Abstract and Applied Analysis

Volume 2012 (2012), Article ID 854517, 22 pages

doi:10.1155/2012/854517

Research Article

Nonlinear Stability and D-Convergence of Additive Runge-Kutta Methods for Multidelay-Integro-Differential Equations

Haiyan Yuan,1,2 Jingjun Zhao,¹ and Yang Xu¹

¹Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

²Department of Mathematics, Heilongjiang Institute of Technology, Harbin 150050, China

Received 30 December 2011; Accepted 19 February 2012

Academic Editor: Muhammad Aslam Noor

Copyright © 2012 Haiyan Yuan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper is devoted to the stability and convergence analysis of the Additive Runge-Kutta methods with the Lagrangian interpolation (ARKLMs) for the numerical solution of multidelay-integro-differential equations (MDIDEs). GDN-stability and D-convergence are introduced and proved. It is shown that strongly algebraically stability gives D-convergence, DA- DAS- and ASI-stability give GDN-stability. A numerical example is given to illustrate the theoretical results.

1. Introduction

Delay differential equations arise in a variety of fields as biology, economy, control theory, electrodynamics (see, e.g., [1–5]). When considering the applicability of numerical methods for the solution of DDEs, it is necessary to analyze the stability of the numerical methods. In the last three decades, many works had dealt with these problems (see, e.g., [6]). For the case of nonlinear delay differential equations, this kind of methodology had been first introduced by Torelli [7] and then developed by [8–12].

In this paper, we consider the following nonlinear multidelay-integro-differential equations (MDIDEs) with delays:

where , , , and are continuous functions such that (1.1) has a unique solution. Moreover, we assume that there exist some inner product and the induced norm such that

, are all nonnegative constants. Throughout this paper, we assume that the problem (1.1) has unique exact solution . Space discretization of some time-dependent delay partial differential equations give rises to such delay differential equations containing additive terms with different stiffness properties. In these situations, additive Runge-Kutta (ARK) methods are used. Some recent works about ARK can refer to [13, 14]. For the additive MDIDEs (1.1), similar to the proof of Theorem 2.1 in [7], it is straightforward to prove that under the conditions (1.2)~(1.4), the analytic solutions satisfy

where is the solution of the perturbed problem to (1.1).

To demand the discrete numerical solutions to preserve the convergence properties of the analytic solutions, Torelli [7] introduced a concept of RN-, GRN-stability for numerical methods applied to dissipative nonlinear systems of DDEs such as (1.1) when , which is the straightforward generalization of the well-known concept of BN-stability of numerical methods with respect to dissipative systems of ODEs (see also [9]). More recently, one has noticed a growing interesting the analysis of delay integro-differential equations (DIDEs). This type of equations have been investigated in various fields, such as mathematical biology and control theory (see [15–17]). The theory of computational methods for delay integro-differential equations (DIDEs) has been studied by many authors, and a great deal of interesting results have been obtained (see [18–22]). Koto [23] dealt with the linear stability of Runge-Kutta (RK) methods for systems of DIDEs; Huang and Vandewalle [24] gave sufficient and necessary stability conditions for exact and discrete solutions of linear Scalar DIDEs. However, little attention has been paid to nonlinear multidelay-integro-differential equations (MDIDEs).

So, the aim of this paper is the study of stability and convergence properties for ARK methods when they are applied to nonlinear multidelay-integro-differential equations (MDIDEs) with delays.

2. The GDN-Stability of the Additive Runge-Kutta Methods

An additive Runge-Kutta method with the Lagrangian interpolation (ARKLM) of stages and levels can be organized in the Butcher tableau:

where , , and .

The adoption of the method (2.1) for solving the problem (1.1) leads to

where , , and , are approximations to the analytic solution , , of (1.1), respectively, and the argument is determined by

with , , integer , , and

We assume is to guarantee that no (unknown) values with are used in the interpolation procedure

which can be computed by a appropriate compound quadrature rule:

As for the quadrature rule (2.6), we usually adopt the compound trapezoidal rule, the compound Simpsons rule or the compound Newton-Cotes rule, and so forth according to the requirement of the convergence of the method (see [19]) and denote and with satisfing , .

In addition, we always put, whenever .

In order to write (2.2), (2.3), (2.5), and (2.6) in a more compact way, we introduce some notations. The identity matrix will be denoted by , , is the Kronecker product of matrix and . For , , we define the inner product and the induced norm in as follows:

Moreover, we also adopt that

With the above notation, method (2.2),(2.3), (2.5), and (2.6) can be written as

In 1997, Zhang and Zhou [25] introduced the extension of RN-stability to GDN-stability as follows.

Definition 2.1. An ARKLM (2.1) for DDEs is called GDN-stable if, numerical approximations and to the solution of (1.1) and its perturbed problem, respectively, satisfy

where constant depends only on the method, the parameter , and the interval length , is the initial function to the perturbed problem of (1.1).

Definition 2.2. An ARKLM (2.1) is called strongly algebraically stable if matrices are nonnegative definite, where

for .

Let

be two sequences of approximations to problems (1.1) and its perturbed problem, respectively. From method (2.1) with the same step size , and write

Then (2.2) and (2.3) read

Our main results about GDN-stability are contained in the following theorem.

Theorem 2.3. Assume ARK method (2.2) is strongly algebraically stable, and then the corresponding ARKLM (2.1) is GDN-stable, and satisfies

where

Proof. From (2.14) and (2.15) we get

If the matrices are nonnegative definite, then

Furthermore, by conditions (1.2)~(1.4) and Schwartz inequality we have

For (2.23), we have

By the same way, we can also get

Substituting (2.25) and (2.24) into (2.19), yields

where .

