# Stepsize Restrictions for Nonlinear Stability Properties of Neutral Delay Differential Equations

Authors: Wei Gu; Ming Wang; Dongfang Li
Journal: Abstract and Applied Analysis
 (2014)

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## 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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## Body

## 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 BNf-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 Section2, 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:(1)ddt[y(t)-Ny(t-τ)]=f(t,y(t),y(t-τ)),t>0,y(t)=ψ(t),-τ≤t≤0,
and the perturbed problem
(2)ddt[z(t)-Nz(t-τ)]=f(t,z(t),z(t-τ)),t>0,z(t)=ϕ(t)-τ≤t≤0.
Here, τ denotes a positive delay term, N∈Cd×d is a constant matrix with ∥N∥<1, ψ(t) and ϕ(t) are continuous, and f: [0,+∞]×X×X→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 〈·,·〉 and the induced norm ∥·∥ such that
(3)Re〈(y-z)-N(u-v),f(t,y,u)-f(t,z,v)〉≤α∥y-z∥2+β∥u-v∥2+δ∥f(t,y,u)-f(t,z,v)∥2,
where α≤0,β≥0, and δ<0 are real constants.WhenN=0, the problem (1) degenerates into nonlinear DDEs of the following type:
(4)y′(t)=f(t,y(t),y(t-τ)),t>0,y(t)=ψ(t)-τ≤t≤0.
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 considers-stage Runge-Kutta methods for (1); the coefficients of the schemes may be organized in Buther tableau as follows:
(5)cAbT,
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
(6)yn+1-Ny~n+1=yn-Ny~n+h∑j=1sbjf(tn+cjh,yj(n),y~j(n)),‍yi(n)-Ny~i(n)=yn-Ny~n+h∑j=1saijf(tn+cjh,yj(n),y~j(n))‍hhhhhhhhhhhhhhhhhhhhhhhhi=1,2,…,s,
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+cih), and y(tn+cih-τ), respectively. We set τ=(m-θ)h with θ∈[0,1), and the arguments y~n and y~j(n) are determined by
(7)y~n=θyn-m+1+(1-θ)yn-m,y~j(n)=θyj(n-m+1)+(1-θ)yj(n-m),
where yi=ψ(ti) for ti≤0 and yj(i)=ψ(ti+cjh) for ti+cjh≤0.Now, letyn 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 constantC such that
(8)∥yn-zn∥≤Cmax⁡-τ≤t≤0∥ψ(t)-ϕ(t)∥
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(9)lim⁡n→∞∥yn-zn∥=0
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≤0, and there exists a positive real number r, such that the matrix
(10)M=diag⁡(b)A+ATdiag⁡(b)-bbT+1rdiag⁡(b)
is nonnegative definite, where bi≥0, i=1,2,…,s. Then the Runge-Kutta method with linear interpolation procedure for NDDEs (1) is globally stable under the stepsize restriction
(11)hr≤-2δ.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
(12)Ui(n)=yi(n)-zi(n),U~0(n)=y~i(n)-z~i(n),U0(n)=yn-zn,U~0(n)=y~n-z~n,Wi=h[f(tn+cih,yi(n),y~i(n))-f(tn+cih,zi(n),z~i(n))].
With the notation, Runge-Kutta methods with the same stepsize h for (1) and (2) yield
(13)U0(n+1)-NU~0(n+1)=U0(n)-NU~0(n)+∑j=1s‍bjWj,Ui(n)-NU~i(n)=U0(n)-NU~0(n)+∑j=1s‍aijWj,i=1,2,…,s.
Thus, we have
(14)∥U0(n+1)-NU~0(n+1)∥2=〈U0(n)-NU~0(n)+∑j=1sbjWj,U0(n)-NU~0(n)+∑i=1sbiWi〉=∥U0(n)-NU~0(n)∥2+2∑i=1sbiRe〈U0(n)-NU~0(n),Wi〉+∑i,j=1sbibj〈Wi,Wj〉=∥U0(n)-NU~0(n)∥2+2∑i=1sbiRe〈Ui(n)-NU~i(n)-∑j=1saijWj,Wi〉+∑i,j=1sbibj〈Wi,Wj〉=∥U0(n)-NU~0(n)∥2+2∑i=1sbiRe〈Ui(n)-NU~i(n),Wi〉-∑i,j=1s‍(biaij+ajibj-bibj)〈Wi,Wj〉.
Now, in view of the nonnegative definite matrix M, we obtain
(15)-∑i,j=1s‍(biaij+ajibj-bibj)〈Wi,Wj〉≤1r∑i=1s‍bi〈Wi,Wi〉.
