A Linear Difference Scheme for Dissipative Symmetric Regularized Long Wave Equations with Damping Term
 Jinsong Hu^{1},
 Youcai Xu^{2}Email author and
 Bing Hu^{2}
DOI: 10.1155/2010/781750
© The Author(s) Jinsong Hu et al. 2010
Received: 24 August 2010
Accepted: 14 November 2010
Published: 30 November 2010
Abstract
We study the initialboundary problem of dissipative symmetric regularized long wave equations with damping term by finite difference method. A linear threelevel implicit finite difference scheme is designed. Existence and uniqueness of numerical solutions are derived. It is proved that the finite difference scheme is of secondorder convergence and unconditionally stable by the discrete energy method. Numerical simulations verify that the method is accurate and efficient.
1. Introduction
Equation (1.3) is explicitly symmetric in the and derivatives and is very similar to the regularized long wave equation that describes shallow water waves and plasma drift waves [2, 3]. The SRLW equation also arises in many other areas of mathematical physics [4–6]. Numerical investigation indicates that interactions of solitary waves are inelastic [7]; thus, the solitary wave of the SRLWE is not a solution. Research on the wellposedness for its solution and numerical methods has aroused more and more interest. In [8], Guo studied the existence, uniqueness, and regularity of the numerical solutions for the periodic initial value problem of generalized SRLW by the spectral method. In [9], Zheng et al. presented a Fourier pseudospectral method with a restraint operator for the SRLWEs and proved its stability and obtained the optimum error estimates. There are other methods such as pseudospectral method, finite difference method for the initialboundary value problem of SRLWEs (see [9–15]).
where are positive constants, is the dissipative coefficient, and is the damping coefficient. Equations (1.4)(1.5) are a reasonable model to render essential phenomena of nonlinear ion acoustic wave motion when dissipation is considered. Existence, uniqueness, and wellposedness of global solutions to (1.4)(1.5) are presented (see [16–20]). But it is difficult to find the analytical solution to (1.4)(1.5), which makes numerical solution important.
We show that this difference scheme is uniquely solvable, convergent, and stable in both theoretical and numerical senses.
Lemma 1.1.
Suppose that , , the solution of (1.4)–(1.7) satisfies , , , and , where is a generic positive constant that varies in the context.
Proof.
So is decreasing with respect to , which implies that , . Then, it indicates that , , and . It is followed from Sobolev inequality that .
2. Finite Difference Scheme and Its Error Estimation
Lemma 2.1.
Lemma 2.2 (discrete Sobolev's inequality [12, 21]).
Lemma 2.3 (discrete Gronwall inequality [12, 21]).
Then .
Theorem 2.4.
Proof.
From Lemma 2.3, we obtain , which implies that, , , and . By Lemma 2.2, we obtain .
Theorem 2.5.
Proof.
By Lemma 2.3, we get , which implies that , . It follows from Theorem 2.4 and Lemma 2.2 that , .
3. Solvability
Theorem 3.1.
The solution of (2.2)–(2.5) is unique.
Proof.
which implies that (3.1)(3.2) have only zero solution. So the solution and of (2.2)–(2.5) is unique.
4. Convergence and Stability
Making use of Taylor expansion, it holds if .
Theorem 4.1.
Assume that , , then the solution and in the senses of norms and , respectively, to the difference scheme (2.2)–(2.5) converges to the solution of problem (1.4)–(1.7) and the order of convergence is .
Proof.
Similarly to Theorem 4.1, we can prove the result as follows.
Theorem 4.2.
Under the conditions of Theorem 4.1, the solution and of (2.2)–(2.5) is stable in the senses of norm and , respectively.
5. Numerical Simulations
Since the threeimplicit finite difference scheme can not start by itself, we need to select other twolevel schemes (such as the CN Scheme) to get , . Then, reusing initial value , , we can work out . Iterative numerical calculation is not required, for this scheme is linear, so it saves computing time.
The error ratios in the sense of at various time steps.