Similar to (2.27), the inequalities:

follows for .

In the following, with the help of inequalities (2.27), (2.28), and induction we shall prove the inequalities:

for , .

In fact, it is clear from (2.27), (2.28), and such that

Suppose for that

Then from (2.27) and (2.28), and , we conclude that

This completes the proof of inequalities (2.29). In view of (2.29), we get for that

As a result, we know that method (2.1) is GDN-stable.

3. D-Convergence

In order to study the convergence of numerical methods for MDIDEs, we have to mention the concept of the convergence for stiff ODEs.

In 1981, Frank et al. [26] introduced the important concept of B-convergence for numerical methods applied to nonlinear stiff initial value problems of ordinary differential equations. Later, there have been rapid developments in the study of B-convergence, and a significant number of important results have already been found for Runge-Kutta methods. In fact, B-convergence result is nothing but a realistic global error estimate based on one-sided Lipschitz constant [27]. In this section, we start discussing the convergence of ARKLM (2.1) for MDIDEs (1.1) with conditions (1.2)–(1.4). The approach to the derivation of these estimates is similar to that used in [25]. We assume the analytic solution of (1.1) is smooth enough, and its derivatives used later are bounded by

where

If we introduce some notations

With the above notations, the local errors in (2.9) can be defined as

with

If we take , , , and

Then we can get the perturbed scheme of (2.9),

With perturbations, , ,, , according to Taylor formula and the formula in [28, pages 205–212], , and can be determined respectively, as follows:

where ,, and satisfy ,, , depends only on the method, and depends only on the method and some in (3.2).

Combining (2.2), (2.3), (2.5), and (2.6) with (3.9), (3.10), (3.11), and (3.12) yields the following recursion scheme for the :

where , ,

here, is the Jacobian matrix .

Assume that () is regular, from (3.16) and (3.17), (3.18), we can get

Now, we introduce the concept of D-convergence from [25].

Definition 3.1. An ARKLM (2.1) with, and is called D-convergence of order if this method, when applied to any given DIDEs (1.1) subject to (1.2)–(1.4); produce an approximation sequence and the global error satisfies a bound of the form:

where the maximum stepsize depends on characteristic parameter and the method, the function depends only on some in (3.2), delay , characteristic parameters , , and the method.

Definition 3.2. The ARKLM (2.2), (2.3), (2.5), and (2.6) is said to be DA-stable if the matrix is regular for , and ,.

Where

Definition 3.3. The ARKLM (2.2), (2.3), (2.5), and (2.6) is said to be ASI-stable if the matrix is regular for, and is uniformly bounded for .

Definition 3.4. The ARKLM (2.2), (2.3), (2.5), and (2.6) is said to be DAS-stable if the matrix is regular for , and is uniformly bounded for .

Lemma 3.5. Suppose the ARKLM (2.2), (2.3), (2.5), and (2.6) is DA- DAS- and ASI-stable, then there exist positive constants , which depend only on the method and the parameter such that

Proof. This Lemma can be proved in the similar way as that of in [29, Lemmas 3.5–3.7].

Theorem 3.6. Suppose the ARKLM (2.2), (2.3), (2.5), and (2.6) is DA- DAS- and ASI-stable, then there exist positive constants , which depend only on the method and the parameters, such that for ,

where ,, ,, .

Proof. Using (3.19) and Lemma 3.5, for , we obtain that

For

Then

For

then

Combine (3.27), (3.29), and (3.25a), we have

Moreover, it follows from (2.5) and (3.11) that

Substituting (3.31) in (3.30), we get

where

By Lemma 3.5, similar to (3.32), we can obtain the inequalities:

where

Setting , , .

Combining (3.32) with (3.34), we immediately obtain the conclusion of this theorem.

Now, we turn to study the convergence of ARKLM (2.1) for (1.1). It is always assumed that the analytic solution of (1.1) is smooth enough on each internal of the form ( is a positive integer) as (3.2) defined.

Theorem 3.7. Assume ARKLM (2.1) with stage order is DA-, DAS- and ASI-stable, then the ARKLM (2.1) is D-convergent of order min, where .

Proof. By Theorem 3.6, we have for

where

It follows from an induction to (3.36) for that

Hence, for, we arrive at

Therefore, the ARKLM (2.1) is D-Convergent of order .

4. Some Examples

Consider the following initial value problem of multidelay-integro-differential equations:

and its perturbed problem:

It an be easily verified that , , and , with analytic solution , where the inner product is standard inner product. We apply the two-stages and two-order additive R-K method:

(1) to the problem (4.1) and its perturbed problem (4.2). Since the order of the method is 2, we adopt the compound trapezoidal rule for computing the integer part. According to the result of Theorems 2.3 and 3.7, the corresponding method for DIDEs is GDN-stable and D-convergent. We denote the numerical solution of problem (4.1) and (4.2) and , where and are approximations to and , respectively. The values and with are listed inFigure 1, (where the abscissa and ordinate denote variable and , resp.) and Figure 2, (where the abscissa and ordinate denote variable and , resp.). It is shown in Table 1 that the numerical solutions are toward to the exact solutions as .

Table 1: Comparison between the numerical solutions for different h and the exact solution.

t
Numerical solution for
Numerical solution for
Numerical solution for
Exact solution y(t)

0.2
2.87569414
1.212021722
0.898712327
0.818730753

0.4
2.751727517
1.165089633
0.801604459
0.670320046

0.6
2.448183694
1.0957815
0.626189996
0.548811636

Figure 1: Values with .

Figure 2: Values with .

It is obvious that the corresponding method for MDIDEs is GDN-stable and D-convergent.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11101109) and the Natural Science Foundationof Hei-long-jiang Province of China (A201107).

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