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

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## 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

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]).

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.

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.

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

Lemma 2.1.

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

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

Then Open image in new window .

Theorem 2.4.

Proof.

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.

Proof.

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.

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

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.

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.

The error ratios in the sense of Open image in new window at various time steps.

| ||||

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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