On the relation between the repetition factor and numerical stability of direct quadrature methods for second kind Volterra integral equations

On the relation between the repetition factor and numerical

stability of direct quadrature methods for second kind Volterra

integral equations

Citation for published version (APA):

Wolkenfelt, P. H. M. (1983). On the relation between the repetition factor and numerical stability of direct quadrature methods for second kind Volterra integral equations. SIAM Journal on Numerical Analysis, 20(5), 1049-1061. https://doi.org/10.1137/0720074

DOI:

10.1137/0720074

Document status and date: Published: 01/01/1983 Document Version:

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ON

THE

RELATION BETWEEN THE REPETITION FACTOR

AND NUMERICAL

STABILITY OF DIRECT QUADRATURE

METHODS

FOR SECOND

KIND

VOLTERRA INTEGRAL

EQUATIONS*

P. H. M. WOLKENFELTt

Abstract. Weconsider directquadrature methodsemployingquadraturerules which are reducibleto

linear multistep methods for ordinary differential equations.Asimple characterization of both the repetition

factorand numerical stability(forsmallh)is given, which enables ustoderivesomeresults with respect toaconjecture of Linz. Inparticular we show that(i) methodswith arepetition factorof one arealways

numericallystable; (it) methods witharepetitionfactorgreater than one arenotnecessarily numerically unstable.Analogousresults are derived with respect to the more general notion of anasymptoticrepetition

factor. We also discuss the concepts of strong stability, absolute stability and relative stability and their

(dis)connectionwith the(asymptotic)repetition factor.Somenumericalresultsarepresentedas a verification.

Keywords, numericalanalysis, Volterra integral equations of thesecondkind, reduciblequadrature methods,numericalstability,repetition factor

1. Introduction. Considerthe secondkindVolterra integral equation

(1.1)

f(x)

g(x)+ K(x, y,f(y))dy, x 20,

where

_f

is the unknown functionand where the forcing function g and the kernelK

aregivensmooth functions.

In

order to define a discretization of (1.1), let x, nh (where h denotes the

stepsize) and let

{w,i}

be the weights associated with thequadrature formula

(1.2) b(y) dy -h

wNb(x),

n >-k.

/’--0

Thena directquadrature methodfor(1.1)is given by

(1.3)

/=0

Here, f, denotes a numerical approximation tof(x,) and k depends on the desired

accuracy. Iftherequired startingvalues

f0,

,

_f-

areknown,the valuesfg,

fk+,

can becomputed in astep-by-stepfashion.Fora detailed discussion ofsuch methods

werefertoBaker

[1].

It is well known (see e.g.

[8],

[11], [1],

[2])

that the structure of the quadrature weightsw,i isimportant for the stability analysisofthemethods (1.3). Inthis connec-tion,thefollowingnotion isrelevant.

DEFINITION 1.1. The weights w,i in (1.3) are said to have an (exact) repetition

factor

r if r is the smallest positive integer such that w,+.i w,i for all nn0 and

n N]Nn-n2, wheren0, n and n2areindependentofn.

A

method (1.3) is said to have arepetition factorr if the associatedweights w,i

have arepetitionfactor r.

*Receivedbythe editors April3, 1981,and in revised form November30, 1982.

tDepartmentof Numerical Mathematics, MathematischCentrum,Kruislaan413, 1098 SJ Amsterdam,

theNetherlands.Presentaddress:DepartmentofMathematicsand Computing Science,Technische

Hoge-school Eindhoven,5600MBEindhoven, the Netherlands.

This paper has been largely motivated by the following conjecture of Linz

[8,

p.

27]

(see also Noble

[11]):

"We may conjecture that (i) methods with a repetition factor ofone tendtobe numericallystable, (ii)those with a repetition factorgreater

than onenumerically unstable."

In order todeal with this conjecture in a proper way, it is necessary to have a

good understanding of the concept of numerical stabilityasdefined by Linz and Noble.