 

μ 







 



 



 



 








 



 



 




From Table 1, it is easy to find that the difference scheme in this paper is secondorder convergent. Figures 1 and 2 show that the height of wave crest is more and more low with time elapsing due to the effect of damping and dissipativeness. It simulates that the continue energy of problem (1.4)–(1.7) in Lemma 1.1 is digressive. Numerical experiments show that the finite difference scheme is efficient.
Declarations
Acknowledgments
The work of Jinsong Hu was supported by the research fund of key disciplinary of application mathematics of Xihua University (Grant no. XZD0910091). The work of Youcai Xu was supported by the Youth Research Foundation of Sichuan University (no. 2009SCU11113).
Authors’ Affiliations
References
 Seyler CE, Fenstermacher DL: A symmetric regularizedlongwave equation. Physics of Fluids 1984,27(1):47. 10.1063/1.864487View ArticleMATH
 Albert J: On the decay of solutions of the generalized BenjaminBonaMahony equations. Journal of Mathematical Analysis and Applications 1989,141(2):527537. 10.1016/0022247X(89)901959MathSciNetView ArticleMATH
 Amick CJ, Bona JL, Schonbek ME: Decay of solutions of some nonlinear wave equations. Journal of Differential Equations 1989,81(1):149. 10.1016/00220396(89)901769MathSciNetView ArticleMATH
 Ogino T, Takeda S: Computer simulation and analysis for the spherical and cylindrical ionacoustic solitons. Journal of the Physical Society of Japan 1976,41(1):257264. 10.1143/JPSJ.41.257MathSciNetView Article
 Makhankov VG: Dynamics of classical solitons (in nonintegrable systems). Physics Reports. Section C 1978,35(1):1128.MathSciNetView Article
 Clarkson PA: New similarity reductions and Painlevé analysis for the symmetric regularised long wave and modified BenjaminBonaMahoney equations. Journal of Physics A 1989,22(18):38213848. 10.1088/03054470/22/18/020MathSciNetView ArticleMATH
 Bogolubsky IL: Some examples of inelastic soliton interaction. Computer Physics Communications 1977,13(3):149155. 10.1016/00104655(77)900091View Article
 Guo B: The spectral method for symmetric regularized wave equations. Journal of Computational Mathematics 1987,5(4):297306.MathSciNetMATH
 Zheng JD, Zhang RF, Guo BY: The Fourier pseudospectral method for the SRLW equation. Applied Mathematics and Mechanics 1989,10(9):801810.MathSciNet
 Zheng JD: Pseudospectral collocation methods for the generalized SRLW equations. Mathematica Numerica Sinica 1989,11(1):6472.MATH
 Shang YD, Guo B: Legendre and Chebyshev pseudospectral methods for the generalized symmetric regularized long wave equations. Acta Mathematicae Applicatae Sinica 2003,26(4):590604.MathSciNetMATH
 Bai Y, Zhang LM: A conservative finite difference scheme for symmetric regularized long wave equations. Acta Mathematicae Applicatae Sinica 2007,30(2):248255.MathSciNetMATH
 Wang T, Zhang L, Chen F: Conservative schemes for the symmetric regularized long wave equations. Applied Mathematics and Computation 2007,190(2):10631080. 10.1016/j.amc.2007.01.105MathSciNetView ArticleMATH
 Wang TC, Zhang LM: Pseudocompact conservative finite difference approximate solution for the symmetric regularized long wave equation. Acta Mathematica Scientia. Series A 2006,26(7):10391046.MathSciNetMATH
 Wang TC, Zhang LM, Chen FQ: Pseudocompact conservative finite difference approximate solutions for symmetric regularizedlongwave equations. Chinese Journal of Engineering Mathematics 2008,25(1):169172.MathSciNetMATH
 Shang Y, Guo B, Fang S: Long time behavior of the dissipative generalized symmetric regularized long wave equations. Journal of Partial Differential Equations 2002,15(1):3545.MathSciNetMATH
 Shang YD, Guo B: Global attractors for a periodic initial value problem for dissipative generalized symmetric regularized long wave equations. Acta Mathematica Scientia. Series A 2003,23(6):745757.MathSciNet
 Guo B, Shang Y: Approximate inertial manifolds to the generalized symmetric regularized long wave equations with damping term. Acta Mathematicae Applicatae Sinica 2003,19(2):191204. 10.1007/s1025500300951MathSciNetView ArticleMATH
 Shang Y, Guo B: Exponential attractor for the generalized symmetric regularized long wave equation with damping term. Applied Mathematics and Mechanics 2005,26(3):259266.MathSciNet
 Shaomei F, Boling G, Hua Q: The existence of global attractors for a system of multidimensional symmetric regularized wave equations. Communications in Nonlinear Science and Numerical Simulation 2009,14(1):6168. 10.1016/j.cnsns.2007.07.001MathSciNetView ArticleMATH
 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 ArticleMATH
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