ArticleMATHGoogle Scholar
 Chang Q, Xu L: A numerical method for a system of generalized nonlinear Schrödinger equations. Journal of Computational Mathematics 1986,4(3):191199.MathSciNetMATHGoogle Scholar
 Chang Q, Jia E, Sun W: Difference schemes for solving the generalized nonlinear Schrödinger equation. Journal of Computational Physics 1999,148(2):397415. 10.1006/jcph.1998.6120MathSciNetView ArticleMATHGoogle Scholar
 Wang TC, Zhang LM: Analysis of some new conservative schemes for nonlinear Schrödinger equation with wave operator. Applied Mathematics and Computation 2006,182(2):17801794. 10.1016/j.amc.2006.06.015MathSciNetView ArticleMATHGoogle Scholar
 Wang T, Guo B, Zhang L: New conservative difference schemes for a coupled nonlinear Schrödinger system. Applied Mathematics and Computation 2010,217(4):16041619. 10.1016/j.amc.2009.07.040MathSciNetView ArticleMATHGoogle Scholar
 Zhang L: A finite difference scheme for generalized regularized longwave equation. Applied Mathematics and Computation 2005,168(2):962972. 10.1016/j.amc.2004.09.027MathSciNetView ArticleMATHGoogle Scholar
 Zhang F, Vázquez L: Two energy conserving numerical schemes for the sineGordon equation. Applied Mathematics and Computation 1991,45(1):1730. 10.1016/00963003(91)900874MathSciNetView ArticleMATHGoogle Scholar
 Wong YS, Chang Q, Gong L: An initialboundary value problem of a nonlinear KleinGordon equation. Applied Mathematics and Computation 1997,84(1):7793. 10.1016/S00963003(96)000653MathSciNetView ArticleMATHGoogle Scholar
 Chang QS, Guo BL, Jiang H: Finite difference method for generalized Zakharov equations. Mathematics of Computation 1995,64(210):537553. 10.1090/S00255718199512846645MathSciNetView ArticleMATHGoogle Scholar
 Hu JS, Zheng KL: Two conservative difference schemes for the generalized Rosenau equation. Boundary Value Problems 2010, 2010:18.Google Scholar
 Hu B, Xu Y, Hu J: CrankNicolson finite difference scheme for the RosenauBurgers equation. Applied Mathematics and Computation 2008,204(1):311316. 10.1016/j.amc.2008.06.051MathSciNetView ArticleMATHGoogle Scholar
 Browder FE: Existence and uniqueness theorems for solutions of nonlinear boundary value problems. In Proceedings of Symposia in Applied Mathematics, vol. 17. American Mathematical Society, Providence, RI, USA; 1965:2449.Google Scholar
 Zhou Y: Applications of Discrete Functional Analysis to the Finite Difference Method. International Academic Publishers, Beijing, China; 1991:vi+260.MATHGoogle Scholar
Copyright
This article is published under license to BioMed Central Ltd. 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.