Stepsize Restrictions for Nonlinear Stability Properties of Neutral Delay Differential Equations

Abstract

The present paper is concerned with the relationship between stepsize restriction and nonlinear stability of Runge-Kutta methods for delay differential equations. We obtain a special stepsize condition guaranteeing global and asymptotical stability properties of numerical methods. Some confirmations of the conditions on Runge-Kutta methods are illustrated at last.

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Wei Gu 1 and Ming Wang 2 and Dongfang Li 3

Academic Editor:Ali H. Bhrawy

1, School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073, China
2, School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, China
3, School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China

Received 22 May 2014; Accepted 9 July 2014; 21 July 2014

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.

1. Introduction

Neutral delay differential equations (NDDEs) are widely used in various kinds of applied disciplines such as biology, ecology, electrodynamics, and physics and hence intrigue lots of researchers in numerical simultation and analysis (see, e.g., [1-3]). Up to now, many researchers have discussed nonlinear stability properties for NDDEs. In 2000, Bellen et al. [4] studied B_Nf -stable continuous Runge-Kutta methods for NDDEs. They extended the contractivity requirements to the numerical stability analysis. Vermiglio and Torelli further pointed out that the numerical solution produced by the methods can preserve the contractivity property of the theoretical solution in [5]. In 2002, Zhang [6] derived nonlinear stability properties for theoretical and numerical solutions of NDDEs based on natural Runge-Kutta schemes. After that, Wang et al. [7, 8] first introduced the concepts of GS(l )- and GAS(l )-stability for nonautonomous nonlinear problems. They proved that (k,l )-algebraically stable Runge-Kutta methods and (k,p,0) -algebraically stable general linear methods lead to GS(l )- and GAS(l )-stability for NDDEs, respectively. Recently, Bhrawy et al. [9-11] studied several kinds of collocation method for some NDDEs. For more analogues results, we refer readers to [12-15]. Useful as these stability results are, however, no conclusions have been found to develop the relationship between nonlinear stability analysis and stepsize restriction with some numerical schemes for NDDEs, especially for some Runge-Kutta methods.

The present paper was in part inspired by the work of Spijker et al. With stepsize restriction to some numerical schemes, they revealed to us some monotonicity and stability properties for ODEs, respectively (see, [16-19]). We extend their study to nonlinear NDDEs in the present paper. With stepsize restriction to Runge-Kutta schemes, global and asymptotical stability results for NDDEs are obtained, respectively.

The rest of the paper is organized as follows. In Section 2, we consider Runge-Kutta schemes with linear interpolation procedure for NDDEs. Some concepts, such as global and asymptotical stability, are also collected. Section 3 is devoted to stability analysis. The given results set up a relationship between the stepsize restriction and nonlinear stability for nonlinear NDDEs. Some examples of Runge-Kutta schemes are presented in Section 4. Finally, we end up with some conclusions and extension in the last section.

2. Runge-Kutta Methods for NDDEs

In the present paper, we consider the following nonlinear NDDEs: [figure omitted; refer to PDF] and the perturbed problem [figure omitted; refer to PDF] Here, τ denotes a positive delay term, N∈^Cd×d is a constant matrix with ||N||<1 , ψ(t) and [varphi](t) are continuous, and f : [0,+∞]×X×X[arrow right]X , such that (1) and (2) own a unique solution, respectively, where X is a real or complex Hilbert space. As in [20, 21], we assume there exist some inner product Y9;·,·YA; and the induced norm ||·|| such that [figure omitted; refer to PDF] where α...4;0, β...5;0 , and δ<0 are real constants.

When N=0 , the problem (1) degenerates into nonlinear DDEs of the following type: [figure omitted; refer to PDF] Nonlinear stability analysis for such systems can be found in [6, 22-25]. Condition (3) can be equivalent to the assumptions in these literatures (see [26], Remark 2.1 ).

Now, let us consider s -stage Runge-Kutta methods for (1); the coefficients of the schemes may be organized in Buther tableau as follows: [figure omitted; refer to PDF] where c=^{[_cl ,...,_cs ]T} , b=^{[_b1 ,...,_bs ]T} , and A=^{(_aij )i,j=1s} .