On the other hand, in terms of condition (3), we find
(16)Re〈Ui(n)-NU~i(n),Wi〉≤αh∥Ui(n)∥2+βh∥U~i(n)∥2+δh∥Wi∥2.
Then, together with (14), (15), and (16) and using the conditions h/r≤-2δ, we get
(17)∥U0(n+1)-NU~0(n+1)∥2≤∥U0(n)-NU~0(n)∥2+2∑i=1shbi‍(αh∥Ui(n)∥2+βh∥U~i(n)∥2+(δh+12r)∥Wi∥2)≤∥U0(n)-NU~0(n)∥2+2∑i=1shbi(α∥Ui(n)∥2+β∥U~i(n)∥2).
Noting that
(18)∥U~i(n)∥2=[θ∥Ui(n-m+1)∥+(1-θ)∥Ui(n-m)∥]2≤θ2∥Ui(n-m+1)∥2+(1-θ)2∥Ui(n-m)∥2+θ(1-θ)(∥Ui(n-m+1)∥2+∥Ui(n-m)∥2)=θ∥Ui(n-m+1)∥2+(1-θ)∥Ui(n-m)∥2
and α+β≤0, we have
(19)∥U0(n+1)-NU~0(n+1)∥2≤∥U0(n)-NU~0(n)∥2+2∑i=1s‍hβbi(∥U~i(n)∥2-∥Ui(n)∥2)≤∥U0(n)-NU~0(n)∥2+2∑i=1s‍hβbi(θ∥Ui(n-m+1)∥2+(1-θ)∥Ui(n-m)∥2-∥Ui(n)∥2)≤∥U0(0)-NU~0(0)∥2+2∑i=1s‍hβbi(∑j=-m+1-1‍θ∥Ui(j)∥2+∑j=-m-1‍(1-θ)∥Ui(j)∥2)≤∥U0(0)-NU~0(0)∥2+2∑i=1s‍mhβbimax⁡-m≤j≤-1∥Ui(j)∥2≤2∥U0(0)∥2+2∥N∥∥U~0(0)∥2+2∑i=1s‍mhβbimax⁡-m≤j≤-1∥Ui(j)∥2≤(2+2∥N∥2+2τ∑i=1s‍βbi)max⁡-τ≤t≤0∥ψ(t)-ϕ(t)∥2=(2+2∥N∥2+2τβ)max⁡-τ≤t≤0∥ψ(t)-ϕ(t)∥2.
This implies that(20)∥U0(n+1)-NU~0(n+1)∥≤C~max⁡-τ≤t≤0∥ψ(t)-ϕ(t)∥,
where C~=(2+2∥N∥2+2τβ).
Note that∥N∥<1; we have
(21)∥U0(n+1)∥≤∥N∥∥U~0(n+1)∥+C~max⁡-τ≤t≤0∥ψ(t)-ϕ(t)∥=∥N∥∥θU0n-m+2+(1-θ)U0n-m+1∥+C~max⁡-τ≤t≤0∥ψ(t)-ϕ(t)∥≤max⁡⁡(∥U0n-m+2∥,∥U0n-m+1∥)+C~max⁡-τ≤t≤0∥ψ(t)-ϕ(t)∥.
An induction to (21) yields
(22)∥U0(n+1)∥≤(1+C~)max⁡-τ≤t≤0∥ψ(t)-ϕ(t)∥.
Therefore, the conclusion is proven.Corollary 4.
Assume condition (3) holds; α+β≤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(23)diag⁡(b)A+ATdiag⁡(b)-bbT
is nonnegative definite and bi≥0(i=1,2,…,s). For example, thes-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≤0, and there exists a positive real number r, such that the matrix
(24)M=diag⁡(b)A+ATdiag⁡(b)-bbT+1rdiag⁡(b)
is nonnegative definite, where bi≥0,  i=1,2,…,s. Then the Runge-Kutta method with linear interpolation procedure for DDEs (4) is globally stable under the stepsize restriction
(25)hr≤-2δ.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
(26)M=diag⁡(b)A+ATdiag⁡(b)-bbT+1rdiag⁡(b)
is nonnegative definite, where bi≥0,  i=1,2,…,s. Then the Runge-Kutta method with linear interpolation procedure for NDDEs (1) is asymptotically stable under the stepsize restriction
(27)hr≤-2δ.Proof.
Like in the proof of Theorem3, let σ=α+β<0, and we can easily find
(28)∥U0(n+1)-NU~0(n+1)∥2≤∥U0(n)-NU~0(n)∥2+2∑i=1s‍hbi(α∥Ui(n)∥2+β∥U~i(n)∥2)≤2(∥U0(n)∥2+∥N∥2∥U~0(n)∥2)+2∑i=1s‍hbi((σ-β)∥Ui(n)∥2+β∥U~i(n)∥2)=2(∥U0(n)∥2+∥N∥2∥U~0(n)∥2)+2∑i=1s‍hβbi(∥U~i(n)∥2-∥Ui(n)∥2)+2∑i=1s‍hbiσ∥Ui(n)∥2≤(2+2∥N∥2+2τ∑i=1s‍βbi)max⁡-τ≤t≤0∥ψ(t)-ϕ(t)∥2+2∑j=1n‍∑i=1s‍hbiσ∥Ui(j)∥2.
Note σ<0 and bi≥0; we have
(29)lim⁡n→∞∑i=1s‍bi∥Ui(n)∥=0.
On the other hand,
(30)∥Wi∥=∥h[f(tn+cih,yi(n),y~i(n))-f(tn+cih,zi(n),z~i(n))]∥≤hL(∥Ui(n)∥+∥U~i(n)∥).
Now, in view of (13), (29), and (30), we obtain
(31)lim⁡n→∞∥U0(n)-NU~0(n)∥=0.
Since
(32)∥U0(n)∥=∥U0(n)-NU~0(n)+NU~0(n)∥≤∥U0(n)-NU~0(n)∥+∥N∥∥U~0(n)∥≤∥U0(n)-NU~0(n)∥+∥N∥max⁡⁡(∥U0n-m+2∥,∥U0n-m+1∥)
and ∥N∥<1, an induction to (32) gives
(33)lim⁡n→∞∥U0(n)∥=0
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
(34)M=diag⁡(b)A+ATdiag⁡(b)-bbT+1rdiag⁡(b)
is nonnegative definite, where bi≥0, i=1,2,…,s. Then the Runge-Kutta method with linear interpolation procedure for DDEs (4) is asymptotically stable under the stepsize restriction
(35)hr≤-2δ.


