Stability analysis of Runge–Kutta methods for …stefan.vandewalle/Papers/...IMA Journal of...

IMA Journal of Numerical Analysis (2004)24, 193–214

Stability analysis of Runge–Kutta methods for nonlinearVolterra delay-integro-differential equations

CHENGJIAN ZHANG† AND STEFAN VANDEWALLE‡Department of Computer Science, Katholieke Universiteit Leuven,

Celestijnenlaan 200A, B-3001 Leuven, Belgium

[Received on 28 October 2002; revised on 16 September 2003]

This paper deals with the stability of Runge–Kutta methods for a class of stiff systemsof nonlinear Volterra delay-integro-differential equations. Two classes of methods areconsidered: Runge–Kutta methods extended with a compound quadrature rule, and Runge–Kutta methods extended with a Pouzet type quadrature technique. Global and asymptoticstability criteria for both types of methods are derived.

Keywords: stability; Volterra delay-integro-differential equations; Runge–Kutta methods;compound quadrature; Pouzet quadrature.

1. Introduction

The last several decades have witnessed a large development in the computationalimplementation and the theoretical analysis of numerical methods for integro-differentialequations of Volterra type. An extensive collection of results has been presented in themonographs by Linz (1985), Brunner & van der Houwen (1986) and Baker (2000). Theaforementioned references focus on integro-differential equations without delay. However,it is well known that certain real-life problems require models of Volterradelay-integro-differential equation type (VDIDEs) for an adequate description (cf. Bochararov & Rihan,2000). Hence, recently, researchers have also turned their attention to the study of VDIDEs.Baker & Ford (1992), Koto (2002), Huang & Vandewalle (2003) and Luzyaninaet al.(2003) have dealt with the linear stability of numerical methods for VDIDEs. Baker &Ford (1988), Brunner (1994) and Enright & Hu (1997) have studied the convergence oflinear multistep methods and continuous Runge–Kutta methods, respectively.

Up to now, only few results have been presented in the literature on thenonlinearstability of numerical methods for VDIDEs. Baker & Tang (1997) investigated thenonlinear stability of continuous Runge–Kutta methods for equations with unboundeddelays. Their results are primarily applicable and relevant for non-stiff problems sincetheir approach is based on a classical Lipschitz condition. Zhang & Vandewalle (2003)considered BDF methods applied to a class of stiff VDIDEs, which they calledclassDI(α, β, (σ1, σ2), γ ), and obtained some analytical and numerical stability results. In thepresent paper, we continue this study and treat the nonlinear stability of two classes ofadapted Runge–Kutta (RK) methods for a class of stiff VDIDEs, which is more general

†Email: [email protected]‡Email: [email protected]

IMA Journal of Numerical Analysis 24(2),c© Institute of Mathematics and its Applications 2004; all rights reserved.

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194 C. ZHANG AND S. VANDEWALLE

than the above class and calledclass RI(α, β, σ, γ ) (which stands forretarded integrals).Both global and asymptotic stability criteria of the presented methods will be derived.

The paper is structured as follows. In Section 2 we define the problem classRI(α, β, σ, γ ), give its stability properties and formulate two classes of discretizationschemes. Among these schemes, one is a RK method with a compound quadrature formula;the other is a RK method with a Pouzet quadrature formula (cf. Brunner & van der Houwen,1986). The latter was first introduced by Koto (2002) and employed to studylinearnumerical stability. Some concepts and lemmas which play a key role in the derivationof the stability results are given in Section 3. The main results are presented in Sections 4and 5, where we describe the global and asymptotic stability criteria for the above twoclasses of methods. In Section 6 we give some applications to classical methods. There wepoint out that the discretization schemes based on the Gauss, Radau IA, Radau IIA andLobatto IIIC formulae are all globally and asymptotically stable under certain conditions.Finally, we end with some concluding remarks in Section 7, where we compare our resultswith some earlier results for the DDE case.

2. A class of VDIDEs and their Runge–Kutta discretization

Consider the following complexN -dimensional system of VDIDEs with constant delayτ > 0:

y′(t) = f (t, y(t), y(t − τ),

∫ t

t−τ

g(t, v, y(v)) dv), t ∈ [t0, +∞),

y(t) = ϕ(t), t ∈ [t0 − τ, t0],(2.1)

where the mappingsf, g andϕ are smooth enough such that system (2.1) has a uniquesmooth solutiony(t) and satisfies the conditions

�〈 f (t, x, y, z) − f (t, x, y, z), x − x〉 � α‖x − x‖2, (2.2)

‖ f (t, x, y, z) − f (t, x, y, z)‖ � β‖y − y‖ + σ‖z − z‖, (2.3)

‖g(t, v, x) − g(t, v, x)‖ � γ ‖x − x‖, (t, v) ∈ D, (2.4)

in which t ∈ [t0, +∞), D = {(t, v) : t ∈ [t0, +∞), v ∈ [t − τ, t]}, x, y, z, x, y, z ∈ CN ,

〈·, ·〉 and‖ · ‖ denote a given inner product and the corresponding induced norm in thecomplex N -dimensional spaceCN . The constants(−α), β, σ andγ are given and non-negative. In the present paper, all the problems of type (2.1) with (2.2)–(2.4) will be calledproblems of class RI(α, β, σ, γ ).

REMARK 2.1 In Zhang & Vandewalle (2003), the authors have proposed a class ofVDIDE problems which they called classDI(α, β, (σ1, σ2), γ ). From the definitions ofboth problem classes, one easily finds that

DI(α, β, (σ1, σ2), γ ) ⊂ RI(α, βσ1, βσ2, γ ).

Hence, the results obtained in this paper will be more generally applicable than those ofZhang & Vandewalle (2003).

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STABILITY ANALYSIS OF RUNGE-KUTTA METHODS 195

For the subsequent stability analysis, we will also need to consider systems withdifferent initial condition

y′(t) = f (t, y(t), y(t − τ),

∫ t

t−τ

g(t, v, y(v)) dv), t ∈ [t0, +∞),

y(t) = ψ(t), t ∈ [t0 − τ, t0].(2.5)

With a similar proof to that of Theorem 3.2 in Zhang & Vandewalle (2003), we canobtain a stability result for system (2.1).

THEOREM 2.2 Assume that systems (2.1) and (2.5) belong to classRI(α, β, σ, γ ) with

β + σγ τ < −α. (2.6)

Then, the following two stability results hold:

‖y(t) − y(t)‖ � maxθ∈[t0−τ, t0]

‖ϕ(θ) − ψ(θ)‖, ∀t � t0, and limt→+∞ ‖y(t) − y(t)‖ = 0.