It

turnsoutthat numerical stabilityin thesense of Linz and Nobleessentially requires

the perturbation sensitivity of the discretization to be "roughly equivalent" to the

perturbation sensitivity of the original continuous problem (compare the concept of

"strong stability" as discussed by Stetter

[13,

p.

54]).

Their analysis is basedon the

asymptotic expansion of the global discretization error (see also Kobayasi

[6]). An

advantage ofthis approach is its applicabilityto generalequations (1.1), i.e. without

anyrestrictions on the kernel and the forcingfunction (exceptfor sufficient

smooth-ness).

A

disadvantage, however, isthat the stepsize h should be sufficiently

small

so

that the conclusions neednothold forlargevalues of h.

As

aconsequence, this kind of stability analysis establishes results with regard to the suitability

_of

a method

_for

general use (but with small h). On the other hand it is not

assumed

that h actually tends to zero. This means that the condition for numerical stability in the sense of Linz and Noble is stronger than the condition forzerostability whichis necessary for

convergence.

To

gain insight into the relationship between the repetition factor of the

quad-rature weightsand the stabilitybehavior of theassociateddirectquadrature method,

we consider the class of methods which are reducible to linear multistep methods

{p,

tr}

forordinary differential equations. For thisclass wederive some properties of

the quadrature weights, and, motivated by these results, we introduce the notion of

the asymptotic repetition

_factor

as anextension of Definition 1.1. Weshall characterize’

(i) the exact and asymptotic repetition factor in terms of the location of the

essentialzerosof thepolynomialp;

(ii) numerical stability in terms of the growth parameters associated with these

zeros.

It

turns outthatwiththesecharacterizations,

results

withregardtothe conjecture

of Linz can be derived in a rather elegant, and almost straightforward, manner. To

bespecific,weshalldemonstrate that:

(i) methods with an asymptotic repetition factorofone are always numerically stablein the senseofLinzand Noble;

(ii) methodswith an exact or asymptotic repetition factor greater than one can

still be numericallystable;

(iii) an asymptotic repetition factor ofone is necessary and sufficient for strong stability (aconcept whichweshall definein 5).

Futhermore, we shall indicate that the stability concept of Linz and Noble is almostidenticaltothe concept ofrelativestability for small h.

Finally, wepresentsome numericalexperimentswhich serveas an illustration of

the theoreticalresults.

2. Preliminaries.

In

this paper we restrict our considerations to a special class ofquadrature methods. We assume thatfor n >-k, 0

<=/’-<

n, the weights wnj in (1.3)

can begenerated bytherecurrencerelation

k _{0 forj}

_0(1)n

_{k 1,}

(2.1)

Y.

aiwn-i.j

whereaiand

bi

(i 0(1)k)are thecoefficients ofalinearmultistep method for ordinary

differentialequations

[7,

p.

11].

Fortheconstruction of theweights by meansof(2.1)

we set wnj 0

for/’

>max(n, k 1) and define a set of starting weights

{w.ln,

f

0(1)k

1}

(see

[15]

for details). Thequadraturerulesgenerated inthisway are called

[9-1

(p,r)-reducible. The direct quadrature method (1.3) employing such quadrature

rlales is also called (p,o-)-reducible.

Here,

p and r denote the first and second

characteristicpolynomialassociated withthe linear multistepmethod; that is,

()

:=

_Y

_a

-,

_r(’)

:=

_Y

_b

"-.

=0 =0

From

(2.1) the following property canbederived:

(2.2) wnj o,_

wherethe sequence

{o,}n_-0

satisfies

(2.3a)

(2.3b)

forn-/=>0, j>=k,

aoo,o bo,

aoool

+

alO)0 bl,

aOOgk

+

alO)k-1

-- -

akO0

bk

aow,

+

alO)n-1"b" -1-akw,,-k O, n -->k

+

1.

Wealso needthe followingdefinitions.

DEFINITION 2.1 (from

[13,

p.

