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

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

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:

Lemma 2.1 (see [12]).

For any two mesh functions, , one has

Furthermore, if , then

Theorem 2.2.

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

Proof.

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

Let

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

where

According to

we have . It follows from (2.12) that

Let

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:

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

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

where are two constants independent of and step length .

Lemma 2.6 (discrete Gronwall's inequality).

Suppose that the discrete function satisfies the inequality

where and are nonnegative constants. Then

where is sufficiently small, such that .

Theorem 2.7.

Suppose that , then the following inequalities

hold.

Proof.

It is follows from (2.9) that

According to Lemma 2.5, we obtain

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:

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

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

we obtain

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

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:

According to the following inequality [11]

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

Let

then (2.32) can be rewritten as

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

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

Then we have

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

Similarly to the proof of Theorem 2.10, we have

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.