Boundary Value Problems

, 2010:781750

# A Linear Difference Scheme for Dissipative Symmetric Regularized Long Wave Equations with Damping Term

Open Access
Research Article

## 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

Solitary Wave Finite Difference Scheme Finite Difference Method Solitary Wave Solution Pseudospectral Method

## 1. Introduction

A symmetric version of regularized long wave equation (SRLWE),
has been proposed to model the propagation of weakly nonlinear ion acoustic and space charge waves [1]. The solitary wave solutions are
The four invariants and some numerical results have been obtained in [1], where is the velocity, . Obviously, eliminating from (1.1), we get a class of SRLWE:

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, 5, 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, 10, 11, 12, 13, 14, 15]).

In applications, the viscous damping effect is inevitable, and it plays the same important role as the dispersive effect. Therefore, it is more significant to study the dissipative symmetric regularized long wave equations with the damping term

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, 17, 18, 19, 20]). But it is difficult to find the analytical solution to (1.4)-(1.5), which makes numerical solution important.

To authors' knowledge, the finite difference method to dissipative SRLWEs with damping term (1.4)-(1.5) has not been studied till now. In this paper, we propose linear three level implicit finite difference scheme for (1.4)-(1.5) with
and the boundary conditions

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.

Let
Multiplying (1.4) by and integrating over , we have
According to
(1.10)
we get
(1.11)
Then, multiplying (1.5) by and integrating over , we have
(1.12)
By
(1.13)
we get
(1.14)
Adding (1.14) to (1.11), we obtain
(1.15)

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

Let and be the uniform step size in the spatial and temporal direction, respectively. Denote , , , , , and . We define the difference operators as follows:
Then, the average three-implicit finite difference scheme for the solution of (1.4)–(1.7) is as follow:

Lemma 2.1.

Summation by parts follows [12, 21] that for any two discrete functions

Lemma 2.2 (discrete Sobolev's inequality [12, 21]).

There exist two constants and such that

Lemma 2.3 (discrete Gronwall inequality [12, 21]).

Suppose that , are nonnegative functions and is nondecreasing. If and

Theorem 2.4.

If , , then the solution of (2.2)–(2.5) satisfies

Proof.

Taking an inner product of (2.2) with   (i.e., ) and considering the boundary condition (2.5) and Lemma 2.1, we obtain
(2.10)
where . Since
(2.11)
we obtain
(2.12)
Taking an inner product of (2.3) with  (i.e., ), we obtain
(2.13)
Adding (2.12) to (2.13), we have
(2.14)
Since
(2.15)
Equation (2.14) can be changed to
(2.16)
Let , and (2.16) is changed to
(2.17)
If is sufficiently small which satisfies , then
(2.18)
Summing up (2.18) from 1 to , we have
(2.19)

From Lemma 2.3, we obtain , which implies that, , , and . By Lemma 2.2, we obtain .

Theorem 2.5.

Assume that , , the solution of difference scheme (2.2)–(2.5) satisfies:
(2.20)

Proof.

Differentiating backward (2.2)–(2.5) with respect to , we obtain
(2.21)
(2.22)
(2.23)
(2.24)
Computing the inner product of (2.21) with   (i.e., ) and considering (2.24) and Lemma 2.1, we obtain
(2.25)
where . It follows from Theorem 2.4 that
(2.26)
By the Schwarz inequality and Lemma 2.1, we get
(2.27)
Noting that
(2.28)
it follows from (2.25) that
(2.29)
Computing the inner product of (2.22) with (i.e., ) and considering (2.24) and Lemma 2.1, we obtain
(2.30)
Since
(2.31)
then (2.30) is changed to
(2.32)
Adding (2.29) to (2.32), we have
(2.33)
Leting , we obtain . Choosing suitable which is small enough to satisfy , we get
(2.34)
Summing up (2.34) from 1 to , we have
(2.35)

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.