(2.7)

REMARK 2.3 The above theorem can be applied to the special but important class ofequations of the following form (cf. Brunner, 1994):

y′(t) = f (t, y(t), y(t − τ)) +∫ t

t−τ

g(t, v, y(v)) dv, t ∈ [t0, +∞),

y(t) = ϕ(t), t ∈ [t0 − τ, t0](2.8)

with (2.4) and with

�〈 f (t, x, y) − f (t, x, y), x − x〉 � α‖x − x‖2, (2.9)

‖ f (t, x, y) − f (t, x, y)‖ � β‖y − y‖. (2.10)

This class of equations belongs to classRI(α, β, 1, γ ). So, they satisfy stability property(2.7) wheneverβ + γ τ < −α. When the integral item

∫ tt−τ

g(t, v, y(v)) dv is removedfrom (2.8), Theorem 2.2 simplifies to a result that was derived in Torelli (1989): (2.7)holds wheneverβ < −α.

To arrive at the discretization schemes for (2.1), we first recall thes-stage underlyingRunge–Kutta (RK) method

y(n)

i = yn + hs∑

j=1

ai j f (tn + c j h, y(n)j ), i = 1,2, . . . , s,

yn+1 = yn + hs∑

j=1

b j f (tn + c j h, y(n)j ), n � 0

(2.11)

for ODEs systems of the formy′(t) = f (t, y(t)), t > t0, with y(t0) = y0. Method (2.11) ischaracterized by the abscissaec j , the weightsb j and the coefficientsai j , where we alwaysassume that the classical consistency conditions are satisfied, together with a commonrestriction on the magnitude ofci , i.e.

∑si=1 bi = 1 and 0� ci � 1, i = 1,2, . . . , s.

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Adapting method (2.11) to VDIDE (2.1) yields the following discretization scheme:

y(n)i = yn + h

s∑j=1

ai j f (t (n)j , y(n)

j , y(n−m)j , z(n)

j ), i = 1,2, . . . , s,

yn+1 = yn + hs∑

j=1

b j f (t (n)j , y(n)

j , y(n−m)j , z(n)

j ), n � 0,

(2.12)

where the time stepsize is given byh = τ/m, with m a given positive integer; thetime points are given bytn = t0 + nh and t (n)

j = tn + c j h, and y(n)i , z(n)

j and yn areapproximations to

y(t (n)i ), z(t (n)

j ) :=∫ t (n)

j

t (n−m)j

g(t (n)j , v, y(v)) dv, and y(tn),

respectively. As to the computation of the integralz(n)j , we distinguish two alternatives.

The first is based on using the compound quadrature (CQ) formula

z(n)j = h

m∑q=0

νq g(t (n)j , t (n−q)

j , y(n−q)j ), j = 1,2, . . . , s, (2.13)

with weights{νq} independent ofm and determined by those of a convergent quadraturerule ∫ τ

0Φ(v) dv ∼= h

m∑q=0

νqΦ((m − q)h) with mh = τ and Φ ∈ C[0, τ]. (2.14)

Such a quadrature formula can be derived from a uniform repeated rule (cf. Baker &Ford, 1992; Zhang & Vandewalle, 2003). By Theorem 2.1.1 in Brunner & van derHouwen (1986) we know that this quadrature formula is convergent iff it converges forall polynomials and if there exits a finite constantν, independent ofm, such that

hm∑

q=0

|νq | < ν with mh = τ . (2.15)

For the stability results in Section 4, we will need a slightly stronger condition than (2.15):

h

√√√√(m + 1)

m∑q=0

|νq |2 < ν with mh = τ . (2.16)

This condition can be fulfilled for most of the common quadrature rules of the form (2.14);some examples have been given in Zhang & Vandewalle (2003). The second approachadopts the so-called Pouzet quadrature (PQ) formula (cf. Brunner & van der Houwen,1986 and Koto, 2002)

z(n)j = h

s∑r=1

a jr g(t (n)j , t (n)

r , y(n)r ) + h

m∑q=1

s∑r=1

br g(t (n)j , t (n−q)

r , y(n−q)r )

− hs∑

r=1

a jr g(t (n)j , t (n−m)

r , y(n−m)r ) (2.17)

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which is produced by discretizing the integral of the functiong(tn + c j h, v, y(v)) over theintervalv ∈ [tn + c j h − τ, tn + c j h], split into the following three parts:∫ tn+c j h

tng(tn + c j h, v, y(v)) dv +

∫ tn

tn−m

g(tn + c j h, v, y(v)) dv

−∫ tn−m+c j h

tn−m

g(tn + c j h, v, y(v)) dv.

Moreover, we always sety0 = y(t0) and, for−m � n < 0, we take

y(n)j = y(t (n)

j ), z(n)j =

∫ t (n)j

t (n)j −τ

g(t (n)j , v, ϕ(v)) dv, yn = y(tn).

As such, we have defined two classes of discretization schemes for (2.1): method (2.12)with (2.13) and method (2.12) with (2.17). The former will further be calledRK methodwith CQ formula and the latterRK method with PQ formula. When any of the abovemethods is applied to system (2.5), the corresponding discrete variables will be denotedwith a tilde superscript, e.g.y(t (n)

j ) is approximated byy(n)j .

3. Introductory concepts and basic lemmas

This section will recall and present some concepts and lemmas that will be important forthe presentation of our main results in Sections 4 and 5.

DEFINITION 3.1 (cf. Burrage & Butcher, 1980) The underlying RK method (2.11) iscalled(k, l)-algebraically stable if there exist real constantsk > 0, l and a non-negativediagonal matrixD = diag(d1, d2, . . . , ds) ∈ R

s×s such that the matrixM is non-negativedefinite, where

M =(

k − 1 − 2leT De eT D − bT − 2leT D ADe − b − 2l AT De D A + AT D − bbT − 2l AT D A

)∈ R

(s+1)×(s+1),

A = (ai j ) ∈ Rs×s, b = (b1, b2 . . . , bs)

T ∈ Rs ande = (1,1, . . . , 1)T ∈ R

s . In particular,a (1,0)-algebraically stable method is called algebraically stable.

Some(k, l)-algebraically stable underlying RK methods are given in the referencesby Burrage & Butcher (1980) and Hairer & Wanner (1991). Using this concept, Huanget al. (1999) investigated the nonlinear stability of RK methods fornon-distributed delaydifferential equations. Here, we will adopt this concept to analyse the nonlinear stability ofRK methods fordistributed delay differential equations (i.e. VDIDEs).

