# A New Conservative Difference Scheme for the General Rosenau-RLW Equation

- Jin-Ming Zuo
^{1}Email author, - Yao-Ming Zhang
^{1}, - Tian-De Zhang
^{2}and - Feng Chang
^{2}

**Received: **28 May 2010

**Accepted: **14 October 2010

**Published: **20 October 2010

## Abstract

A new conservative finite difference scheme is presented for an initial-boundary value problem of the general Rosenau-RLW equation. Existence of its difference solutions are proved by Brouwer fixed point theorem. It is proved by the discrete energy method that the scheme is uniquely solvable, unconditionally stable, and second-order convergent. Numerical examples show the efficiency of the scheme.

## Keywords

## 1. Introduction

It is known the conservative scheme is better than the nonconservative ones. Zhang et al. [1] point out that the nonconservative scheme may easily show nonlinear blow up. In [2] Li and Vu-Quoc said " in some areas, the ability to preserve some invariant properties of the original differential equation is a criterion to judge the success of a numerical simulation". In [3–11], some conservative finite difference schemes were used for a system of the generalized nonlinear Schrödinger equations, Regularized long wave (RLW) equations, Sine-Gordon equation, Klein-Gordon equation, Zakharov equations, Rosenau equation, respectively. Numerical results of all the schemes are very good. Hence, we propose a new conservative difference scheme for the general Rosenau-RLW equation, which simulates conservative laws (1.4) and (1.5) at the same time. The outline of the paper is as follows. In Section 2, a nonlinear difference scheme is proposed and corresponding convergence and stability of the scheme are proved. In Section 3, some numerical experiments are shown.

## 2. A Nonlinear-Implicit Conservative Scheme

In this section, we propose a nonlinear-implicit conservative scheme for the initial-boundary value problem (1.1)–(1.3) and give its numerical analysis.

### 2.1. The Nonlinear-Implicit Scheme and Its Conservative Law

and in the paper, denotes a general positive constant, which may have different values in different occurrences.

Lemma 2.1 (see [12]).

Theorem 2.2.

Proof.

Then (2.8) is gotten from (2.10).

Then (2.9) is gotten from (2.15). This completes the proof of Theorem 2.2.

### 2.2. Existence and Prior Estimates of Difference Solution

To show the existence of the approximations
for scheme (2.3)–(2.5), we introduce the following *Brouwer* fixed point theorem [13].

Lemma 2.3.

Let be a finite-dimensional inner product space, be the associated norm, and be continuous. Assume, moreover, that there exist , for all , , . Then, there exists a such that and .

Let , , then have the following.

Theorem 2.4.

There exists which satisfies scheme (2.3)–(2.5).

Proof.

It follows from the original problem (1.1)–(1.3) that satisfies scheme (2.3)–(2.5). Assume there exists which satisfy scheme (2.3)–(2.5), as , now we try to prove that , satisfy scheme (2.3)–(2.5).

Hence, for all , there exists . It follows from Lemma 2.3 that exists which satisfies . Let , then it can be proved that is the solution of scheme (2.3)–(2.5). This completes the proof of Theorem 2.4.

Next we will give some priori estimates of difference solutions. First the following two lemmas [14] are introduced:

Lemma 2.5 (discrete Sobolev's estimate).

where are two constants independent of and step length .

Lemma 2.6 (discrete Gronwall's inequality).

where is sufficiently small, such that .

Theorem 2.7.

hold.

Proof.

This completes the proof of Theorem 2.7.

Remark 2.8.

Theorem 2.7 implies that scheme (2.3)–(2.5) is unconditionally stable.

### 2.3. Convergence and Uniqueness of Difference Solution

then from Taylor's expansion, we obtain the following.

Theorem 2.9.

Theorem 2.10.

Suppose that the conditions of Theorem 2.9 are satisfied, then the solution of scheme (2.3)–(2.5) converges to the solution of problem (1.1)–(1.3) with order in the norm.

Proof.

It follows from Lemma 2.5, we have . This completes the proof of Theorem 2.10.

Theorem 2.11.

Scheme (2.3)–(2.5) is uniquely solvable.

Proof.

This completes the proof of Theorem 2.11.

Remark 2.12.

All results above in this paper are correct for initial-boundary value problem of the general Rosenau-RLW equation with finite or infinite boundary.

## 3. Numerical Experiments

It follows from (3.4) that the initial-boundary value problem (3.1)–(3.3) is consistent to the boundary value problem (3.3) for . In the following examples, we always choose , .

## Declarations

### Acknowledgments

The authors would like to express their sincere thanks to the referees for their valuable suggestions and comments. This paper is supported by the National Natural Science Foundation of China (nos. 10871117 and 10571110).

## Authors’ Affiliations

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