On Third Order Stable Difference Scheme for Hyperbolic Multipoint Nonlocal Boundary Value Problem

Abstract

This paper presents a third order of accuracy stable difference scheme for the approximate solution of multipoint nonlocal boundary value problem of the hyperbolic type in a Hilbert space with self-adjoint positive definite operator. Stability estimates for solution of the difference scheme are obtained. Some results of numerical experiments that support theoretical statements are presented.

1. Introduction

Hyperbolic partial differential equations (PDEs) are of growing interest in several areas of engineering and natural sciences such as acoustic, electromagnetic, hydrodynamic, elasticity, fluid mechanics, and other areas of physics (see, e.g., [1–5] and the references given therein). In some cases, classical boundary conditions cannot describe process or phenomenon precisely. Therefore, mathematical models of various physical, chemical, biological, or environmental processes often involve nonclassical conditions. Such conditions are usually identified as nonlocal boundary conditions and project situations when the data on the domain boundary cannot be measured directly or when the data on the boundary depend on the data inside the domain. Many researchers study nonlocal boundary value problems (NBVPs) for hyperbolic and mixed types of partial differential equations. There are many results (see [6–10]) on integral inequalities with two dependent limits to theory of integral-differential equations of the hyperbolic type and of difference schemes for the approximate solution of these problems. Stability has been actively studied in the development of numerical methods for the approximate solutions of PDEs (see [5–31]). In particular, an appropriate model for the analysis of stability is provided by a suitable unconditionally stable difference scheme with an unbounded operator.

It is known (see [29, 30]) that various multipoint NBVPs for hyperbolic equations can be reduced to the problem Here is a self-adjoint positive definite operator with domain in a Hilbert space .

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 the last decades, many scientists worked and published scientific papers in the field of finite difference method for the numerical solutions of hyperbolic PDEs. In this field, first order and two types of second order stable difference schemes for the solution of abstract hyperbolic problem (1) were presented in [17] and high order of accuracy difference schemes for the solution of the same problem were presented in [18]. However, the difference schemes presented in these references are generated by square roots of . This action is not good for realization. Therefore, in spite of theoretical results the role of their application to a numerical solution of (1) is not great.

In the present paper, a third order of accuracy unconditionally stable difference scheme generated by the integer power of for the approximate solution of the multipoint NBVP (1) is presented. Stability estimates for the solution of this difference scheme are established.

Some results of this paper, without proof, are published in [16].

Note that many scientists have studied the solutions of boundary value problems such as parabolic equations, elliptic equations, and equations of mixed types extensively (see, e.g., [6–10, 21–31] and the references therein).

2. Third-Order Accurate Stable Difference Scheme

In the present section a third order of accuracy stable difference scheme for the approximate solution of multipoint NBVP (1) is presented. We associate problem (1) with the corresponding third order of accuracy difference scheme: Here Note that throughout this paper for clarity and will be considered.

We are interested in studying the stability of solution of difference scheme (2) under assumption Throughout this section for simplicity we denote First, let us present some lemmas. We will consider the following operators: and their conjugate and their conjugate , and their conjugate , and their conjugate , Now we consider the following lemma which was presented before in [14].

Lemma 1. The following estimates hold:

Proof. Using the spectral property of self-adjoint positive definite operators, we write Since we have that In exactly the same manner, one can easily obtain all the other estimates of Lemma 1.

Lemma 2. Suppose that assumption (4) holds. Then, the operator has an inverse . From the symmetry and positivity properties of the operator the following estimate holds:

Proof. Using the definitions of , , , estimates in (11), and the triangle inequality, we get where Since , the operator has a bounded inverse and Lemma 2 is proved.

Note that the stability estimates in the sequel are obtained by the method presented in the proofs of Lemmas 1 and 2 to obtain stability estimates. For clarity and avoiding long equations the proofs of estimates are not put.

Now, applying the following formula (see [14]) and the multipoint nonlocal boundary conditions in (2), we obtain

Formulas (19) and (20) give a solution of problem (2).

Theorem 3. Suppose that assumptions (4) and (11) hold and and . Then, for the solution of difference scheme (2), the following stability estimates hold: where does not depend on , , , and , .