First, we introduce the following notational conventions:

Yn = yn − yn, Y(n)j = y(n)

j − y(n)j , Z(n)

j = z(n)j − z(n)

j ,

F (n)j = f (t (n)

j , y(n)j , y(n−m)

j , z(n)j ) − f (t (n)

j , y(n)j , y(n−m)

j , z(n)j ).

The following stability behaviour of RK methods (2.12) with CQ formula (2.13) andRK methods with PQ formula (2.17) will be investigated in the subsequent sections.

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DEFINITION 3.2 Method (2.12) with CQ formula (2.13) or PQ formula (2.17) is calledglobally stable for problems of classRI(α, β, σ, γ ) if there exists a constantH > 0, whichdepends only onα, β, σ, γ, τ and the method, such that

‖Yn‖ � H maxt0−τ�t�t0

‖ϕ(t) − ψ(t)‖, ∀n � 1. (3.1)

Since the stability constantH does not depend on the time variablet , the above conceptof global stability is a type of long-term stability. In other studies, when a similar numericalstability for other differential equations was investigated, one often askedH = 1. However,it is our opinion that this excludes many excellent numerical methods.

DEFINITION 3.3 Method (2.12) with CQ formula (2.13) or PQ formula (2.17) is calledasymptotically stable for problems of classRI(α, β, σ, γ ) if

limn→∞ ‖Yn‖ = 0. (3.2)

When presenting our stability analysis, the two (very technical) lemmas proven belowwill play a key role.

LEMMA 3.4 Suppose that the underlying RK method (2.11) is(k, l)-algebraically stablewith 0 < k � k � 1, wherek is independent of the indexn of the method, and theconditions (2.2)–(2.3) hold. Then, the induced method (2.12) satisfies for alln � 0

‖Yn+1‖2 � kn+1‖Y0‖2 + {[h(2α + β + σ) − 2l]km + hβ}n∑

i=0

kn−m−is∑

j=1

d j‖Y(i)j ‖2

+βτ kn−m+1s∑

j=1

d j max−m�i�−1

{‖Y(i)j ‖2} + hσ

n∑i=0

kn−is∑

j=1

d j‖Z(i)j ‖2. (3.3)

REMARK 3.5 In particular, whenk = 1, (3.3) can be read as

‖Yn+1‖2 � ‖Y0‖2 + {h[2(α + β) + σ ] − 2l}n∑

i=0

s∑j=1

d j‖Y(i)j ‖2

+βτ

s∑j=1


{‖Y(i)j ‖2} + hσ

n∑i=0

s∑j=1

d j‖Z(i)j ‖2. (3.4)

Proof. By a direct computation and(k, l)-algebraic stability, one has (see also Burrage &Butcher, 1980)

‖Yn+1‖2 − k‖Yn‖2 − 2s∑

j=1

d j�〈Y(n)j , hF (n)

j − lY(n)j 〉 = −

s+1∑i=1

s+1∑j=1

mi j 〈ωi , ω j 〉 � 0,

(3.5)

whereM = (mi j ), ω1 = Yn, ωi+1 = hF (n)i (i = 1,2, . . . , s). Hence, one has

‖Yn+1‖2 � k‖Yn‖2 + 2s∑

j=1

d j�〈Y(n)j , hF (n)

j − lY(n)j 〉. (3.6)

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It follows from (2.2)–(2.3) that

2�〈Y(n)j , hF (n)

j 〉 = 2h�〈Y(n)j , f (t (n)

j , y(n)j , y(n−m)

j , z(n)j ) − f (t (n)

j , y(n)j , y(n−m)

j , z(n)j )〉

+2h�〈Y(n)j , f (t (n)

j , y(n)j , y(n−m)

j , z(n)j ) − f (t (n)

j , y(n)j , y(n−m)

j , z(n)j )〉

� 2hα‖Y(n)j ‖2 + 2h‖Y(n)

j ‖‖ f (t (n)j , y(n)

j , y(n−m)j , z(n)

j )

− f (t (n)j , y(n)

j , y(n−m)j , z(n)

j )‖� 2hα‖Y(n)

j ‖2 + 2h‖Y(n)j ‖(β‖Y(n−m)

j ‖ + σ‖Z(n)j ‖)

� h(2α + β + σ)‖Y(n)j ‖2 + hβ‖Y(n−m)

j ‖2 + hσ‖Z(n)j ‖2 (3.7)

where the latter is obtained by using the inequality 2uv � u2+v2 (∀u, v ∈ R). Substitutingboth (3.7) and the condition 0< k � k into (3.6) gives

‖Yn+1‖2 � k‖Yn‖2 + [h(2α + β + σ) − 2l]s∑

j=1

d j‖Y(n)j ‖2

+hs∑

j=1

d j [β‖Y(n−m)j ‖2 + σ‖Z(n)

j ‖2].

An induction to the above inequality generates the following result:

‖Yn+1‖2 � kn+1‖Y0‖2 + [h(2α + β + σ) − 2l]n∑

i=0

kn−is∑

j=1

d j‖Y(i)j ‖2

+ hn∑

i=0

kn−is∑

j=1

d j [β‖Y(i−m)j ‖2 + σ‖Z(i)

j ‖2]. (3.8)

Also, it holds that

hn∑

i=0

kn−is∑

j=1

d j‖Y(i−m)j ‖2 = h

n−m∑i=0

kn−m−is∑

j=1

d j‖Y(i)j ‖2 + h

−1∑i=−m

kn−m−i

×s∑

j=1

d j‖Y(i)j ‖2 � h

n∑i=0

kn−m−is∑

j=1

d j‖Y(i)j ‖2

+τ kn−m+1s∑

j=1


‖Y(i)j ‖2, (3.9)

where we have used conditionsmh = τ and 0< k � 1. A combination of (3.8) and (3.9)infers (3.3). This completes the proof. �LEMMA 3.6 Suppose that{Ai }n

i=0 and {Bi }ni=−m are two arbitrary non-negative real

sequences. Then, the following inequalities hold:

n∑i=0

(Ai

m∑j=0

Bi− j

)�

m∑j=0

n∑i=0

Ai+ j Bi +(

m∑j=1

j∑i=1

A j−i

)max

−m�q�−1{Bq}, ∀n, m � 0

(3.10)

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andn∑

i=0

m∑j=0

Bi− j � (m + 1)

n∑i=0

Bi + m(m + 1)

2max

−m�q�−1{Bq}, ∀n, m � 0. (3.11)

Proof. Inequality (3.10) is proven first, by rewriting its left-hand side as follows:

n∑i=0

(Ai

m∑j=0

Bi− j

)=

n∑i=0

Ai Bi +m∑

j=1

n∑i=0

Ai Bi− j

=n∑

i=0

Ai Bi +m∑

j=1

(n∑

i= j

Ai Bi− j +j−1∑i=0

Ai Bi− j

)

=n∑

i=0

Ai Bi +m∑

j=1

(n− j∑i=0

Ai+ j Bi +j∑

i=1

A j−i B−i

)

=m∑

j=0

n− j∑i=0

Ai+ j Bi +m∑

j=1

j∑i=1

A j−i B−i .

