Boundary Value Problems

, 2010:781750 | Cite as

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 Open image in new window solitary wave solutions are
The four invariants and some numerical results have been obtained in [1], where Open image in new window is the velocity, Open image in new window . Obviously, eliminating Open image in new window from (1.1), we get a class of SRLWE:

Equation (1.3) is explicitly symmetric in the Open image in new window and Open image in new window 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 Open image in new window are positive constants, Open image in new window is the dissipative coefficient, and Open image in new window 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 Open image in new window , Open image in new window , the solution of (1.4)–(1.7) satisfies Open image in new window , Open image in new window , Open image in new window , and Open image in new window , where Open image in new window is a generic positive constant that varies in the context.

Proof.

Multiplying (1.4) by Open image in new window and integrating over Open image in new window , we have
According to
Then, multiplying (1.5) by Open image in new window and integrating over Open image in new window , we have
Adding (1.14) to (1.11), we obtain

So Open image in new window is decreasing with respect to Open image in new window , which implies that Open image in new window , Open image in new window . Then, it indicates that Open image in new window , Open image in new window , and Open image in new window . It is followed from Sobolev inequality that Open image in new window .

2. Finite Difference Scheme and Its Error Estimation

Let Open image in new window and Open image in new window be the uniform step size in the spatial and temporal direction, respectively. Denote Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , and Open image in new window . 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 Open image in new window

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

There exist two constants Open image in new window and Open image in new window such that

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

Suppose that Open image in new window , Open image in new window are nonnegative functions and Open image in new window is nondecreasing. If Open image in new window and

Then Open image in new window .

Theorem 2.4.

If Open image in new window , Open image in new window , then the solution of (2.2)–(2.5) satisfies

Proof.

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

From Lemma 2.3, we obtain Open image in new window , which implies that, Open image in new window , Open image in new window , and Open image in new window . By Lemma 2.2, we obtain Open image in new window .

Theorem 2.5.

Assume that Open image in new window , Open image in new window , the solution of difference scheme (2.2)–(2.5) satisfies:

Proof.

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

By Lemma 2.3, we get Open image in new window , which implies that Open image in new window , Open image in new window . It follows from Theorem 2.4 and Lemma 2.2 that Open image in new window , Open image in new window .

3. Solvability

Theorem 3.1.

The solution Open image in new window of (2.2)–(2.5) is unique.

Proof.

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

which implies that (3.1)-(3.2) have only zero solution. So the solution Open image in new window and Open image in new window of (2.2)–(2.5) is unique.

4. Convergence and Stability

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

Making use of Taylor expansion, it holds Open image in new window if Open image in new window .

Theorem 4.1.

Assume that Open image in new window , Open image in new window , then the solution Open image in new window and Open image in new window in the senses of norms Open image in new window and Open image in new window , 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 Open image in new window .

Proof.

Subtracting (2.2) from (4.1) subtracting (2.3) from (4.2), and letting Open image in new window , Open image in new window , we have
Computing the inner product of (4.3) with Open image in new window , we get
According to
it follow from Lemma 1.1, Theorems 2.4, and 2.5 that
By the Schwarz inequality, we obtain
it follows from (4.9)–(4.10) and (4.6) that
Computing the inner product of (4.4) with Open image in new window , we obtain
Adding (4.12) to (4.11), we have
If Open image in new window is sufficiently small which satisfies Open image in new window , then
Summing up (4.16) from 1 to Open image in new window , we have
Select appropriate second-order methods (such as the C-N Schemes), and calculate Open image in new window and Open image in new window , which satisfies
Noticing that
we then have
By Lemma 2.3, we get
This yields
By Lemma 2.2, we have

Similarly to Theorem 4.1, we can prove the result as follows.

Theorem 4.2.

Under the conditions of Theorem 4.1, the solution Open image in new window and Open image in new window of (2.2)–(2.5) is stable in the senses of norm Open image in new window and Open image in new window , 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 Open image in new window , Open image in new window . Then, reusing initial value Open image in new window , Open image in new window , we can work out Open image in new window . Iterative numerical calculation is not required, for this scheme is linear, so it saves computing time.

When Open image in new window , 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 Open image in new window , Open image in new window , Open image in new window , and Open image in new window . 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 Open image in new window as the reference solution. In Table 1, we give the ratios in the sense of Open image in new window at various time steps.
When Open image in new window , a wave figure comparison of Open image in new window and Open image in new window at various time steps is as in Figures 1 and 2.
Figure 1

When Open image in new window , the wave graph of Open image in new window at various times.

Figure 2

When Open image in new window , the wave graph of Open image in new window at various times.

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 Open image in new window 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).

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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

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