### Abstract

The abstract Cauchy problem for the hyperbolic equation in a Hilbert space *H* with self-adjoint positive definite operator *A* is considered. The third and fourth orders of accuracy difference schemes for the
approximate solution of this problem are presented. The stability estimates for the
solutions of these difference schemes are established. A finite difference method
and some results of numerical experiments are presented in order to support theoretical
statements.

**MSC: **
65J10, 65M12, 65N12, 35L30.

##### Keywords:

abstract hyperbolic equation; stability; initial boundary value problem### 1 Introduction

Partial differential equations of the hyperbolic type play an important role in many
branches of science and engineering. For example, acoustics, electromagnetics, hydrodynamics,
elasticity, fluid mechanics, and other areas of physics lead to partial differential
equations of the hyperbolic type (see, *e.g.*, [1-5] and the references given therein). The stability has been an important research topic
in the development of numerical techniques for solving these equations (see [6-27]). Particularly, a convenient model for analyzing the stability is provided by a proper
unconditionally absolutely stable difference scheme with an unbounded operator.

A large cycle of works on difference schemes for hyperbolic partial differential equations,
in which stability was established under the assumption that the magnitude of the
grid steps *τ* and *h* with respect to time and space variables are connected (see, *e.g.*, [5-7] and the references therein). Of great interest is the study of absolute stable difference
schemes of a high order of accuracy for hyperbolic partial differential equations,
in which stability was established without any assumptions in respect of the grid
steps *τ* and *h*. Such type stability inequalities for the solutions of the first order of accuracy
difference scheme for the differential equations of hyperbolic type were established
for the first time in [20].

It is known (see [25,26]) that various initial boundary value problems for a hyperbolic equation can be reduced to the initial value problem

where *A* is a self-adjoint positive definite linear operator with the domain
*H*.

A function

(i)

(ii) The element

(iii)

In recent decades, many scientists have worked in the field of a finite difference
method for the numerical solutions of hyperbolic PDEs and have published many scientific
papers. For problem (1), the first and two types of second order difference schemes
were presented and the stability estimates for the solution of these difference schemes
and for the first- and second-order difference derivatives were obtained in [11]. The high order of accuracy two-step difference schemes generated by an exact difference
scheme or by the Taylor decomposition on the three points for the numerical solution
of the same problem were presented and the stability estimates for approximate solution
of these difference schemes were obtained in [10]. However, the difference methods developed in these references are generated by square
roots of *A*. This action is very difficult for the realization. Therefore, in spite of theoretical
results, the role of their application to a numerical solution for an initial value
problem is not great. In this paper, the third order of accuracy difference scheme

and the fourth order of accuracy difference scheme

for the approximate solution of initial value problem (1) are constructed using the
integer powers of the operator *A*, and stability estimates for the solution of these difference schemes are obtained.
Here,

and

Some results of this paper without proof are accepted and will be published in 2013 (see [14]).

Note that boundary value problems for parabolic equations, elliptic equations, and
equations of mixed type have been studied extensively by many scientists (see, *e.g.*, [17-27] and the references therein).

### 2 The stability estimates

In this section, construction of difference schemes (2), (3) and stability estimates for the solutions of these difference schemes are presented.

Let us obtain the third and fourth orders of approximation formulas for the solution
of problem (1). If the function

gives a solution of problem (1). Throughout this paper,

Then, from the definition of the sine operator-function

it follows that

For the theory of cosine operator-function, we refer to [25] and [27].

A uniform grid is considered on the segment

In the construction of two-step difference schemes for the solution of initial value
problem (1), it is necessary to approximate differential equation (1) and the derivative

In the first step, the approximation of differential equation (1) is considered. Using Taylor’s decomposition on three points, the following formulas for the third order of approximation and the fourth order of approximation of (1) are obtained respectively:

Applying equation (1), one can write

Using (11) and (13), the following formula:

for the third order of approximation of (1), and using (12), (13), the following formula

for the fourth order of approximation of (1) are obtained.

Neglecting the last small term, we get

and

for the third and fourth orders of approximations of (1), respectively.

In the second step, the approximation of

From (18) it is obvious that for the approximation of

Using the definitions of

for the third order of approximation of (1), where

and

for the fourth order of approximation of (1), where

Here

Using (22), (20), (21), (23), we obtain the third order of approximations of

and the fourth order of approximations of

Let us remark that in constructing the approximation of

and formulas of

and for the fourth order of approximation

For simplicity, denote

and

where

and

Thus, we have the following formula for the approximation of

Using the approximation formulas above, difference schemes (2) and (3) are constructed.

