Numerical Stability and Oscillations of Runge-Kutta Methods for Dif f erential Equations with Piecewise Constant Arguments of Advanced Type

WANG QI

(School of Applied Mathematics,Guangdong University of Technology,Guangzhou,510006)

Communicated by Ma Fu-ming

Numerical Stability and Oscillations of Runge-Kutta Methods for Dif f erential Equations with Piecewise Constant Arguments of Advanced Type

WANG QI

(School of Applied Mathematics,Guangdong University of Technology,Guangzhou,510006)

Communicated by Ma Fu-ming

For dif f erential equations with piecewise constant arguments of advanced type,numerical stability and oscillations of Runge-Kutta methods are investigated. The necessary and sufficient conditions under which the numerical stability region contains the analytic stability region are given.The conditions of oscillations for the Runge-Kutta methods are obtained also.We prove that the Runge-Kutta methods preserve the oscillations of the analytic solution.Moreover,the relationship between stability and oscillations is discussed.Several numerical examples which conf i rm the results of our analysis are presented.

numerical solution,Runge-Kutta method,asymptotic stability,oscillation

1 Introduction

In this paper,we consider the dif f erential equations with piecewise constant arguments (EPCA)of advanced type,given bywhere a,b,u0are real constants and[·]denotes the greatest integer function.The general form of(1.1)is

where the argument α(t)has intervals of constancy.

The theory of EPCA was initiated in[1–3].In the literature,there are many papers dealing with the properties of EPCA,such as Wiener and Cooke[4],Xia et al.[5],Muroya[6]and Akhmet[7].Signif i cant parts of pioneer results for EPCA can be found in[8].For more details of EPCA,the reader can see[9–11]and the references therein.

In recent years,much research focused on the numerical solutions of EPCA.The stability and the oscillations of numerical solutions of EPCA was investigated in[12–16].As far as we know,very few results were obtained on combining the stability with the oscillations of the numerical solutions in the paper except for[17].Dif f erent from[17],the novel idea of our paper is that we study both stability and oscillations of the numerical solutions by using the Runge-Kutta methods for the problem(1.1),and their relationships are analyzed quantitatively.

2 Preliminaries

In this section,we introduce some def i nitions and theorems which are useful for our paper.

Def i nition 2.1[8]A solution of the problem(1.1)on[0,∞)is a function u(t)which satisf i es the conditions:

(i)u(t)is continuous on[0,∞);

(ii)The derivative u′(t)exists at each point t∈[0,∞)with the possible exception of the points[t]∈[0,∞),where one-sided derivatives exist;

(iii)(1.1)is satisf i ed on each interval[n,n+1)?[0,∞)with integral end-points.

Theorem 2.1[8]

,then the problem(1.1)has a unique solution on[0,∞)

where{t}is the fractional part of t and

Theorem 2.2[8]The solution of the problem(1.1)is asymptotically stable for all u0,if and only if()

Def i nition 2.2A non-trivial solution of the problem(1.1)is said to be oscillatory if there exists a sequencesuch that tk→∞as k→∞and u(tk)u(tk-1)＜0,otherwise,it is called non-oscillatory.We say that the problem(1.1)is oscillatory if all the non-trivial solutions of(1.1)are oscillatory.We say that the problem(1.1)is non-oscillatory if all the non-trivial solutions of(1.1)are non-oscillatory.

Theorem 2.3[8]Every solution of the problem(1.1)is non-oscillatory if and only if

3 Numerical Stability

3.1The Runge-Kutta Methods

For the ν-stage Runge-Kutta methods(A,B,C)with the matrixthe vectorsandwe always assume that B1+B2+ ···+Bν=1 and 0≤C1≤C2≤···≤Cν≤1.By use of similar process as in[13],letbe a given stepsize with integer m≥1 and the grid points tnbe def i ned by tn=nh (n=1,2,3,···).Applying the Runge-Kutta methods to the problem(1.2)leads to the following numerical process:

where unanddenote the approximation to u(t)at the grid points tn(n=1,2,3,···) and the approximation to,respectively. Application of the process(3.1)to the problem(1.1)yields

If we denote n=km+l(l=0,1,···,m-1),then can be def i ned as ukm.Let.Then(3.2)reduces to

Assume that a/=0 and I-xA is invertible.By(3.3)we have

It is not difficult to see that(3.4)is equivalent to

Similarly to[13],we can easily prove that the Runge-Kutta methods for the problem (1.1)preserve their order of convergence.

3.2Stability Analysis

For given Runge-Kutta methods,we assume δ1＜0＜δ2such that

which implies

Def i nition 3.1The process(3.1)for the problem(1.1)is called asymptotically stable at (a,b)for all sufficiently small h if and only if there exists a constant h0,such that for any given u0,the relations(3.5)and(3.6)def i ne unthat satisf i es un→0 for n→∞whenever 0＜h＜h0.