The non-negativity ofAi andBi allows us to bound the latter expression by the right-handside of (3.10). Inequality (3.11) is obtained by settingAi = 1 for all i in inequality (3.10).

�A trivial modification of inequalities (3.10) and (3.11) will also be used further on in

this paper, namely the inequalities

n∑i=0

(Ai

m∑j=1

Bi− j

)�

m∑j=1

n∑i=0

Ai+ j Bi +(

m∑j=1

j∑i=1

A j−i

)max

−m�q�−1{Bq}, ∀n, m � 0

(3.12)

andn∑

i=0

m∑j=1

Bi− j � mn∑

i=0

Bi + m(m + 1)

2max

−m�q�−1{Bq}, ∀n, m � 0. (3.13)

4. Stability of RK methods with CQ formula

This section will deal with both the global and the asymptotic stability of Runge–Kuttamethods with compound quadrature formula.

THEOREM 4.1 Suppose the underlying RK method (2.11) is(k, l)-algebraically stablefor a non-negative diagonal matrixD = diag(d1, d2, . . . , ds) ∈ R

s×s , where 0< k � 1,and suppose the quadrature formula (2.13) satisfies condition (2.16). Then, the induced RKmethod (2.12) with CQ formula (2.13) is globally stable for class(α, β, σ, γ ) with stabilityconstant

H =√√√√1 + τ(2β + σγ 2ν2)

2

s∑j=1

d j (4.1)

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whenever

h[2(α + β) + σ(1 + γ 2ν2)] � 2l. (4.2)

Proof. By conditions (2.4) and (2.16), we have

‖Z(i)j ‖2 � (hγ )2

(m∑

q=0

|νq |‖Y(i−q)j ‖

)2

� (hγ )2

(m∑

q=0

|νq |2) (

m∑q=0

‖Y(i−q)j ‖2

)

� (γ ν)2

m + 1

m∑q=0

‖Y(i−q)j ‖2, (4.3)

in which the Cauchy inequality has been used. Substituting (4.3) into (3.4) yields

‖Yn+1‖2 � ‖Y0‖2 + [h(2(α + β) + σ) − 2l]n∑

i=0

s∑j=1

d j‖Y(i)j ‖2

+βτ

s∑j=1


{‖Y(i)j ‖2} + hσγ 2ν2

m + 1

n∑i=0

s∑j=1

d j

m∑q=0

‖Y(i−q)j ‖2. (4.4)

With inequality (3.11) in Lemma 3.6, it holds that

n∑i=0

m∑q=0

‖Y(i−q)j ‖2 � (m + 1)

n∑i=0

‖Y(i)j ‖2 + m(m + 1)

2max

−m�i�−1{‖Y(i)

j ‖2}. (4.5)

Embedding (4.5) and conditionmh = τ into (4.4) yields

‖Yn+1‖2 � ‖Y0‖2 + [h(2(α + β) + σ) − 2l]n∑

i=0

s∑j=1

d j‖Y(i)j ‖2

+βτ

s∑j=1


{‖Y(i)j ‖2}

+hσγ 2ν2s∑

j=1

d j

n∑i=0

‖Y(i)j ‖2 + mhσγ 2ν2

2

s∑j=1


{‖Y(i)j ‖2}

= ‖Y0‖2 + [h(2(α + β) + σ(1 + γ 2ν2)) − 2l]n∑

i=0

s∑j=1

d j‖Y(i)j ‖2

+τ(2β + σγ 2ν2)

2

s∑j=1


{‖Y(i)j ‖2}.

Hence (4.2) implies that

‖Yn+1‖2 �[

1 + τ(2β + σγ 2ν2)

2

s∑j=1

d j

]max

t0−τ�t�t0‖ϕ(t) − ψ(t)‖2, ∀n � 0. (4.6)

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This leads to the value ofH given in (4.1); hence the method is globally stable. �

Since an underlying RK method is algebraically stable iffb j � 0 ( j = 1,2, . . . , s) andmatrix D A+ AT D−bbT is non-negative definite, whereD = diag(b1, b2, . . . , bs) ∈ R

s×s

(cf. Hairer & Wanner, 1991), we can derive the following corollary based on Theorem 4.1.

COROLLARY 4.2 Suppose that the underlying RK method (2.11) is algebraically stableand suppose the quadrature formula (2.13) satisfies condition (2.16). Then, the inducedRK method (2.12) with CQ formula (2.13) is globally stable for the classRI(α, β, σ, γ )

with stability constant (4.1) withd j = b j whenever

2β + σ(1 + γ 2ν2) � −2α. (4.7)

Next, we study theasymptotic stability. For this we have the following theorem.

THEOREM 4.3 Suppose the underlying RK method (2.11) is(k, l)-algebraically stable fora non-negative diagonal matrixD = diag(d1, d2, . . . , ds) ∈ R

s×s , where 0< k < 1, andsuppose the quadrature formula (2.13) satisfies condition (2.16). Then, the induced RKmethod (2.12) with CQ formula (2.13) is asymptotically stable for the classRI(α, β, σ, γ )

whenever

h[2(α + β) + σ(1 + γ 2ν2)] < 2l. (4.8)

Proof. Wedefine the quantityθ as

θ = max

k,

[h(β + σγ 2ν2)

2l − h(2α + β + σ)

] 1m

. (4.9)

With 0 < k < 1 and (4.8), it can be deduced that 0< θ < 1. Because 0< k � θ , itfollows from (3.3) in Lemma 3.4 that

‖Yn+1‖2 � θn+1‖Y0‖2 + {[h(2α + β + σ) − 2l]θm + hβ}n∑

i=0

θn−m−is∑

j=1

d j‖Y(i)j ‖2

+βτθn−m+1s∑

j=1


{‖Y(i)j ‖2} + hσ

n∑i=0

θn−is∑

j=1

d j‖Z(i)j ‖2

� θn+1‖Y0‖2 + {[h(2α + β + σ) − 2l]θm + hβ}n∑

i=0

θn−m−is∑

j=1

d j‖Y(i)j ‖2

+βτθn−m+1s∑

j=1

d j maxt0−τ�t�t0

‖ϕ(t) − ψ(t)‖2 + hσ

n∑i=0

θn−is∑

j=1

d j‖Z(i)j ‖2.

