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

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.