According to Liu in [27], Runge-Kutta methods for NDDEs can be written as [figure omitted; refer to PDF] where h is stepsize and _tn =nh,_yn ,_y~n ,^yi(n) and ^y~i(n) are approximations to the analytic solutions y(_tn ) , y(_tn -τ) , y(_tn +_ci h) , and y(_tn +_ci h-τ) , respectively. We set τ=(m-θ)h with θ∈[0,1) , and the arguments _y~n and ^y~j(n) are determined by [figure omitted; refer to PDF] where _yi =ψ(_ti ) for _ti ...4;0 and ^yj(i) =ψ(_ti +_cj h) for _ti +_cj h...4;0 .

Now, let _yn and _zn be two sequences of approximations to problems (1) and (2), respectively. Following Definitions 9.1.1 and 9.1.2 in [1] for delay systems, we introduce some stability concepts.

Definition 1.

A numerical method for DDEs or NDDEs is called globally stable, if there exists a constant C such that [figure omitted; refer to PDF] holds when the method is applied to (1) and (2) under some assumptions.

Definition 2.

A numerical method for DDEs or NDDEs is said to be asymptotically stable, if [figure omitted; refer to PDF] holds when the method is applied to (1) and (2) under some assumptions.

3. Stability Analysis

In the section, we will discuss the relationship between the stepsize restriction and nonlinear stability of the method.

Theorem 3.

Assume condition (3) holds, α+β...4;0 , and there exists a positive real number r , such that the matrix [figure omitted; refer to PDF] is nonnegative definite, where _bi ...5;0 , i=1,2,...,s . Then the Runge-Kutta method with linear interpolation procedure for NDDEs (1) is globally stable under the stepsize restriction [figure omitted; refer to PDF]

Proof.

Let {_yn ,^yi(n) ,^y~i(n) } and {_zn ,^zi(n) ,^z~i(n) } be two sequences of approximations to problems (1) and (2), respectively, and write [figure omitted; refer to PDF] With the notation, Runge-Kutta methods with the same stepsize h for (1) and (2) yield [figure omitted; refer to PDF] Thus, we have [figure omitted; refer to PDF] Now, in view of the nonnegative definite matrix M , we obtain [figure omitted; refer to PDF] On the other hand, in terms of condition (3), we find [figure omitted; refer to PDF]

Then, together with (14), (15), and (16) and using the conditions h/r...4;-2δ , we get [figure omitted; refer to PDF] Noting that [figure omitted; refer to PDF] and α+β...4;0 , we have [figure omitted; refer to PDF]

This implies that [figure omitted; refer to PDF] where C~=(2+2^||N||2 +2τβ) .

Note that ||N||<1 ; we have [figure omitted; refer to PDF] An induction to (21) yields [figure omitted; refer to PDF] Therefore, the conclusion is proven.

Corollary 4.

Assume condition (3) holds; α+β...4;0 . Then an algebraically stable Runge-Kutta method with linear interpolation procedure for DDEs (4) or NDDEs (1) is globally stable.

Remark 5.

A Runge-Kutta method is algebraically stable if the matrix [figure omitted; refer to PDF] is nonnegative definite and _bi ...5;0 (i=1,2,...,s) . For example, the s -stage Gauss, Radau IA , Radau IIA , and Lobatto IIIC methods are algebraically stable. Corollary 4 can be derived for r=∞ . This implies that the stepsize restriction for DDEs disappears.

Corollary 6.

Assume condition (3) holds, α+β...4;0 , and there exists a positive real number r , such that the matrix [figure omitted; refer to PDF] is nonnegative definite, where _bi ...5;0 , i=1,2,...,s . Then the Runge-Kutta method with linear interpolation procedure for DDEs (4) is globally stable under the stepsize restriction [figure omitted; refer to PDF]

Theorem 7.

Assume condition (3) holds, α+β<0 , the function f(t,u,v) is uniformly Lipschitz continuous with constant L in variables u and v , and there exists a positive real number r , such that the matrix [figure omitted; refer to PDF] is nonnegative definite, where _bi ...5;0 , i=1,2,...,s . Then the Runge-Kutta method with linear interpolation procedure for NDDEs (1) is asymptotically stable under the stepsize restriction [figure omitted; refer to PDF]

Proof.