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By (4.3) and (3.10), we have

hn∑

i=0

θn−is∑

j=1

d j‖Z(i)j ‖2 � hγ 2ν2θn

m + 1

s∑j=1

d j

n∑i=0

θ−im∑

q=0

‖Y(i−q)j ‖2

� hγ 2ν2θn

m + 1

s∑j=1

d j

[n∑

i=0

m∑q=0

θ−(i+q)‖Y(i)j ‖2 +

(m∑

q=1

q∑i=1

θ−(q−i)

)max

−m�q�−1{‖Y(q)

j ‖2}]

� hγ 2ν2n∑

i=0

θn−m−is∑

j=1

d j‖Y(i)j ‖2 + m2hγ 2ν2θn−m+1

m + 1

s∑j=1

d j max−m�q�−1

{‖Y(q)j ‖2}

� hγ 2ν2n∑

i=0

θn−m−is∑

j=1

d j‖Y(i)j ‖2 + τγ 2ν2θn−m+1

s∑j=1


‖ϕ(t) − ψ(t)‖2.

(4.10)

Substituting (4.10) into the bound for‖Yn+1‖2 leads to a further upper bound for‖Yn+1‖2:

θn+1‖Y0‖2 + {[h(2α + β + σ) − 2l]θm + h(β + σγ 2ν2)}n∑

i=0

θn−m−is∑

j=1

d j‖Y(i)j ‖2

+τθn−m+1(β + σγ 2ν2)

s∑j=1


‖ϕ(t) − ψ(t)‖2. (4.11)

Since by

h(2α + β + σ) � h[2(α + β) + σ(1 + γ 2ν2)] < 2l

and (4.9) we have

[h(2α + β + σ) − 2l]θm + h(β + σγ 2ν2) < 0.

Bound (4.11) therefore implies

‖Yn+1‖2 � θn+1‖Y0‖2 + τθn−m+1(β + σγ 2ν2)

s∑j=1


‖ϕ(t) − ψ(t)‖2. (4.12)

This, together with 0< θ < 1, leads to limn→∞ ‖Yn‖ = 0. �

5. Stability of RK methods with PQ formula

This section will focus on the global and the asymptotic stability of RK methods (2.12)with PQ formula (2.17). First, we define a quantity related to theD-matrix that appears inDefinition 3.1; this quantity is frequently used later on:

L := min1�r�s

{dr }.

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THEOREM 5.1 Suppose the underlying RK method (2.11) is(k, l)-algebraically stablefor a positive diagonal matrixD = diag(d1, d2, . . . , ds) ∈ R

s×s , where 0 < k � 1.Then, the induced RK method (2.12) with PQ formula (2.17) is globally stable for theclassRI(α, β, σ, γ ) with stability constant

H =√√√√1 + τ

[β + 3σγ2τ2

L

(s∑

j=1

d j

s∑r=1

(|a jr |2 + |br |2))]

s∑r=1

dr (5.1)

whenever

h

{2(α + β) + σ

[1 + 3γ2τ2

Ls∑

j=1

d j

s∑r=1

(2|a jr |2 + |br |2)]}

� 2l. (5.2)

Proof. With condition (2.4) and PQ formula (2.17), we have

‖Z(i)j ‖2 �

[hγ

(s∑

r=1

|a jr |‖Y(i)r ‖ +

m∑q=1

s∑r=1

|br |‖Y(i−q)r ‖ +

s∑r=1

|a jr |‖Y(i−m)r ‖

)]2

�3h2γ 2

(

s∑r=1

|a jr |‖Y(i)r ‖

)2

+(

m∑q=1

s∑r=1

|br |‖Y(i−q)r ‖

)2

+(

s∑r=1

|a jr |‖Y(i−m)r ‖

)2

�3h2γ 2

[(s∑

r=1

|a jr |2) (

s∑r=1

‖Y(i)r ‖2

)+ m

m∑q=1

(s∑

r=1

|br |2) (

s∑r=1

‖Y(i−q)r ‖2

)

+(

s∑r=1

|a jr |2) (

s∑r=1

‖Y(i−m)r ‖2

)], (5.3)

where we have repeatedly used the Cauchy inequality and the induced inequality(n∑

i=1

Ai

)2

� nn∑

i=1

A2i , ∀n ∈ N, ∀Ai ∈ R. (5.4)

Inserting (5.3) into (3.4) generates

‖Yn+1‖2 �‖Y0‖2 + [h(2(α + β) + σ) − 2l]n∑

i=0

s∑j=1

d j‖Y(i)j ‖2

+ βτ

s∑j=1


{‖Y(i)j ‖2}

+ 3h3σγ 2

[(s∑

j=1

d j

s∑r=1

|a jr |2) (

n∑i=0

s∑r=1

‖Y(i)r ‖2

)

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+(

s∑j=1

d j

s∑r=1

|a jr |2) (

n∑i=0

s∑r=1

‖Y(i−m)r ‖2

)

+m

(s∑

j=1

d j

) (s∑

r=1

|br |2) (

s∑r=1

n∑i=0

m∑q=1

‖Y(i−q)r ‖2

)]. (5.5)

It follows from inequality (3.13) that

n∑i=0

m∑q=1

‖Y(i−q)r ‖2 � m

n∑i=0

‖Y(i)r ‖2 + m(m + 1)

2max

−m�i�−1{‖Y(i)

r ‖2}. (5.6)

Moreover, it holds that

n∑i=0

s∑r=1

‖Y(i−m)r ‖2 =

n−m∑i=−m

s∑r=1

‖Y(i)r ‖2 �

n∑i=0

s∑r=1

‖Y(i)r ‖2 + m

s∑r=1

max−m�i�−1

{‖Y(i)r ‖2}.