206]). A

polynomial is said to satisfy the root

condition if it has no zeros outside the closed unit disk and only simple zeros onthe unitcircle, itis saidto satisfy thestrong rootcondition if it satisfies therootcondition

and 1 isitsonly zero on the unit circle.

DEFINITION 2.2.

A

nonvanishing zero

"

of a polynomial

P

is called essential if

[’]

1 and nonessential if

I

r]

<

1.

A

possible vanishing zero of

P

is called the trivial

zeroofP.

Furthermore,weshallassumethroughout thispaperthat0 andcrhaveno common

factors and thatthe method {p,

r}

is convergent (thatis,

0(1)=0,

p’(1)=r(1) and 0

satisfies therootcondition).

We

remark that the property(2.2)ischaracteristic for(p,r)-reduciblequadrature

rules.

To

be precise, if (2.2) does not hold, thenthe quadrature rules are not (p

r)-reducible. Suchasituation occurs forexampleinthestable Simpson method (Simpson

#2 in

[11]).

For this method w2n.2, and w2n-.2,- and therefore it is not

(p,r)-reducible.

It

turns out that the stableSimpson methodis reducible to a2-cyclic

linearmultistep methodfor

ODEs.

Theaboveremarksgiveinsightinto thestructure of(p,

r)-reducible

quadrature

rules.

Moreover,

they also suggest generalizations of (2,1).

A

possible extension of

(2.1)

is to define quadrature rules which are reducible to cyclic linear multistep

methods. Occasionally, we shall state some results with respect to such reducible

quadrature methods, but for reasons of clarity our results are mainly related to

(p,cr

)-

reducible quadraturemethods.

3. Properties of thequadratureweights

o.

We shall derive some properties of

thesequence

{o,}

definedin (2.3). Thissequence satisfies ahomogeneous difference equation with characteristic polynomialp. Since, by assumption, p satisfies the root

condition, the essential zeros of p are simple.

In

order to simplify the presentation of the results and their proofs, we assume subsequently that the nonessential zeros

of p are alsosimple.Withoutthis assumptionourresults remain true, however, unless the assumption is given explicitly in the statement ofthe theorems.

First we give the explicit form of the solution of a difference equation with constant coefficients.

LEMMA3.1. Letthe sequence

{y,’}=o

satisfy the

_difference

equation

k

(3.1)

Y.

aiy’,-i-0, n_->k _(ao0),

i--0

with starting values y0,’" ",_Yk-l.

Assume

that the characteristic polynomial p()--k

aik-ihas nonvanishing simplezerosanda zero 0

_of

multiplicity

mo

(too 0

)

isallowed). Furthermore,let the

_{coecients a}

be

_defined

by

k-1

()k-l-j

(3.2)

_E

_a

:=p()/(-), i=1,2,...,t.

i=0

Then the solution {y,}is given by

(3.3) y,

(?a/o’(),

n too, i=1 where k-1

(i)y

(3.4)

_Ai

/=0

Proof.

Proceed alongthe lines indicatedbyHenrici

[3,

p.

238].

[3

In

view of (3.2) the coefficients a can be expressed in

_’i

and the coefficients

ao, ",ak tobespecific,

(i) ]--u

(3.5)

_a

avi

i=1,2, t.

Wenow return to the recurrence relation (2.3b). Duetothe specialstructureof the starting valuestoa,

,

togdefinedby(2.3a),wecan prove the followingbasic result.

THEOREM 3.1.

Let

the linear multistepmethod

{p, tr}

beconvergent.

Assume

that the nonvanishingzeros

₁

1,2,

_{t Of}

[9aresimple and let

mo

denotethemultiplicity

of

the trivial zero

r

0 (too_->0). Then thesolution

{to,,}

of

(2.3b) with startingvalues (2.3a)is givenby

(3.6) to,,

’),ti",

n _-->

mo

+

1,

i=1

where

(3.7)

"/i O’(i)l(i[9’(i)) O, 1,2,’" ",t.

Proof.

