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.