Def i nition 3.2The set,denote by S,of all points(a,b)at which the process(3.1)for the problem(1.1)is asymptotically stable,for all sufficiently small h,is called the asymptotic stability region.

Corollary 3.1un→0 as n→∞if and only if ukm→0 as k→∞.

Theorem 3.1The numerical solution of the problem(1.1)is asymptotically stable if and only if

Proof.By Def i nition 3.1,Corollary 3.1 and(3.5),we have that un→0 as n→∞if and only if

where

So we have

which is equivalent to

This ends the proof of the theorem.

By Theorem 2.2,we have the following corollary.

Corollary 3.2The analytic solution of the problem(1.1)is asymptotically stable for all u0,if and only if

Let us def i ne the following set of all points(a,b)∈R2which satisfy condition(3.8)by H: Similarly,by Theorem 3.1,we have

{ and then we study which conditions lead to H?S.The following corollary is useful to determine the stability and oscillatory conditions.

Corollary 3.3[13]Suppose that Q(x)=φ(x)/ψ(x)(where φ(x),ψ(x)are polynomials)is the(r,s)-Padˊe approximation to ex.Then

(i)Q(x)＜exif and only if s is even for all x＞0;

(ii)Q(x)＞exif and only if s is odd for 0＜x＜ξ;

(iii)Q(x)＞exif and only if r is even for all x＜0;

(iv)Q(x)＜exif and only if r is odd for all η＜x＜0,

where ξ is a real zero of ψs(x)and η is a real zero of φr(x).

Then the main stability results of this paper are as follows.

Theorem 3.2Suppose that the stability function Q(x)of the Runge-Kutta method is given by the(r,s)-Padˊe approximation to ex.Then H?S if and only if s is odd for a＞0 and r is odd for a＜0.

Proof.In view of(3.9)and(3.10),we know that H?S if and only if

which is equivalent to

As a consequence of Corollary 2.1,the proof is completed.

Theorem 3.3If a=0 for all Runge-Kutta methods,we have H=S.

Remark 3.1For the A-stable higher order Runge-Kutta method,it is easy to obtain the corresponding results from Theorem 3.2(see Table 3.1).

4 Numerical Oscillations and Non-oscillations

In this section,we discuss the oscillations and non-oscillations of numerical solution.

Theorem 4.1Let{un}and{ukm}be given by(3.6)and(3.5),respectively.Then{un} is non-oscillatory if and only if{ukm}is non-oscillatory.

Proof.It is not difficult to know that the necessity is obvious.Next,we consider the sufficiency.If{ukm}is non-oscillatory,we assume that{ukm}is an eventually negative solution of(3.5).That is,there exists a K0∈R such that ukm＜0 for k＞K0.We prove ukm+l＜0 for all k＞K0+1 and l=0,1,···,m-1.Suppose that b＜0.According to (3.6),if a＞0,then 1＜Q(x)＜∞and Q(x)-m≤Q(x)-l.Therefore

So ukm+l＜0.The proof is completed.

From Theorem 4.1 we get the following corollary.

Corollary 4.1Let{un}and{ukm}be given by(3.6)and(3.5),respectively.Then{un} is oscillatory if and only if{ukm}is oscillatory.

Theorem 4.2(3.5)is oscillatory if and only if

Proof.(3.5)is oscillatory if and only if the roots of the characteristic equation satisfy

which is equivalent to

The proof is completed.

Let

We have the following lemma.

Lemma 4.1

(R1)Λ(m)→Λ as h→0;

(R2)Λ≥Λ(m)if either of the following conditions is satisf i ed:

(R3)Λ＜Λ(m)if either of the following conditions is satisf i ed:

Proof.(R2)If a＞0,ex≤Q(x),then ea≤Q(x)m,which is equivalent to

So we have Λ≥Λ(m).

Similarly,we can prove the other cases.

Def i nition 4.1[15]We say that the Runge-Kutta method preserves the oscillations of the problem(1.1)if the problem(1.1)oscillates implying that there is an h0＞0 such that(3.6) oscillates for h＜h0.Similarly,we say that the Runge-Kutta method preserves the nonoscillations of the problem(1.1)if the problem(1.1)non-oscillates implying that there is an h0＞0 such that(3.6)non-oscillates for h＜h0.

By Theorems 2.3,4.1,4.2 and Corollary 4.1,we have

Theorem 4.3(i)The Runge-Kutta method preserves the oscillations of the problem(1.1) if and only if Λ＞Λ(m);

(ii)The Runge-Kutta method preserves the non-oscillations of the problem(1.1)if and only if Λ≤Λ(m).