We replace y’, in (3.3) by (o,’+1 and determine

Ai. In

view of (3.4) k-1

mi

E/=0

oi)tok_

j. Substitution of (3.5) gives

m

aotok 4-(aoi4-al)tok_l4-"

4-(ao’

-

+..

+

a_a)to. Collecting powers of

_’i

and using(2.3a) yields

Ai

(b, a,too)

+

(i(b,- atc-ltoO)+"

+

,/-1

(bl a

k k

E

bi

sr

/k

-J tOo

E

ai’/k-i

i=1

o"(’i)

bor

_tooO(r,)

+

aoto

or/

o.(," ),

since

a0w0=b0

and p((i) 0. As a result, o,+1

=i__1

(’cr((i)/p’((i), n>=too, and its

equivalencewith(3.6)isreadilyseen.Since,by assumption,pandcrhavenocommon

factor, r(’i) 0,whichproves thatyi 0. 13

Notethatyl r(1)/p’(1)=1 byvirtueof consistency.

As

aconsequence of(3.6)

wehave

COROLLARY 3.1.

_If

p

_satisfies

thestrongrootcondition, then

(3.8) lim o, 1.

In

particular,

_if

p

(’)

a0

"’-a

("

1),then

(3.9) w,, 1

_for

alln >-_k.

Property (3.9) holds, for example, for the Adams-Moulton methods (which

generate the well-known

Gregory

quadraturerules). Ontheotherhand,thebackward

differentiationmethods generate asequence

{o,}

satisfying(3.8).

From (3.6)

wecanalsoderivethe following periodicity property.

COROLLARY3.2. Letthe weightsobe

_defined

by (2.3). Then

(3.10)

w,+a=o,,

_foralln

>=mo+l

if

andonly

_if

the nonvanishingzeros

_of

psatisfy 1.

Proof.

In

viewof(3.6),w,+-w,

₌₁

/’

(r

a 1)forall n

=> mo+

1.Since y 0 for 1,2,.

.,

t,a,+a-o, 0ifand onlyif

Obviously,theperiodicity ofthesequence

{o,}

islostif p has anonessentialzero.

We

can, however,derivethe following asymptotic result.

COROLLARY3.3. Letthe weightsobe

_defined

by (2.3). Then

(3.11) lim (O,+d --o,) 0

if

andonly

_if

the essentialzeros

_of

psatisfy

(a

1.

Proof.

Let

’,...,

_’s

denote the essential zeros ofp. The weights o, are given by(3.6)and canbe writtenas u,

+

v,where

Therefore lim(o,+a-o,) lim(u,+a- u,). Using the sameargument asin the proof

ofCor. 3.2, this limit is zero ifand onlyif

r(

1, 1,...,s.

The properties derived in this section enable us to characterize the repetition factor intermsof thelocationof the essentialzerosof

4. Characterization of the (asymptotic) repetition factor.

In

view ofDefinition

1.1 and property(2.2), the weights

w,.

of a (p,r)-reducible quadrature method have

an exact repetition factor r if artd only if r is the smallestpositive integer such that

On/r O,, n

=>

no.

This observation togetherwith Corollary 3.2 yields the following

characterization.

THEOREM 4.1. The weights

_of

a (p,r)-reducible quadraturemethodhavean exact

repetition

factor

r

_if

andonly

_if

risthe smallest positiveintegersuch thatthe nonvanishing

zeros

_of

psatisfy

r=

1.

Werecallthatthepolynomialp associated witha linearmultistep method derived

from interpolatory quadrature has the form

r

_,-r

(compare the Adams family

(r 1)orthe Milne-Simpson family

(r

2)). For

such methodswehave the following

resultas animmediateconsequence of Theorem 4.1.

COROLIARY4.1.

_If

p()=

ao(

k

_k-r)

then the weights havean exactrepetition

Weshallnow consider the case wherep has also nonessentialzeros.

In

thiscase

the weights donothave anexactrepetition factorr in view of Theorem 4.1.