Proof. Using formulas (20), estimates (11) and (15), and the method of Lemmas 1 and 2, we get Applying to formulas (20), we obtain Now, applying Abel’s formula to (20), we get Let us obtain estimates for and using the method of lemmas. First, applying to formula (24), using estimates (11) and (15) and the triangle inequality, we obtain Second, applying to formula (25), using estimates (11) and (15) and the triangle inequality, we obtain The estimates (21) are proved for . Now, we give the proofs of estimates (21) which were also presented in [14] for . Using formula (19), estimates (11), (22), and (23), the triangle inequality, and method of Lemmas 1 and 2, we obtain Applying to (19), we get Applying Abel’s formula to (19), we get Applying to formula (30), using estimates (11) and the triangle inequality, we obtain Theorem 3 is proved.

We denote the mesh function of approximation by . Then if we assume that tends to 0 as not slower than . It takes place in applications by supplementary restriction of the smooth property of the data of space variables. Let us give applications of the abstract results presented in Theorem 3.

First, the mixed multipoint NBVP for hyperbolic equation is considered. Problem (32) has a unique smooth solution , , and the smooth functions (, ), , () and (). This allows us to reduce mixed problem (32) to multipoint NBVP (1) in a Hilbert space with a self-adjoint positive definite operator defined by (32).

The discretization of problem (32) is carried out in two steps. In the first step, we present the grid space in a Hilbert space , and of the grid functions defined on , equipped with the norms respectively. To the differential operator generated by the problem (32), we assign the difference operator by the formula acting in the space of grid functions satisfying the conditions , . With the help of we arrive at the multipoint NBVP for an infinite system of ordinary differential equations.

In the second step, we replace problem (36) by difference scheme (37):

Theorem 4. Let and be sufficiently small numbers. Then, the solution of difference scheme (37) satisfies the following stability estimates: Here does not depend on , , , , and , .

The proof of Theorem 4 is based on the proof of abstract Theorem 3 and the symmetry property of the operator defined by (35).

Second, let be the unit open cube in the -dimensional Euclidean space with boundary , . In , the mixed boundary value problem for the multidimensional hyperbolic equation is considered. Here , (), , (), and (, ) are given smooth functions and .

We introduce the Hilbert space of all square integrable functions defined on , equipped with the norm Problem (39) has a unique smooth solution , and , , , and are smooth functions. This allows us to reduce the mixed problem (39) to multipoint NBVP (1) in a Hilbert space with a self-adjoint positive definite operator defined by (39).

The discretization of problem (39) is carried out in two steps. In the first step, we define the grid sets We introduce the Banach space , , and of the grid functions defined on , equipped with norms respectively. To the differential operator generated by problem (39), we assign the difference operator by the formula acting in the space of grid functions , satisfying the conditions for all . It is known that is a self-adjoint positive definite operator in . With the help of we arrive at the multipoint NBVP: for an infinite system of ordinary differential equations.

In the second step, we replace problem (44) by difference scheme (45):

Theorem 5. Let and be sufficiently small numbers. Then, the solution of difference scheme (45) satisfies the following stability estimates: Here does not depend on , , , , , and , .

Proof of Theorem 5 is based on the proof of abstract Theorem 3, the symmetry property of the operator defined by formula (43), and the following theorem on the coercivity inequality for the solution of the elliptic difference problem in .

Theorem 6. For the solutions of the elliptic difference problem the following coercivity inequality holds [25]: Here does not depend on , .

3. Numerical Results

In this section some numerical results are presented. The multipoint NBVP for one-dimensional hyperbolic equation is considered. The exact solution of (49) is For the approximate solution of problem (49) the third order of accuracy difference scheme (2) is used, and the set of a family of grid points depending on the small parameters and is considered: 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 Errors are presented in Table 1.

Table 1 

Errors for the approximate solutions of problem (49).

In the table, the results are presented for different and values which are the step numbers for time and space variables, respectively. The results in Table 1 indicate that error decreases as and values get larger. Therefore it can be verified by the experimental results (Table 1) that the difference scheme is stable.

4. Conclusion

In this paper a third order of accuracy unconditionally stable difference scheme for the approximate solution of hyperbolic multipoint NBVP in a Hilbert space with self-adjoint positive definite operator is presented. The stability is established without any assumptions in respect of the grid steps and . Numerical results verifying the theoretical statements are presented. It is not easy to obtain arbitrary order stable difference schemes. Nevertheless one can obtain fourth order or higher order stable difference schemes using our results and method. It is possible to consider the same problem with fractional derivative or with small parameter .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank Professor Dr. A. Ashyralyev and Professor Dr. P. E. Sobolevskii for their helpful suggestions to the improvement of this paper.