From Theorem 4.3,Lemma 4.1 and Corollary 3.3,we can obtain the following main theorems about oscillations.

Theorem 4.4Suppose that Q(x)is the(r,s)-Padˊe approximation to ex.Then the Runge-Kutta method preserves oscillations of the problem(1.1)if either of the following conditions is satisf i ed:

(i)a＞0,h＜h1and s is odd;

(ii)a＜0,h＜h2and r is odd,

where h1=-δ1/a,h2=-δ2/a.

Theorem 4.5Suppose that Q(x)is the(r,s)-Padˊe approximation to ex.Then the Runge-Kutta method preserves non-oscillations of the problem(1.1)if either of the following conditions is satisf i ed:

(i)a＞0,h＜h1and s is even;

(ii)a＜0,h＜h2and r is even,

where h1=-δ1/a,h2=-δ2/a.

We give the conditions that some higher order Runge-Kutta method preserves oscillations and non-oscillations of the problem(1.1)(see Tables 4.1 and 4.2).

Table 4.1Preservation of oscillations for higher order Runge-Kutta method

Table 4.2Preservation of non-oscillations for higher order Runge-Kutta method

5 Relationships Between Stability and Oscillations

Set

By Theorems 2.3,3.1,4.2 and Corollary 3.2,we can get the other main theorems as follows.

Theorem 5.1The analytic solution of the problem(1.1)is

(i)non-oscillatory and asymptotically stable if b∈(-∞,-a);

(ii)non-oscillatory and unstable if b∈(-a,Λ);

(iii)oscillatory and unstable if b∈(Λ,?Λ);

(iv)oscillatory and asymptotically stable if b∈(?Λ,+∞).

Theorem 5.2The numerical solution of the problem(1.1)is

(i)non-oscillatory and asymptotically stable if b∈(-∞,-a);

(ii)non-oscillatory and unstable if b∈(-a,Λ(m));

(iii)oscillatory and unstable if b∈(Λ(m),?Λ(m));

(iv)oscillatory and asymptotically stable if b∈(?Λ(m),+∞).

6 Numerical Experiments

In this section,we use the following six equations to demonstrate the main theorems:

By straightforward computing to(6.1)and(6.2),it is not difficult to see that(3,4), (-4,-3)∈H.

We use 1-Gauss-Legendre method,2-Radau IA method and 2-Lobatto IIIC method with stepsize h=1/m to get the numerical solution at t=15,where the true solutions are u(15)≈-0.0525 and u(15)≈2.2289×10-30for(6.1)and(6.2),respectively.

In Tables 6.1 and 6.2 we have listed the absolute errors(AE)and the relative errors(RE) at t=15 and the Ratio of the errors of the case m=50 over that of m=100.We can see from these tables that the methods preserve their order of convergence.

Table 6.1

Table 6.2

Furthermore,in Figs.6.1 and 6.2,we draw the numerical solutions by 1-Gauss-Legendre method and 2-Radau IA method with m=20 for(6.1)and(6.2),respectively.It is easy to see that the numerical solutions are asymptotically stable.

The analytic solutions of(6.3)and(6.4)are oscillatory;the analytic solutions of(6.5) and(6.6)are non-oscillatory according to Theorem 2.3.From Figs.6.3–6.6,we can see that the numerical solutions of(6.3)and(6.4)are oscillatory;the numerical solutions of(6.5) and(6.6)are non-oscillatory,which are in accordance with Theorems 4.4 and 4.5.

Fig.6.1The numerical solution of(6.1)by 1-Gauss-Legendre method with m=20.

Fig.6.2The numerical solution of(6.2)by 2-Radau IA method with m=20.

Fig.6.3The analytic and numerical solution of(6.3)by 1-Gauss-Legendre method with m=50.

Fig.6.4The analytic and numerical solution of(6.4)by 2-Radau IA method with m=50.

Fig.6.5The analytic and numerical solution of(6.5)by 2-Radau IA method with m=50.

Fig.6.6The analytic and numerical solution of(6.6)by 2-Lobatto IIIC method with m=50.

From Fig.6.3,we can also see that the analytic solutions and the numerical solutions of (6.3)are both oscillatory and asymptotically stable,which are in agreement with Theorems 5.1 and 5.2.For(6.1)–(6.2)and(6.4)–(6.6),we can verify them in the same way(see Figs. 6.1–6.2,6.4–6.6).

AcknowledgementThe author would like to thank Professor Liu Ming-zhu for his helpful comments and constructive suggestions to improve the quality of the paper.

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tion:65L07,65L20

A

1674-5647(2013)02-0131-12

Received date:April 26,2011.

The NSF(11201084,51008084)of China.

E-mail address:bmwzwq@126.com(Wang Q).