We

have seen, however,inCorollary3.3 thattO,/d---to, forn sufficientlylarge, iftheessential zeros ofpsatisfysrd

!.

In

particular, if the weights arecomputedusing finite-precision

arithmetic, we havetheidentitytO,/d to, for large n.These observations suggest the following extension of Definition 1.1.

DEFINITION4.1. The weightswjin(1.3) aresaidtohaveanasymptotic repetition

factor

r if r isthesmallestpositive integersuch that lim_.

(w/r..- w)=

0for all/, n <--/" <-n n2,where n andn2areindependent ofn.

WiththisdefinitionandCorollary 3.3 the following theoremisself-evident. THEOREM 4.2. The weights

_of

a (p, cr)-reducible quadrature method have an

asymptoticrepetition

factor

r

_if

andonly

_if

r is thesmallestpositive integersuchthat the

essential zeros

_of

psatisfy

=

1.

As

an example, the quadrature weights generated by the polynomials p(’)=

(’-

1)((2-(+

1)

and p(sr) (sr-

_1)(s

r2

₊

_1)(r-)

have an exact repetition factor of 6 and an asymptotic repetition factor of4, respectively.

As

an important special case of Theorems 4.2 and 4.1, we have the following

result whichweshallusein 6 in connection with the conjecture of Linz.

COROLLARY 4.2. The weights

of

a (p, tr)-reducible quadrature method have an

asymptotic repetition

factor

of

one

_if

andonly

if

p

_satisfies

the strong rootcondition.

In

particular, the weights have an exact repetition

factor

of

one

_if

and only

_if

p(()=

aosr-(

"-1).

5. Characterization of numerical stability(forsmallh).

In

the following, numeri-calstabilityin the senseof Linz and Noble willbecalled numericalstability (forsmall

h).

We

touched

upon

the concept of numerical stability (for small h) already in

I

in connectionwith the conjecture of Linz. For the sake of completeness we repeat

here the stability definitions of both Linz and Noble.

DEFINrrON

5.1 (Linz

[8,

p.

20]). A

step-by-step method for

(1.1)

isnumerically. stable iftheerrorgrowthisroughlyequivalenttothat ofthe solutionofthe variational equation of

(1.1).

If there existsome equations forwhichthe error grows much faster

than the solution of the variational equation of (1.1), then the method must be

considerednumerically unstable.

DEFINrrON5.2 (Noble

[11,

p.

25];

see also

[1,

p.

823]). A

step-by-stepmethod

for solving a Volterra integral equation is said to be unstable if the error in the

computed solution has dominant spurious components introduced by the numerical

scheme.

We shall now explain how these definitions must be interpreted. For a (p,

or)-reducible quadrature method (of order p), the asymptotic expansion of the global

discretization error

e(x)=f,-f(x,)

assumestheform

([4])

(i)(x)+O(h,/)

(5.1)

e(x,)=h ie,

i=1

()

(x)satisfies where

_’1

1, st2,

..,

sr

arethe essential zeros of p and where

e,

(i)

Here,

K(x,

y)=(o/Of)K(x,

y,f(y)) and the quantities yi are the so-called growth

parameters

([3])

defined as

(5.3)

y, tr(i)/(,p’(,)), 1, 2,...,s.

(i)

(x)

in (5 2) are related to the (local) quadrature errors and to the

Thefunctions gp

errors in the starting values.

(1)

(x)

associated with

’1

1 is called the principal error

corn-The component ep

ponent. Since 3’1 1, thiscomponent satisfies, in viewof (5.2), an equation which is

identical tothe variational equation ofthe continuous problem

(1.1).

The remaining

()(x)

associated with "Y2, ",y are called the

components (if any)

e(2)

(x),.

.,

ep

spuriouserrorcomponents introducedbythe discretizationmethod. These components

satisfyequations(5.2)which

are

differentfrom thevariationalequationof(1.1),unless

")(x)

is dominant and the

yi 1. If

[e,(i)(x)[

>>

[e(o

1)

(x)l

for some

(2

<=

<s), then ep

method is numerically unstable (in the sense ofLinzandNoble).