Using the mathematical induction, clearly, , are uniquely determined by initial conditions (2.4). then select appropriate second-order methods (such as the C-N Schemes) and calculate and (i.e. , , and , are uniquely determined). Assume that and are the only solution, now consider and in (2.2) and (2.3):
Taking an inner product of (3.1) with , we have
Since
then it holds
Taking an inner product of (3.2) with and adding to (3.5), we have

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

Let and be the solution of problem (1.4)–(1.7); that is, , , then the truncation of the difference scheme (2.2)–(2.5) is

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.

Subtracting (2.2) from (4.1) subtracting (2.3) from (4.2), and letting , , we have
where
Computing the inner product of (4.3) with , we get
According to
it follow from Lemma 1.1, Theorems 2.4, and 2.5 that
By the Schwarz inequality, we obtain
Since
(4.10)
it follows from (4.9)–(4.10) and (4.6) that
(4.11)
Computing the inner product of (4.4) with , we obtain
(4.12)
Adding (4.12) to (4.11), we have
(4.13)
Leting
(4.14)
we get
(4.15)
If is sufficiently small which satisfies , then
(4.16)
Summing up (4.16) from 1 to , we have
(4.17)
Select appropriate second-order methods (such as the C-N Schemes), and calculate and , which satisfies
(4.18)
Noticing that
(4.19)
we then have
(4.20)
By Lemma 2.3, we get
(4.21)
This yields
(4.22)
By Lemma 2.2, we have
(4.23)

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.

When , the damping does not have an effect and the dissipative will not appear. So the initial conditions of (1.4)–(1.7) are same as those of (1.1):
Let , , , and . Since we do not know the exact solution of (1.4)-(1.5), an error estimates method in [21] is used: a comparison between the numerical solutions on a coarse mesh and those on a refine mesh is made. We consider the solution on mesh as the reference solution. In Table 1, we give the ratios in the sense of at various time steps.
Table 1

The error ratios in the sense of at various time steps.

μ

When , a wave figure comparison of and at various time steps is as in Figures 1 and 2.

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.

## Notes

### 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).

### References

1. 1.
Seyler CE, Fenstermacher DL: A symmetric regularized-long-wave equation. Physics of Fluids 1984,27(1):4-7. 10.1063/1.864487
2. 2.
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)90195-9
3. 3.
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)90176-9
4. 4.
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.41.257
5. 5.
Makhankov VG: Dynamics of classical solitons (in non-integrable systems). Physics Reports. Section C 1978,35(1):1-128.
6. 6.
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. 10.1088/0305-4470/22/18/020
7. 7.
Bogolubsky IL: Some examples of inelastic soliton interaction. Computer Physics Communications 1977,13(3):149-155. 10.1016/0010-4655(77)90009-1
8. 8.
Guo B: The spectral method for symmetric regularized wave equations. Journal of Computational Mathematics 1987,5(4):297-306.
9. 9.
Zheng JD, Zhang RF, Guo BY: The Fourier pseudo-spectral method for the SRLW equation. Applied Mathematics and Mechanics 1989,10(9):801-810.
10. 10.
Zheng JD: Pseudospectral collocation methods for the generalized SRLW equations. Mathematica Numerica Sinica 1989,11(1):64-72.
11. 11.
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.
12. 12.
Bai Y, Zhang LM: A conservative finite difference scheme for symmetric regularized long wave equations. Acta Mathematicae Applicatae Sinica 2007,30(2):248-255.
13. 13.
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.2007.01.105
14. 14.
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.
15. 15.
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.
16. 16.
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.
17. 17.
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.
18. 18.
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/s10255-003-0095-1
19. 19.
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.
20. 20.
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.2007.07.001
21. 21.
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.2008.06.051

## Copyright information

© The Author(s) Jinsong Hu et al. 2010

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.

## Authors and Affiliations

1. 1.School of Mathematics and Computer EngineeringXihua UniversityChengduChina
2. 2.School of MathematicsSichuan UniversityChengduChina