## 4. Some Examples

As it is shown in the theorems, the parametersδ 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
(36)∥ρ((y-Nu)-(z-Nv))+f~(t,y-Nu)-f~(t,z-Nv)∥≤ρ∥((y-Nu)-(z-Nv))∥
with ρ>0, we have the following form in an inner product norm:
(37)Re〈(y-Nu)-(z-Nv),f~(t,y-Nu)-f~(t,z-Nv)〉≤δ∥f~(t,y-Nu)-f~(t,z-Nv)∥2
with δ=-1/(2ρ)<0.In particular, letf(t,y,u)=-a(My-Nu), where a>0,  M<1 are constants independent of t, respectively. We have
(38)Re〈y-Nu,f(t,y,u)〉=Re〈My-Nu-(M-1)y,-a(My-Nu)〉=-1a∥a(My+Nu)∥2+aM(M-1)∥y∥2+Re〈a(1-M)y,Nu〉≤-1a∥a(My+Nu)∥2+aM(M-1)∥y∥2+12a(1-M)∥N∥(∥y∥2+∥u∥2)=(aM(M-1)+12a(1-M)∥N∥)∥y∥2+12a(1-M)∥N∥∥u∥2-1a∥f(t,y,u)∥2.Next, we give some examples on how to calculate the parameterr.Example 1.
Considers-stage 1-order Runge-Kutta methods (see [17], section 2.7)
(39)001s1s02s1s1s0⋮⋮⋮⋮s-1s1s1s1s…1s01s1s1s1s1s1s
and we have
(40)M=diag⁡(b)A+ATdiag⁡(b)-bbT+1rdiag⁡(b)=(1rs-1s2)Is.
Thus, the matrix M is nonnegative definite for 0<r≤s. They imply that these methods for DDE with interpolation are stable with stepsize restriction h≤-2δs.Example 2.
Consider 2-stage 2-order Runge-Kutta method:(41)0001101212
and we obtain
(42)M=[12r-14141412r-14].
Therefore, the matrix M is nonnegative definite for 0<r≤1. They imply that the stepsize h≤-2δ is feasible under the assumptions (3) for NDDEs (1).For more Runge-Kutta methods with the nonnegative definite matrixM, 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
(43)r=-λmin⁡-1([diag⁡(b)]-1/2[ATdiag⁡(b)A+ATdiag⁡(b)-bbT]×[diag⁡(b)]-1/2),
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:(44)ddt[y(t)-∑i=1l‍Niy(t-τi)]=f(t,y,y(t-τ1),…,y(t-τl)),t≥0,y(t)=ψ(t),t≤0,
under the following assumption:
(45)Re〈(y-z)-∑i=1lNi(yi-zi),f(t,y,y1,…,yN)-f(t,z,z1,…,zN)∑i=1lNi(yi-zi)〉≤α∥y-z∥2+∑i=1Nβi∥yi-zi∥2‍+δ∥f(t,y,y1,…,yN)-f(t,z,z1,…,zN)∥2,
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].



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