From

the above explanation we conclude thatthevalues ofthegrowth parameters are crucial for numericalstability ofa (p,tr)-reduciblequadrature method.

In

order

tomake thisevenmore transparentweconsider the integral equation

(5.4) f(x)=g(x)+A exp(l(x-y))f(y)dy,

whose solution isgiven by

(5.5)

f(x)

g(x)

+

exp((A +tx)(x-y))g(y)dy.

Clearly, the problem (5.4)is well-conditionedwithrespecttobounded perturbations ofg if

Re (A +/x)

is nonpositive.

Suppose

that foragiven method, y 1 forsome i;

(i)

(X)

is

then, in view of

(5.2)

and (5.5), the associated spurious error component

_e

givenby

(5.6)

ep

(i)(x)-’g(i)(x)+yiA

p exp((TiA

+)(x-y))gi)(y)dy.

Sinceyi 1one canalways chooseh andtxsuchthatRe (h

+

tz) <

0andRe(yih

+

Ix) >

(i)

(x) which is

0.

As

a consequence, the global error has a spurious component e_o

exponentially increasing in general, whereas the continuous problem (5.4) is

well-conditioned.

From

the foregoingthefollowingcharacterization isreadily deduced.

THEOREM 5.1.

A

reducible quadrature method

_of

the

_form

(1.3)

is numerically stable

(for

small h) (in thesense

_of

Linz andNoble)

if

each essentialzero

_of

phas a

growthparameterequaltoone; themethodisweakly stable

(for

small h) (ornumerically unstable in the terminology

_of

Linz andNoble)

_if

thereexists atleastoneessentialzero

of

pwhosegrowthparameter is

_different

from

one.

Essentially, this theorem is an equivalent, but more quantitative, definition of

the numerical stability concept. We have used the term weak stability rather than

numerical

instability, because a weakly stable method does not always display an unstable behavior.

We recall that in the numerical treatment of ordinary differential equations a linearmultistep methodisweakly stableif p has an essential zero with y

<

0 (cf.

[13,

p.

246]). In

the context of integral equations however, weak stability can also occur

For

the expansion (5.1) we also observe that ingeneral the terms

_’7

will cause

the global error to be nonsmooth at consecutive grid points. This situation cannot

occur if

’1

1 is the only essential zero ofp.

In

order toemphasize and distinguish

thisimportant featurewegive the followingdefinition.

DEFINrrION 5.3.

A

numerically stable reducible quadrature method is called

strongly stable (for small h) if the associated polynomial p satisfies the strong root condition.

Remark 5.1. The terms strong and weakstability are adoptedfrom Henrici

[3]

and Stetter

[13].

Numerical stability (for small h) which is not strong is sometimes

called harmlessweakstability (cf.

[12]).

Remark 5.2. The growth parameters of (p, o-)-reducible quadrature methods

were defined in (5.3). Since Theorem 5.1 is not restricted to this class of methods,

weshall nowbrieflyindicatehow thevalues of thegrowth parameterscan be obtained

formoregeneralquadraturemethods.

In

general, the application of a (step-by-step) direct quadrature method to the

test equation

f(x)=

1+h

f(y)dy

(cf.

[2])

is equivalent to the application of an

m-cyclic linear multistep method to the ODE test equation

f’=,tf.

Let P(h;

)

(h mhA be the associated characteristicpolynomial andlet

sr

(0),

,

sr

(0)be the

essential zeros of p(’):= P(0;

st);

then the growth parameters 3’ are given by the

expansion

’i(h)=’(0)(1 +3,h) ash 0.

For m 1 the equivalencewith

(5.3)

is wellknown.

6. Numerical stabilityversus repetition factor. In 4wehave characterized the

asymptotic repetition factor in terms of the location of the essential zeros of the

polynomialp, and in 5 numerical stabilitywascharacterized in terms ofthegrowth parameters associated with these zeros.