Like in the proof of Theorem 3, let σ=α+β<0 , and we can easily find [figure omitted; refer to PDF] Note σ<0 and _bi ...5;0 ; we have [figure omitted; refer to PDF] On the other hand, [figure omitted; refer to PDF] Now, in view of (13), (29), and (30), we obtain [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] and ||N||<1 , an induction to (32) gives [figure omitted; refer to PDF] which completes the proof.

Corollary 8.

Assume condition (3) holds, α+β<0 , the function f(t,u,v) is uniformly Lipschitz continuous with constant L in variables u and v . Then an algebraically stable Runge-Kutta method with linear interpolation procedure for DDEs (4) or NDDEs (1) is asymptotically stable.

Corollary 9.

Assume condition (3) holds, α+β<0 , the function f(t,u,v) is uniformly Lipschitz continuous with constant L in variables u and v , and there exists a positive real number r , such that the matrix [figure omitted; refer to PDF] is nonnegative definite, where _bi ...5;0 , i=1,2,...,s . Then the Runge-Kutta method with linear interpolation procedure for DDEs (4) is asymptotically stable under the stepsize restriction [figure omitted; refer to PDF]

4. Some Examples

As it is shown in the theorems, the parameters δ and r in the matrix M play a key role in the stability analysis. The larger the existed parameter r is, the larger stepsize we could choose. In this section, we will show some examples.

Consider the following case, like the conditions in [22] or [28], if f(t,y,u)=f~(t,y-Nu) and [figure omitted; refer to PDF] with ρ>0 , we have the following form in an inner product norm: [figure omitted; refer to PDF] with δ=-1/(2ρ)<0 .

In particular, let f(t,y,u)=-a(My-Nu) , where a>0 , M<1 are constants independent of t , respectively. We have [figure omitted; refer to PDF]

Next, we give some examples on how to calculate the parameter r .

Example 1.

Consider s -stage 1-order Runge-Kutta methods (see [17], section 2.7 ) [figure omitted; refer to PDF] and we have [figure omitted; refer to PDF] Thus, the matrix M is nonnegative definite for 0<r...4;s . They imply that these methods for DDE with interpolation are stable with stepsize restriction h...4;-2δs .

Example 2.

Consider 2-stage 2-order Runge-Kutta method: [figure omitted; refer to PDF] and we obtain [figure omitted; refer to PDF] Therefore, the matrix M is nonnegative definite for 0<r...4;1 . They imply that the stepsize h...4;-2δ is feasible under the assumptions (3) for NDDEs (1).

For more Runge-Kutta methods with the nonnegative definite matrix M , we refer readers to Section 2.2.4 in [28]. Higueras revealed to us how to find the largest r such that the matrix M is nonnegative definite. He pointed that if the matrix diag...(b) is positive definite, the largest r can be determined by [figure omitted; refer to PDF] where _λmin... (·) denotes the smallest eigenvalue of the matrix (·) .

5. Conclusions and Discussions

In this study, we show that the Runge-Kutta methods with stepsize restrictions can preserve global and asymptotical stability of the continuous DDEs or NDDEs. An algebraically stable Runge-Kutta method with linear interpolation procedure for DDEs or NDDEs is globally stable and asymptotically stable. These results can be easily extended to the following equation with several delays: [figure omitted; refer to PDF] under the following assumption: [figure omitted; refer to PDF] where _τi >0, i=1,2,...,l , _yi =y(t-_τi ) , and _zi =z(t-_τi ) . We do not list them here for the sake of brevity.

Moreover, the present results have certain instructive effect in numerical simulation. In the future, we hope to apply the results to some real-world problems, for example, reaction-diffusion dynamical systems with time delay [24] and non-Fickian delay reaction-diffusion equations [25, 29].

Acknowledgment

This work is supported by NSFC (Grant nos. 11201161, 11171125, and 91130003).

Conflict of Interests

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

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