(5.7)

Substituting (5.6), (5.7) andh = τm � τ into (5.5) yields a new upper bound for‖Yn+1‖2:

‖Y0‖2 + [h(2(α + β) + σ) − 2l]n∑

i=0

s∑j=1

d j‖Y(i)j ‖2 + βτ

s∑j=1


{‖Y(i)j ‖2}

+ 3h3σγ 2

[2

(s∑

j=1

d j

s∑r=1

|a jr |2) (

n∑i=0

s∑r=1

‖Y(i)r ‖2

)

+ m2

(s∑

j=1

d j

) (s∑

r=1

|br |2) (

n∑i=0

s∑r=1

‖Y(i)r ‖2

)

+ m2(m + 1)

2

(s∑

j=1

d j

) (s∑

r=1

|br |2) (

s∑r=1

max−m�i�−1

{‖Y(i)r ‖2}

)

+m

(s∑

j=1

d j

s∑r=1

|a jr |2) (

s∑r=1

max−m�i�−1

{‖Y(i)r ‖2}

)]

� ‖Y0‖2 + [h(2(α + β) + σ) − 2l]n∑

i=0

s∑j=1


s∑j=1


{‖Y(i)j ‖2}

+ 3h3σγ 2

L

[2

(s∑

j=1

d j

s∑r=1

|a jr |2) (

n∑i=0

s∑r=1

dr‖Y(i)r ‖2

)

+ m2

(s∑

j=1

d j

) (s∑

r=1

|br |2) (

n∑i=0

s∑r=1

dr‖Y(i)r ‖2

)

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+ m3

(s∑

j=1

d j

) (s∑

r=1

|br |2) (

s∑r=1

dr max−m�i�−1

{‖Y(i)r ‖2}

)

+m

(s∑

j=1

d j

s∑r=1

|a jr |2) (

s∑r=1


{‖Y(i)r ‖2}

)]

� ‖Y0‖2 + [h(2(α + β) + σ) − 2l]n∑

i=0

s∑j=1


s∑j=1


{‖Y(i)j ‖2}

+ 3hσγ 2τ2

L

[(s∑

j=1

d j

s∑r=1

(2|a jr |2 + |br |2)] (

n∑i=0

s∑r=1

dr‖Y(i)r ‖2

)

+ 3σγ2τ3

L

[s∑

j=1

d j

s∑r=1

(|a jr |2 + |br |2)] (

s∑r=1


{‖Y(i)r ‖2}

)

= ‖Y0‖2 +{

h

[2(α + β) + σ

(1 + 3γ2τ2

Ls∑

j=1

d j

s∑r=1

(2|a jr |2 + |br |2))]

− 2l

}

×(

n∑i=0

s∑j=1

d j‖Y(i)j ‖2

)

+ τ

{β + 3σγ2τ2

L

[s∑

j=1

d j

s∑r=1

(|a jr |2 + |br |2)]} (

s∑r=1


{‖Y(i)r ‖2}

).

(5.8)

Applying (5.2) to (5.8) shows

‖Yn+1‖2 �‖Y0‖2 + τ

{β + 3σγ2τ2

L

[s∑

j=1

d j

s∑r=1

(|a jr |2 + |br |2)]}

×(

s∑r=1


{‖Y(i)r ‖2}

)

�{

1 + τ

[β + 3σγ2τ2

L

(s∑

j=1

d j

s∑r=1

(|a jr |2 + |br |2))]

s∑r=1

dr

}

× maxt0−τ�t�t0

‖ϕ(t) − ψ(t)‖2.

This implies inequality (5.1) and hence the method is globally stable. �

COROLLARY 5.2 Suppose the underlying RK method (2.11) withb j > 0 is algebraicallystable. Then, the induced RK method (2.12) with PQ formula (2.17) is globally stable forthe classRI(α, β, σ, γ ) with stability constant (5.1) withd j = b j , whenever

2β + σ

[1 + 3γ2τ2

L

s∑j=1

b j

s∑r=1

(2|a jr |2 + |br |2)]

� −2α, with L = min1�r�s

{br }. (5.9)

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Next, we study theasymptotic stability of the RK methods with PQ formula. For this,we have the following theorem.

THEOREM 5.3 Suppose the underlying RK method (2.11) is(k, l)-algebraically stable fora positive diagonal matrixD = diag(d1, d2, . . . , ds) ∈ R

s×s , where 0< k < 1. Then, theinduced RK method (2.12) with PQ formula (2.17) is asymptotically stable for the classRI(α, β, σ, γ ) whenever

h

{2(α + β) + σ

[1 + 3γ2τ2

Ls∑

j=1

d j

s∑r=1

(2|a jr |2 + |br |2)]}

< 2l. (5.10)

Proof. Wedefine the quantityη as

η = max

k,

h[β + 3σγ2τ2

Ls∑

j=1d j

s∑r=1

(2|a jr |2 + |br |2)]

2l − h(2α + β + σ)

1m

. (5.11)

It follows from both 0< k < 1 and (5.10) that 0< η < 1. Since 0� k � η, we can bound‖Yn+1‖2 by using (3.3) in Lemma 3.4,

‖Yn+1‖2 � ηn+1‖Y0‖2 + {[h(2α + β + σ) − 2l]ηm + hβ}n∑

i=0

ηn−m−is∑

j=1

d j‖Y(i)j ‖2

+βτηn−m+1s∑

j=1


{‖Y(i)j ‖2} + hσ

n∑i=0

ηn−is∑

j=1

d j‖Z(i)j ‖2

� ηn+1‖Y0‖2 + {[h(2α + β + σ) − 2l]ηm + hβ}n∑

i=0

ηn−m−is∑

j=1

d j‖Y(i)j ‖2

+βτηn−m+1s∑

j=1


‖ϕ(t)−ψ(t)‖2 + hσ

n∑i=0

ηn−is∑

j=1

d j‖Z(i)j ‖2.

(5.12)

Also, by (5.3), (3.12) and by using the conditionh = τm � τ we can bound one of the

terms in the above inequality,

hs∑

i=0

ηn−is∑

j=1

d j‖Z(i)j ‖2 �3h3γ 2

s∑i=0

ηn−is∑

j=1

d j

[(s∑

r=1

|a jr |2) (

s∑r=1

‖Y(i)r ‖2

)

+ mm∑

q=1

(s∑

r=1

|br |2) (

s∑r=1

‖Y(i−q)r ‖2

)

+(

s∑r=1

|a jr |2) (

s∑r=1

‖Y(i−m)r ‖2

)]

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�3h3γ 2

{(s∑

j=1

d j

s∑r=1

|a jr |2) (

n∑i=0

ηn−is∑

r=1

‖Y(i)r ‖2

)

+ m

(s∑

r=1

|br |2) (

s∑j=1

d j

)s∑

r=1

[m∑

q=1

n∑i=0

ηn−(i+q)‖Y(i)r ‖2

+(

m∑q=1

q∑i=1

ηn−(q−i)

) (max

−m�i�−1{‖Y(i)

r ‖2})]

+(

s∑j=1

d j

s∑r=1

|a jr |2) (

n−m∑i=−m

ηn−m−is∑

r=1

‖Y(i)r ‖2

)}

�3h3γ 2

{(s∑

j=1

d j

s∑r=1

|a jr |2) (

n∑i=0

ηn−is∑

r=1

‖Y(i)r ‖2

)

+ m

(s∑

r=1

|br |2) (

s∑j=1

d j

)

