# Two Conservative Difference Schemes for the Generalized Rosenau Equation

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## Abstract

Numerical solutions for generalized Rosenau equation are considered and two energy conservative finite difference schemes are proposed. Existence of the solutions for the difference scheme has been shown. Stability, convergence, and priori error estimate of the scheme are proved using energy method. Numerical results demonstrate that two schemes are efficient and reliable.

## Keywords

Difference Scheme Fixed Point Theorem Finite Difference Scheme Finite Difference Method Discontinuous Galerkin Method## 1. Introduction

Consider the following initial-boundary value problem for generalized Rosenau equation:

with an initial condition

and boundary conditions

where Open image in new window is a integer.

When Open image in new window , (1.1) is called as usual Rosenau equation proposed by Rosenau [1] for treating the dynamics of dense discrete systems. Since then, the Cauchy problem of the Rosenau equation was investigated by Park [2]. Many numerical schemes have been proposed, such as Open image in new window -conforming finite element method by Chung and Pani [3], discontinuous Galerkin method by Choo et al. [4], orthogonal cubic spline collocation method by Manickam [5], and finite difference method by Chung [6] and Omrani et al. [7]. As for the generalized case, however, there are few studies on theoretical analysis and numerical methods.

It can be proved easily that the problem (1.1)–(1.3) has the following conservative law:

Hence, we propose two conservative difference schemes which simulate conservative law (1.4). 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, a linearized difference scheme is proposed and theoretical results are obtained. In Section 4, some numerical experiments are shown.

## 2. Nonlinear Finite Difference Scheme

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 , and Open image in new window . Define

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

Since Open image in new window , then the following finite difference scheme is considered:

Lemma 2.1 (see [8]).

Theorem 2.2.

Proof.

According to

By the definition of Open image in new window , (2.7) holds.

To prove the existence of solution for scheme (2.2)–(2.4), the following Browder fixed point Theorem should be introduced. For the proof, see [9].

Lemma 2.3 (Browder fixed point Theorem).

Let Open image in new window be a finite dimensional inner product space. Suppose that Open image in new window is continuous and there exists an Open image in new window such that Open image in new window for all Open image in new window with Open image in new window . Then there exists Open image in new window such that Open image in new window and Open image in new window .

Theorem 2.4.

There exists Open image in new window satisfying the difference scheme (2.2)–(2.4).

Proof.

By the mathematical induction, for Open image in new window , assume that Open image in new window satisfy (2.2)–(2.4). Next we prove that there exists Open image in new window satisfying (2.2)–(2.4).

Define a operator Open image in new window on Open image in new window as follows:

Obviously, for all Open image in new window , Open image in new window with Open image in new window . It follows from Lemma 2.3 that there exists Open image in new window which satisfies Open image in new window . Let Open image in new window , it can be proved that Open image in new window is the solution of the scheme (2.2)–(2.4).

Next, we discuss the convergence and stability of the scheme (2.2)–(2.4). Let Open image in new window be the solution of problem (1.1)–(1.3), Open image in new window , then the truncation of the scheme (2.2)–(2.4) is

Using Taylor expansion, we know that Open image in new window holds if Open image in new window .

Lemma 2.5.

Proof.

Hence, Open image in new window . According to Sobolev inequality, we have Open image in new window .

Lemma 2.6 (Discrete Sobolev's inequality [10]).

Lemma 2.7 (Discrete Gronwall inequality [10]).

Theorem 2.8.

Suppose Open image in new window , then the solution Open image in new window of (2.2) satisfies Open image in new window , which yield Open image in new window .

Proof.

According to Lemma 2.6, we have Open image in new window .

Theorem 2.9.

Suppose Open image in new window , then the solution Open image in new window of the scheme (2.2)–(2.4) converges to the solution of problem (1.1)–(1.3) and the rate of convergence is Open image in new window .

Proof.

This completes the proof of Theorem 2.9.

Similarly, the following theorem can be proved.

Theorem 2.10.

Under the conditions of Theorem 2.9, the solution of the scheme (2.2)–(2.4) is stable by Open image in new window .

## 3. Linearized Finite Difference Scheme

In this section, we propose a linear-implicit finite difference scheme as follows:

Theorem 3.1.

Proof.

By the definition of Open image in new window , (3.2) holds.

Theorem 3.2.

The difference scheme (3.1) is uniquely solvable.

Proof.

That is, there uniquely exists trivial solution satisfying (3.8). Hence, Open image in new window in (3.1) is uniquely solvable.

To discuss the convergence and stability of the scheme (3.1), we denote the truncation of the scheme (3.1):

Using Taylor expansion, we know that Open image in new window holds if Open image in new window .

Theorem 3.3.

Suppose Open image in new window , then the solution of (3.1) satisfies Open image in new window , which yield Open image in new window .

Proof.

Theorem 3.4.

Suppose Open image in new window , then the solution Open image in new window of the schemes (3.1), (2.3), and (2.4) converges to the solution of problem (1.1)–(1.3) and the rate of convergence is Open image in new window .

Proof.

This completes the proof of Theorem 3.4.

Similarly, the following theorem can be proved that.

Theorem 3.5.

Under the conditions of Theorem 3.4, the solution of the schemes (3.1), (2.3), and (2.4) are stable by Open image in new window .

## 4. Numerical Experiments

Consider the generalized Rosenau equation:

with an initial condition

and boundary conditions

The errors estimates in the sense of Open image in new window , when Open image in new window and Open image in new window .

Scheme (2.2) | Scheme (3.1) | Scheme (2.2) | Scheme (3.1) | Scheme (2.2) | Scheme (3.1) | |

The errors estimates in the sense of Open image in new window , when Open image in new window and Open image in new window .

Scheme (2.2) | Scheme (3.1) | Scheme (2.2) | Scheme (3.1) | Scheme (2.2) | Scheme (3.1) | |

The errors estimates in the sense of Open image in new window , when Open image in new window and Open image in new window .

Scheme (2.2) | Scheme (3.1) | Scheme (2.2) | Scheme (3.1) | Scheme (2.2) | Scheme (3.1) | |

From the numerical results, two finite difference schemes of this paper are efficient.

## Notes

### Acknowledgment

This work was supported by the Youth Research Foundation of SWUST (no. 08zx3125).

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