Abstract
The abstract Cauchy problem for the hyperbolic equation in a Hilbert space H with selfadjoint 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 problem1 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., [15] and the references given therein). The stability has been an important research topic in the development of numerical techniques for solving these equations (see [627]). 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., [57] 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 selfadjoint positive definite linear operator with the domain in a Hilbert space H.
A function is called a solution of problem (1) if the following conditions are satisfied:
(i) is twice continuously differentiable on the segment . The derivatives at the endpoints of the segment are understood as the appropriate unilateral derivatives.
(ii) The element belongs to for all and the function is continuous on the segment .
(iii) satisfies the equations and the initial conditions (1).
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 secondorder difference derivatives were obtained in [11]. The high order of accuracy twostep 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., [1727] 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 is not only continuous, but also continuously differentiable on , and , it is easy to show that (see [25]) the formula
gives a solution of problem (1). Throughout this paper, is a strongly continuous cosine operatorfunction defined by the formula
Then, from the definition of the sine operatorfunction
it follows that
For the theory of cosine operatorfunction, we refer to [25] and [27].
A uniform grid is considered on the segment
In the construction of twostep 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 is considered. Applying (6), we can write
From (18) it is obvious that for the approximation of , it is necessary to approximate the expressions
Using the definitions of , , and Padé fractions for the function (see [9]), the following approximation formulas are obtained:
for the third order of approximation of (1), where
and
for the fourth order of approximation of (1), where
Using (22), (20), (21), (23), we obtain the third order of approximations of and
and the fourth order of approximations of and
Let us remark that in constructing the approximation of , it is important to know how to construct and such that
and formulas of and are sufficiently simple. The choice of and is not unique. Using Taylor’s formula and integration, we obtain the following formulas for the third order of approximation:
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, by (21) and the following operators:
Let us give one lemma, without proof, that will be needed below.
Lemma 1The following estimates hold:
Now, we will give the first main theorem of the present paper on the stability of difference scheme (2).
Theorem 1Let, , . Then, for the solution of difference scheme (2), the following stability estimates hold:
whereMdoes not depend onτ, 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 , . Here and are roots of equation (47), and , . We can rewrite (47) into the following difference problem:
Replacing a with , b with , c with , and ψ with
with and applying formula (48), we obtain the following formulas for , :
Second, we will prove estimates (44), (45), and (46). Using (43) and the formula for , and the following simple estimates:
we get
Applying to the formula for and using estimates (43), (50), (51), and (52), we get
Using the formula
and estimates (43), (52), and the following simple estimates:
we get
Applying to (53) and using estimates (43), (52), (54), (55), we obtain
Applying A to the formula for and using estimates (43), (50), (51), (52), we get
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 , the following estimates are proved:
Now, we will establish these estimates for any . Using formula (49), estimate (43), and the triangle inequality, we obtain
for any . Combining the estimates for any k, we obtain (44).
Applying to (49), using estimate (43) and the triangle inequality, we get
for . Combining the estimates for for any k, we obtain
Applying formula (49), we get
Using (65), estimate (43), and the triangle inequality, we obtain
Combining the estimates for for any k, we obtain
From estimates (64), (67), estimate (45) follows.
Now, applying Abel’s formula to (65), we obtain
Next, applying to formula (68) and using estimate (43) and the triangle inequality, we get
Combining the estimates for for any 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 . Combining the estimates for for any 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 for any 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, by (23) and the following operators:
which will be used in the sequel. It is obvious that , , and , , and , , and , are conjugates.
Let us give one lemma, without proof, that will be needed below.
Lemma 2The following estimates hold:
Now, we will give the first main theorem of the present paper on the stability of difference scheme (3).
Theorem 2Let, , . Then, for the solution of difference scheme (3), the following stability estimates hold:
whereMdoes not depend onτ, 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
with and applying formula (48), we obtain the following formula for , :
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 , and the following simple estimates:
we get
Applying to the formula for and using estimates (76), (83), (84), and (85), we obtain
Applying A to the formula for and using estimates (76), (83), (84), and (85), we have
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 to (87) and using estimates (76), (85), (89), (90), we obtain
So, for , the following estimates are proved:
Now, we will establish these estimates for any . Using formula (81), we obtain
Using formula (93), estimate (76), and the triangle inequality, we get
for any . Combining the estimates for any k, we obtain (77).
Applying to (93), using estimate (76) and the triangle inequality, we get
for . Combining the estimates for for any 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 for any k, we obtain
Using formula (81), we get
Next, using formula (97), estimate (76), and the triangle inequality, we obtain
Combining the estimates for for any k, we get
From estimates (94) and (99), we obtain (78).
Now, applying Abel’s formula to (97), we have
Next, applying to formula (100), using estimate (76) and the triangle inequality, we obtain
Combining the estimates for for any 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 onedimensional hyperbolic equation. The exact solution of this problem is .
We consider the set of a family of grid points depending on the small parameters τ and h: . First, for the approximate solution of problem (102), we have applied the third and fourth orders of accuracy difference schemes respectively.
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
for the approximate solution of Cauchy problem (102). We have system of linear equations in (103) which can be written in the matrix form as follows:
where
and
Note that
For a solution of difference equation (103), we have applied the procedure of the modified Gauss elimination method with respect to n with matrix coefficients. Therefore, we seek a solution of the matrix equation by using the following iteration formula:
where , () are square matrices and are column matrices. Now, we obtain the formulas of , , as
we get
From that, it follows
Thus, using the formulas and matrices above, the third order of approximation difference scheme with matrix coefficients for the approximate solution of Cauchy problem (102) is obtained.
Second, let us consider the fourth order of accuracy in time difference scheme (3) for the approximate solution of Cauchy problem (102).
Using difference scheme (3), we obtain
for an approximate solution of Cauchy problem (102). We have again the same system of linear equations (104), where
and
Note that
For the solution of difference equation (127), we use exactly the same method that we have used for the solution of difference equation (103). Thus, using the formulas and matrices above, the fourth order of approximation difference scheme with matrix coefficients for the approximate solution of Cauchy problem (102) is obtained. The implementations of the numerical experiments are carried out by Matlab.
The errors are computed by the following formula:
Here, represents the exact solution and represents the numerical solution at . The errors are presented in Table 1.
Table 1. Errors for the approximate solutions of problem (102)
In the table, the results are presented for different M and N values which are the step numbers for time and space variables respectively.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
OY carried out the studies, participated in the sequence alignment and drafted the manuscript and AA carried out the studies, participated in the sequence alignment.
Acknowledgements
The authors would like to thank Prof. Dr. P. E. Sobolevskii for his helpful suggestions to the improvement of this paper.
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