×(

mn∑

i=0

ηn−m−is∑

r=1

‖Y(i)r ‖2 + m2ηn−m+1

s∑r=1

max−m�i�−1

{‖Y(i)r ‖2}

)

+(

s∑j=1

d j

s∑r=1

|a jr |2) (

n∑i=0

ηn−m−is∑

r=1

‖Y(i)r ‖2 +

−1∑i=−m

ηn−m−is∑

r=1

‖Y(i)r ‖2

)}

�3hγ 2τ2

L

[s∑

j=1

d j

s∑r=1

(2|a jr |2 + |br |2)] (

n∑i=0

ηn−m−is∑

r=1

dr‖Y(i)r ‖2

)

+ 3γ2τ3ηn−m+1

L

[s∑

j=1

d j

s∑r=1

(|a jr |2 + |br |2)] (

s∑r=1

dr max−1�i�−m

{‖Y(i)r ‖2}

)

�3hγ 2τ2

L

[s∑

j=1

d j

s∑r=1

(2|a jr |2 + |br |2)] (

n∑i=0

ηn−m−is∑

r=1

dr‖Y(i)r ‖2

)

+ 3γ2τ3ηn−m+1

L

[s∑

j=1

d j

s∑r=1

(|a jr |2 + |br |2)] (

s∑r=1

dr

)max

t0−τ�t�t0‖ϕ(t) − ψ(t)‖2.

(5.13)

Inserting (5.13) into (5.12) generates

‖Yn+1‖2 � ηn+1‖Y0‖2 +{

[h(2α + β + σ) − 2l]ηm

+ h

[β + 3σγ2τ2

L

(s∑

j=1

d j

s∑r=1

(2|a jr |2 + |br |2))]}

n∑i=0

ηn−m−is∑

j=1

d j‖Y(i)j ‖2

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+ τηn−m+1

[β + 3σγ2τ2

Ls∑

j=1

d j

s∑r=1

(|a jr |2 + |br |2)]

×(

s∑r=1

dr

)max

t0−τ�t�t0‖ϕ(t) − ψ(t)‖2. (5.14)

Moreover, by (5.10) it holds that

h(2α + β + σ) � h

{2(α + β) + σ

[1 + 3γ2τ2

Ls∑

j=1

d j

s∑r=1

(2|a jr |2 + |br |2)]}

< 2l.

(5.15)

Combining this with (5.11) leads to

[h(2α + β + σ) − 2l]ηm + h

[β + 3σγ2τ2

L

(s∑

j=1

d j

s∑r=1

(2|a jr |2 + |br |2))]

< 0.

(5.16)

Hence, it follows from (5.14) that‖Yn+1‖2 is bounded by

ηn+1‖Y0‖2 + τηn−m+1

[β + 3σγ2τ2

Ls∑

j=1

d j

s∑r=1

(|a jr |2 + |br |2)]

×s∑

r=1

dr maxt0−τ�t�t0

‖ϕ(t) − ψ(t)‖2.

This, together with 0< η < 1, shows limn→∞ ‖Yn‖ = 0: the method is asymptotically stable.

�

6. Application to some classical underlying RK schemes

The global and asymptotic stability results derived above are applicable to the VDIDEmethods induced by various common RK methods. Based on Corollaries 4.2 and 5.2, wecan summarize our findings in the two theorems given below. For their proof we need onlynote the fact that the underlying RK methods of type Gauss, Radau IA, Radau IIA andLobatto IIIC are all algebraically stable and satisfyb j > 0 ( j = 1,2, . . . , s) (cf. Hairer &Wanner, 1991).

THEOREM 6.1 Suppose the underlying RK method (2.11) is of type Gauss, Radau IA,Radau IIA or Lobatto IIIC and suppose the quadrature formula (2.13) satisfies condition(2.16). Then, the induced RK method (2.12) with CQ formula (2.13) is globally stable forthe classRI(α, β, σ, γ ) with stability constant (4.1) withd j = b j whenever condition (4.7)holds.

THEOREM 6.2 Suppose the underlying RK method (2.11) is of type Gauss, Radau IA,Radau IIA or Lobatto IIIC. Then, the induced RK method (2.12) with PQ formula (2.17)is globally stable for the classRI(α, β, σ, γ ) with stability constant (5.1) withd j = b j

whenever condition (5.9) holds.

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As for the asymptotic stability of the methods, we can rely on Theorems 4.3 and 5.3to provide us with effective criteria for judging the asymptotic behaviour of the methods.We cannot, however, derive results as general as those in Theorems 6.1 and 6.2. Indeed,algebraic stability means(1,0)-stability, i.e. withk = 1, for which Theorems 4.3 and 5.3do not apply. Here, we present some examples of conditions for asymptotic stability.

EXAMPLE 6.1 First, we consider the two-stage Lobatto IIIC formula:

0 1/2 −1/21 1/2 1/2

1/2 1/2

. (6.1)

Burrage & Butcher (1980) showed that this formula is(1/(1− l)2, l)-algebraically stablefor the non-negative diagonal matrixD = diag(1/(2− 2l), 1/(2 − 2l)), wherel < 1.Wheneverl < 0, we have 0< k < 1, and, hence, condition (4.8) holds as a condition forasymptotic stability. We immediately have the following result as a corollary of Theorem4.3.

COROLLARY 6.3 Method (2.12) generated by the CQ formula (2.13) and the LobattoIIIC formula (6.1) is asymptotically stable for the classRI(α, β, σ, γ ) when (2.16) and thefollowing condition hold:

2(α + β) + σ(1 + γ 2ν2) < 0.

A similar argument together with Theorem 5.3 leads to a result for PQ-based methods.

COROLLARY 6.4 The method generated by the PQ formula (2.17) and the Lobatto IIICformula (6.1) is asymptotically stable for the classRI(α, β, σ, γ ) when the followingcondition holds:

2(α + β) + σ(1 + 9γ2τ2) < 0.

