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 ).
For any two mesh functions, , one has
Furthermore, if , then
Suppose that , then scheme (2.3)–(2.5) is conservative in the senses:
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 .
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.
There exists which satisfies scheme (2.3)–(2.5).
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  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 .
Suppose that , then the following inequalities
It is follows from (2.9) that
According to Lemma 2.5, we obtain
This completes the proof of Theorem 2.7.
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.
Suppose that and , then the truncation errors of scheme (2.3)–(2.5) satisfy
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.
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 
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
Then we have
It follows from Lemma 2.5, we have . This completes the proof of Theorem 2.10.
Scheme (2.3)–(2.5) is uniquely solvable.
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.
All results above in this paper are correct for initial-boundary value problem of the general Rosenau-RLW equation with finite or infinite boundary.