# 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}

**2010**:781750

https://doi.org/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 initial-boundary problem of dissipative symmetric regularized long wave equations with damping term by finite difference method. A linear three-level implicit finite difference scheme is designed. Existence and uniqueness of numerical solutions are derived. It is proved that the finite difference scheme is of second-order convergence and unconditionally stable by the discrete energy method. Numerical simulations verify that the method is accurate and efficient.

## Keywords

## 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 initial-boundary 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]).

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 three-implicit finite difference scheme can not start by itself, we need to select other two-level schemes (such as the C-N 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.

From Table 1, it is easy to find that the difference scheme in this paper is second-order 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. XZD0910-09-1). 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 regularized-long-wave equation.***Physics of Fluids*1984,**27**(1):4-7. 10.1063/1.864487View ArticleMATHGoogle Scholar - Albert J:
**On the decay of solutions of the generalized Benjamin-Bona-Mahony equations.***Journal of Mathematical Analysis and Applications*1989,**141**(2):527-537. 10.1016/0022-247X(89)90195-9MathSciNetView ArticleMATHGoogle Scholar - Amick CJ, Bona JL, Schonbek ME:
**Decay of solutions of some nonlinear wave equations.***Journal of Differential Equations*1989,**81**(1):1-49. 10.1016/0022-0396(89)90176-9MathSciNetView ArticleMATHGoogle Scholar - Ogino T, Takeda S:
**Computer simulation and analysis for the spherical and cylindrical ion-acoustic solitons.***Journal of the Physical Society of Japan*1976,**41**(1):257-264. 10.1143/JPSJ.41.257MathSciNetView ArticleGoogle Scholar - Makhankov VG:
**Dynamics of classical solitons (in non-integrable systems).***Physics Reports. Section C*1978,**35**(1):1-128.MathSciNetView ArticleGoogle Scholar - Clarkson PA:
**New similarity reductions and Painlevé analysis for the symmetric regularised long wave and modified Benjamin-Bona-Mahoney equations.***Journal of Physics A*1989,**22**(18):3821-3848. 10.1088/0305-4470/22/18/020MathSciNetView ArticleMATHGoogle Scholar - Bogolubsky IL:
**Some examples of inelastic soliton interaction.***Computer Physics Communications*1977,**13**(3):149-155. 10.1016/0010-4655(77)90009-1View ArticleGoogle Scholar - Guo B:
**The spectral method for symmetric regularized wave equations.***Journal of Computational Mathematics*1987,**5**(4):297-306.MathSciNetMATHGoogle Scholar - Zheng JD, Zhang RF, Guo BY:
**The Fourier pseudo-spectral method for the SRLW equation.***Applied Mathematics and Mechanics*1989,**10**(9):801-810.MathSciNetGoogle Scholar - Zheng JD:
**Pseudospectral collocation methods for the generalized SRLW equations.***Mathematica Numerica Sinica*1989,**11**(1):64-72.MATHGoogle Scholar - Shang YD, Guo B:
**Legendre and Chebyshev pseudospectral methods for the generalized symmetric regularized long wave equations.***Acta Mathematicae Applicatae Sinica*2003,**26**(4):590-604.MathSciNetMATHGoogle Scholar - Bai Y, Zhang LM:
**A conservative finite difference scheme for symmetric regularized long wave equations.***Acta Mathematicae Applicatae Sinica*2007,**30**(2):248-255.MathSciNetMATHGoogle Scholar - Wang T, Zhang L, Chen F:
**Conservative schemes for the symmetric regularized long wave equations.***Applied Mathematics and Computation*2007,**190**(2):1063-1080. 10.1016/j.amc.2007.01.105MathSciNetView ArticleMATHGoogle Scholar - Wang TC, Zhang LM:
**Pseudo-compact conservative finite difference approximate solution for the symmetric regularized long wave equation.***Acta Mathematica Scientia. Series A*2006,**26**(7):1039-1046.MathSciNetMATHGoogle Scholar - Wang TC, Zhang LM, Chen FQ:
**Pseudo-compact conservative finite difference approximate solutions for symmetric regularized-long-wave equations.***Chinese Journal of Engineering Mathematics*2008,**25**(1):169-172.MathSciNetMATHGoogle Scholar - 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):35-45.MathSciNetMATHGoogle Scholar - 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):745-757.MathSciNetGoogle Scholar - 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):191-204. 10.1007/s10255-003-0095-1MathSciNetView ArticleMATHGoogle Scholar - Shang Y, Guo B:
**Exponential attractor for the generalized symmetric regularized long wave equation with damping term.***Applied Mathematics and Mechanics*2005,**26**(3):259-266.MathSciNetGoogle Scholar - Shaomei F, Boling G, Hua Q:
**The existence of global attractors for a system of multi-dimensional symmetric regularized wave equations.***Communications in Nonlinear Science and Numerical Simulation*2009,**14**(1):61-68. 10.1016/j.cnsns.2007.07.001MathSciNetView ArticleMATHGoogle Scholar - Hu B, Xu Y, Hu J:
**Crank-Nicolson finite difference scheme for the Rosenau-Burgers equation.***Applied Mathematics and Computation*2008,**204**(1):311-316. 10.1016/j.amc.2008.06.051MathSciNetView ArticleMATHGoogle 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.