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

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

we have

. It follows from (2.12) that

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

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

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

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.