Now, we will obtain the stability estimates for the solution of difference scheme
(2). We consider operators *R*,

and its conjugate

and its conjugate

and its conjugate

Let us give one lemma, without proof, that will be needed below.

**Lemma 1***The following estimates hold*:

Now, we will give the first main theorem of the present paper on the stability of difference scheme (2).

**Theorem 1***Let*
*Then*, *for the solution of difference scheme* (2), *the following stability estimates hold*:

*where**M**does not depend on**τ*, *φ*, *ψ*,
*and*

*Proof* First, we will obtain the formula for the solution of problem (1). It is clear that
there exists a unique solution of the initial value problem

and for the solution of (47), the following formula is satisfied (see [11]):

where

Replacing *a* with
*b* with
*c* with
*ψ* with

Second, we will prove estimates (44), (45), and (46). Using (43) and the formula
for

we get

Applying

Using the formula

and estimates (43), (52), and the following simple estimates:

we get

Applying

Applying *A* to the formula for

Using difference scheme (2) and formula (53), we obtain

Using formula (59), estimates (43), (51), (52), (55), and the following simple estimates:

we have

So, for

Now, we will establish these estimates for any

for any
*k*, we obtain (44).

Applying

for
*k*, we obtain

Applying formula (49), we get

Using (65), estimate (43), and the triangle inequality, we obtain

Combining the estimates for
*k*, we obtain

From estimates (64), (67), estimate (45) follows.

Now, applying Abel’s formula to (65), we obtain

Next, applying

Combining the estimates for
*k*, we obtain

Now, applying Abel’s formula to (49), we have

Next, applying *A* to formula (70) and using (43) and the triangle inequality, we get

for
*k*, we obtain

Now, applying formula (49), we get

First, applying Abel’s formula to (72), we have

Second, using formulas (73), (43), and the triangle inequality, we get

Combining the estimates for
*k*, we obtain

From estimates (69), (71), and (75), we obtain (46). Theorem 1 is proved. □

Now, we will obtain the stability estimates for the solution of difference scheme
(3). We consider operators *R*,

which will be used in the sequel. It is obvious that

Let us give one lemma, without proof, that will be needed below.

**Lemma 2***The following estimates hold*:

Now, we will give the first main theorem of the present paper on the stability of difference scheme (3).

**Theorem 2***Let*
*Then*, *for the solution of difference scheme* (3), *the following stability estimates hold*:

*where**M**does not depend on**τ*, *φ*, *ψ*,
*and*

*Proof* First, we will obtain the formula for the solution of problem (1). In exactly the
same manner as in Theorem 1, one establishes formula (48) for the solution of initial
value problem (47).

We can rewrite (47) into the following difference problem:

Replacing *a* with
*b* with
*c* with

and *ψ* with

Now, we will prove estimates (77), (78), and (78). First, using the formula of difference scheme (3), we obtain the formula

Using estimates (76), the formula for

we get

Applying

Applying *A* to the formula for

Using the formula for the solution of difference scheme (3), we obtain

Using formula (87), estimates (76), (83), (85), and the following simple estimates:

we get

Applying

So, for

Now, we will establish these estimates for any

Using formula (93), estimate (76), and the triangle inequality, we get

for any
*k*, we obtain (77).

Applying

for
*k*, we get

Now, applying Abel’s formula to (93), we obtain

Applying *A* to formula (95), using estimate (76) and the triangle inequality, we get

Combining the estimates for
*k*, we obtain

Using formula (81), we get

Next, using formula (97), estimate (76), and the triangle inequality, we obtain

Combining the estimates for
*k*, we get

From estimates (94) and (99), we obtain (78).

Now, applying Abel’s formula to (97), we have

Next, applying

Combining the estimates for
*k*, we get

From estimates (96) and (101), estimate (78) follows. Theorem 2 is proved. □

### 3 Numerical analysis

In the present section, finite difference method is used and two numerical examples and some numerical results are presented in order to support our theoretical statements.

Generally, we have not been able to determine a sharp estimate for the constant figuring in the stability inequalities. However, the numerical results are presented by considering the Cauchy problem

for a one-dimensional hyperbolic equation. The exact solution of this problem is

We consider the set
*τ* and *h*:

First, let us consider the third order of accuracy in time difference scheme for the approximate solution of Cauchy problem (102). Using difference scheme (2), we obtain