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

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.

In this paper, we consider the following initial-boundary value problem of the general Rosenau-RLW equation:

(11)

with an initial condition

(12)

and boundary conditions

(13)

where is a integer and is a known smooth function. When , (1.1) is called as usual Rosenau-RLW equation. When , (1.1) is called as modified Rosenau-RLW (MRosenau-RLW) equation. The initial boundary value problem (1.1)–(1.3) possesses the following conservative quantities:

(14)

(15)

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.

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

For convenience, we introduce the following notations

(21)

where and denote the spatial and temporal mesh sizes, , , respectively,

(22)

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

Since , then the finite difference scheme for the problem (1.1)–(1.3) is written as follows:

(23)

(24)

(25)

Lemma 2.1 (see [12]).

For any two mesh functions, , one has

(26)

Furthermore, if , then

(27)

Theorem 2.2.

Suppose that , then scheme (2.3)–(2.5) is conservative in the senses:

(28)

(29)

Proof.

Multiplying (2.3) with , according to boundary condition (2.5), and then summing up for from 1 to , we have

(210)

Let

(211)

Then (2.8) is gotten from (2.10).

Computing the inner product of (2.3) with , according to boundary condition (2.5) and Lemma 2.1, we obtain

(212)

where

(213)

According to

(214)

we have . It follows from (2.12) that

(215)

Let

(216)

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).

We define on as follows:

(217)

where . Computing the inner product of (2.17) with and considering and , we obtain

(218)

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).

For any discrete function on the finite interval , there is the inequality

(219)

where are two constants independent of and step length .

Lemma 2.6 (discrete Gronwall's inequality).

Suppose that the discrete function satisfies the inequality

(220)

where and are nonnegative constants. Then

(221)

where is sufficiently small, such that .

Theorem 2.7.

Suppose that , then the following inequalities

(222)

hold.

Proof.

It is follows from (2.9) that

(223)

According to Lemma 2.5, we obtain

(224)

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

First, we consider the convergence of scheme (2.3)–(2.5). We define the truncation error as follows:

(225)

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

Theorem 2.9.

Suppose that and , then the truncation errors of scheme (2.3)–(2.5) satisfy

(226)

as ,

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.

Subtracting (2.3) from (2.25) letting

(227)

we obtain

(228)

Computing the inner product of (2.28) with , we obtain

(229)

From the conservative property (1.5), it can be proved by Lemma 2.5 that . Then by Theorem 2.7 we can estimate (2.29) as follows:

(230)

According to the following inequality [11]

(231)

Substituting (2.30)–(2.31) into (2.29), we obtain

(232)

Let

(233)

then (2.32) can be rewritten as

(234)

Choosing suitable which is small enough, we obtain by Lemma 2.6 that

(235)

From the discrete initial conditions, we know that is of second-order accuracy, then

(236)

Then we have

(237)

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.

Assume that and both satisfy scheme (2.3)–(2.5), let , we obtain

(238)

Similarly to the proof of Theorem 2.10, we have

(239)

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.

In order to test the correction of the numerical analysis in this paper, we consider the following initial-boundary value problems of the general Rosenau-RLW equation:

(31)

with an initial condition

(32)

and boundary conditions

(33)

where . Then the exact solution of the initial value problem (3.1)-(3.2) is

(34)

where is wave velocity.

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 , .

Tables 1, 2, and 3 give the errors in the sense of -norm and -norm of the numerical solutions under various steps of and at for and . The three tables verify the second-order convergence and good stability of the numerical solutions. Tables 4, 5, and 6 shows the conservative law of discrete mass and discrete energy computed by scheme (2.3)–(2.5) for and .

Figures 1, 2, and 3 plot the exact solutions at and the numerical solutions computed by scheme (2.3)–(2.5) with at , which also show the accuracy of scheme (2.3)–(2.5).

Exact solutions of at and numerical solutions computed by scheme (2. 3)–(2.5) at for .

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Exact solutions of at and numerical solutions computed by scheme (2. 3)–(2.5) at for .

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Exact solutions of at and numerical solutions computed by scheme (2. 3)–(2.5) at for .

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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).

Affiliations

1. School of Science, Shandong University of Technology, Zibo, 255049, China

   • Jin-Ming Zuo
   •  & Yao-Ming Zhang

2. School of Mathematics, Shandong University, Jinan, 250100, China

   • Tian-De Zhang
   •  & Feng Chang

Corresponding author

Correspondence to Jin-Ming Zuo.

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