EXAMPLE 6.2 Next, we consider the two-stage Radau IIA formula

1/3 5/12 −1/121 3/4 1/4

3/4 1/4

. (6.2)

Burrage & Butcher (1980) have deduced that this formula is(

16(5−2l)2 , l

)-algebraically

stable for the non-negative diagonal matrixD = diag(

9(3−l)(5−2l) ,

25−2l

)when

l � 9−3√

178 . Furthermore, the method is

((3+4l)2

(3−2l)(3+4l−2l2), l

)-algebraically stable for

the non-negative diagonal matrix given byD = diag(

(3+4l)2

4(3+4l−2l2), 3+4l

4(3+4l−2l2)

)when

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9−3√

178 < l < 2

3. Also, we have that

0 < k := 16

(5 − 2l)2< 1 if l <

1

2or l >

9

2(6.3)

0 < k := (3 + 4l)2

(3 − 2l)(3 + 4l − 2l2)< 1 if

15− 3√

33

4< l < 0 or l >

15+ 3√

33

4·

(6.4)

Combining all of the above, we know that the Radau IIA formula is(

16(5−2l)2 , l

)-

algebraically stable and satisfies 0< k := 16(5−2l)2 < 1 wheneverl � 9−3

√17

8 , and((3+4l)2

(3−2l)(3+4l−2l2), l

)-algebraically stable and satisfies 0< k := (3+4l)2

(3−2l)(3+4l−2l2)< 1

whenever9−3√

178 < l <0. Therefore, our theorems imply that the method with CQ formula

(2.13) is asymptotically stable for the classRI(α, β, σ, γ ) whenever there exists anl suchthat (2.16) and one of the following two conditions hold:

(I) h[2(α + β) + σ(1 + γ 2ν2)] < 2l � 9−3√

174 ,

(II) max{

9−3√

174 , h[2(α + β) + σ(1 + γ 2ν2)]

}< 2l < 0.

One can easily see that such anl can always be found, under the condition given in thefollowing corollary.

COROLLARY 6.5 Method (2.12) generated by the CQ formula (2.13) and the Radau IIAformula (6.2) is asymptotically stable for the classRI(α, β, σ, γ ) when (2.16) and thefollowing condition hold:

2(α + β) + σ(1 + γ 2ν2) < 0.

The method with PQ formula (2.17) is asymptotically stable for the problem classRI(α, β, σ, γ ) whenever there exists anl such that one of the following two conditionsholds:

(III) h[2(α + β) + σ

(1 + 3γ2τ2(161−30l)

8L(5−2l)(3−l)

)]< 2l � 9−3

√17

4 with

L ={

25−2l , −3

2 � l � 9−3√

178 ,

9(3−l)(5−2l) , l � −3

2;

(IV) max{

9−3√

174 , h

[2(α + β) + σ

(1 +

(29+22l

2

)γ 2τ2

)]}< 2l < 0.

The corollary below states a condition which, when satisfied, guarantees the existence ofsuch anl-value.

COROLLARY 6.6 The method generated by the PQ formula (2.17) and the Radau IIAformula (6.1) is asymptotically stable for the classRI(α, β, σ, γ ) when the followingcondition holds:

2(α + β) + σ(1 + 292 γ 2τ2) < 0.

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Indeed, when the left-hand side of the condition evaluates to a valueε, with ε < 0, one cantakel = max{(9− 3

√17)/8,hε/4} to satisfy condition (IV).

In a similar way, we can show that methods induced by Gauss and Radau IA typeformulae, with CQ or PQ methods, possess asymptotic stability under certain suitableconditions.

REMARK 6.7 Obviously, all of the previous results hold true withσ set to 1, for equationsof the form (2.8) with (2.9), (2.10) and (2.4).

REMARK 6.8 Some of the underlying Runge–Kutta methods considered in this paper canbe viewed as continuous collocations schemes, e.g. the Gauss and Radau IIA methods.Their extension to VDIDEs naturally leads one to consider so-calledexact collocationschemes (Brunner, 1994), i.e. schemes of the form (2.12) where the integral termsz(t (n)

j )

are evaluatedexactly. For θ ∈ [0,1], such schemes can be formulated as

u(tn + θh) = u(tn) + hs∑

j=1

a j (θ) f (t (n)j , u(t (n)

j ), u(t (n−m)j ),

∫ t (n)j

t (n)j −τ

g(t (n)j , v, u(v)) dv),

wherea j (θ) = ∫ θ

0 l j (v) dv andl j (v) = ∏sq=1, q �= j

v−cqc j −cq

. The results of the present paperare not directly applicable to this scheme. Instead, we have only considered the case wherethe additional discretization step for the approximation of the integrals is explicitly takeninto account.

7. Concluding remarks

In the present paper, we have investigated the numerical stability of two groups of adaptedRK methods for a general class of nonlinear VDIDEs. Here we would like to compare ourfindings with some of the results from the existing DDE literature. Specializing our resultsto the DDE equationy′(t) = f (t, y(t), y(t − τ)), we find our results to be consistent withthose in Huanget al. (1999), slightly weaker than those in Torelli (1989) and different fromthose in Reverdy (1990) and Bellen & Zennaro (1992).

In paper Torelli (1989), the author has studied the implicit Euler method for DDEs andproved that this method satisfies the following stability property:

‖yn − yn‖ � maxt0−τ�t�t0

‖ϕ(t) − ψ(t)‖, ∀n � 0. (7.1)

This excellent property could only be derived thanks to the simple structure of theEuler scheme. It cannot be generalized to more complex time integration methods. Inpaper Reverdy (1990), DDEs are considered that satisfy a weak one-sided Lipschitzcondition with a characteristic parameterρ > 0. For underlying RK methods that arealgebraically stable, a generalized stability result is derived, calledBρ stability. For theDDEs considered, the induced RK methods satisfy a stability inequality of the form

�yn+1 − yn+1� � �yn − yn�, (7.2)

where the following notation is used:

�yn�2 = 1

2‖yn‖2 + hρ

∑n−m� j�n−1

s∑i=1

bi‖y( j)i ‖.

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This clearly only shows the perturbation behaviour of a combination of the numericalsolutionyn and the stage value approximationsy( j)

i . In practice one usually wishes to checkthe perturbation behaviour of numerical solutions, i.e.‖yn − yn‖. Hence, the above stabilityresult needs to be sharpened. In paper Bellen & Zennaro (1992), based on the concept of(semi)B N f -stability of the underlying continuous RK methods, the authors derived (RN -)G RN -stability results for RK methods for DDEs, with a stability inequality characterizedby (7.1). More extended results can also be found in the book by Bellen & Zennaro (2003).The assumption of (semi)B N f -stability is very strong, however, as it leads to an orderbarrier for the underlying RK methods. So far, only twoB N f -stable RK methods havebeen identified: implicit Euler and Lobatto IIIC with linear interpolation. Also, Zennaro(1997) has shown that four is a sure order barrier for anyB N f -stable RK method. Ourstudy on the contrary is based on algebraic stability, which, in general, does not bring onan order barrier. Hence, our research results extend and improve on the earlier results.

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BAKER, C. T. H. & FORD, N. J. (1988) Convergence of linear multistep methods for a class ofdelay-integro-differential equations.International Series of Numerical Mathematics, Vol. 86.Basel: Birkhauser, pp. 47–59.

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