Let
and
be the uniform step size in the spatial and temporal direction, respectively. Denote
,
, and
. Define

and in the paper,
denotes a general positive constant which may have different values in different occurrences.

Since
, then the following finite difference scheme is considered:

Lemma 2.1 (see [8]).

For any two mesh functions,

, one has

Furthermore, if

, then

Theorem 2.2.

Suppose

, then the scheme (2.2)–(2.4) is conservative for discrete energy, that is,

Proof.

Computing the inner product of (2.2) with

, according to boundary condition (2.4) and Lemma 2.1, we have

According to

By the definition of
, (2.7) holds.

To prove the existence of solution for scheme (2.2)–(2.4), the following Browder fixed point Theorem should be introduced. For the proof, see [9].

Lemma 2.3 (Browder fixed point Theorem).

Let
be a finite dimensional inner product space. Suppose that
is continuous and there exists an
such that
for all
with
. Then there exists
such that
and
.

Theorem 2.4.

There exists
satisfying the difference scheme (2.2)–(2.4).

Proof.

By the mathematical induction, for
, assume that
satisfy (2.2)–(2.4). Next we prove that there exists
satisfying (2.2)–(2.4).

Define a operator
on
as follows:

Taking the inner product of (2.13) with

, we get

Obviously, for all
,
with
. It follows from Lemma 2.3 that there exists
which satisfies
. Let
, it can be proved that
is the solution of the scheme (2.2)–(2.4).

Next, we discuss the convergence and stability of the scheme (2.2)–(2.4). Let
be the solution of problem (1.1)–(1.3),
, then the truncation of the scheme (2.2)–(2.4) is

Using Taylor expansion, we know that
holds if
.

Lemma 2.5.

Suppose that

, then the solution of the initial-boundary value problem (1.1)–(1.3) satisfies

Proof.

It follows from (1.4) that

Using Hölder inequality and Schwartz inequality, we get

Hence,
. According to Sobolev inequality, we have
.

Lemma 2.6 (Discrete Sobolev's inequality [10]).

There exist two constant

and

such that

Lemma 2.7 (Discrete Gronwall inequality [10]).

Suppose

are nonnegative mesh functions and

is nondecreasing. If

and

Theorem 2.8.

Suppose
, then the solution
of (2.2) satisfies
, which yield
.

Proof.

It follows from (2.7) that

Using Lemma 2.1 and Schwartz inequality, we get

According to Lemma 2.6, we have
.

Theorem 2.9.

Suppose
, then the solution
of the scheme (2.2)–(2.4) converges to the solution of problem (1.1)–(1.3) and the rate of convergence is
.

Proof.

Subtracting (2.15) from (2.2) and letting

, we have

Computing the inner product of (2.24) with

, and using

, we get

According to Lemma 2.5, Theorem 2.8, and Schwartz inequality, we have

Substituting (2.27)–(2.29) into (2.25), we get

Similarly to the proof of (2.23), we have

and (2.30) can be rewritten as

Let

, then (2.32) is written as follows:

If

is sufficiently small which satisfies

, then

Summing up (2.34) from

to

, we have

and

, we have

. Hence

According to Lemma 2.7, we get

, that is,

It follows from (2.31) that

By using Lemma 2.6, we have

This completes the proof of Theorem 2.9.

Similarly, the following theorem can be proved.

Theorem 2.10.

Under the conditions of Theorem 2.9, the solution of the scheme (2.2)–(2.4) is stable by
.