In

otherwords, numerical stability is

deter-mined by therate ofchange (relative toh)of the essential zeros and notsomuch by

their location. It is intuitively clear therefore that numerical stability cannot be characterized completely by the repetition factor. We can indicate, however, some connectionsbetweenthetwoconcepts.

THEOREM 6.1 (Noble

[11]).

Step-by-step methods (1.3) with an exact repetition

factor of

onearenumerically stable

(for

small h).

With the more general notion of the asymptotic repetition factorintroduced in

4,the above resultcan beextended.

THEOREM 6.2.

A

(p, o’)-reducible quadrature methodwith an asymptotic repetition

factor of

one isnumerically stable

(for

small h).

Proof.

In

viewof Corollary 4.2, the polynomialp satisfies thestrongroot

condi-tion, orequivalently,

rl

1 is the only essential zero.

Its

growth parameter is equal

to oneby virtue of consistency. Application of Theorem 5.1 yields the result. [3

The reversestatementsofthe theorems aboveare nottrue; that is,

THEOREM 6.3. Methods withanexactorasymptoticrepetition

factor

greaterthan

onecan be numericallystable.

Proof.

Itissufficienttoconsiderspecificexamples.Consider the(p,o-)-reducible

quadrature method with p(’)=(r2-

1)(r-1/2)

and

cr(’)=

r(r2-

r

₊

_1).

_In

view of

Theorem 4.2, the weights have an asymptotic repetition factor of two. The method

is numerically stablesince the growth parameters associated withthe essential zeros

’1

1 and

_r2

=-1 arebothequaltoone.

An

example of anumerically stable method

which has an exact repetition factor of two is obtained by taking p(’)=

r2-1

and

,r(’)

We

emphasize, however, that there existsanequivalence between anasymptotic repetition factor of one and strong, stability in the sense of Definition 5.3. This

important resultis given inthefollowing theorem.

THEOREM 6.4.

A

(p,tr)-reducible quadrature method

_for

solving second kind

Volterra integralequations isstrongly stable

(for

smallh)

_if

andonly

_if

the quadrature weights havean asymptotic repetition

factor

of

one.

Proof.

See theproofof Theorem 6.2.

We remark that Theorem 6.4 does not hold ifthe asymptotic repetition factor

isreplaced by anexactrepetition factor. Thisclearly showstherelevance oftheformer

notion.

We conjecture that formore general quadrature methods (e.g., methods which

arereducibletocyclic linear multistepmethods)aresult analogoustothat of Theorem 6.4 canbederived. Suchaderivation,however,isbeyondthescope ofthispaper.

7. Absoluteand relativestability. McKeeandBrunner

[10]

have interpreted the

stability concept of Linz and Noble in a different way. With reference to the test

equation (cf.

[2])

(7.1)

f(x)

1

+

A f(y) dy, A

<

O,

(whose solution f(x)--exp

(Ax)

decays to zero as xa3), they give the following definition.

DEFINITION 7.1 (from

[1-0]). A

method for (1.1) is called numerically stable

if when appliedto (7.1) the discretized solution

_fn

tends tozero as n

-

c for some fixedh.

Note that this definition is reminiscent ofthe definition of absolute stability in

the numericaltreatmentofODEs.

Withthisdefinitionof numerical stability,

McKee

andBrunnergive the following

example to demonstrate that the conjecture of Linz is incorrect. They consider the

(second order)methodgenerated bythequadrature weights

(7.2)

W0

-0 33 444 366 448 366 _448 3 44 66 6...63 484...844_

The weights in (7.2) are not (p,tr)-reducible, but are reducible to a 2-cyclic

linear multistep method. Clearly,

W0

has a repetition factor of two. Furthermore,

McKee

and Brunnershow that the method has a nonvanishing interval of absolute stability ofthe form (-a,0), and therefore

(7.2)

is numerically stable in the sense of

Definition7.1.

We recall from 1 that the asymptotic analysis ofLinz and Noble is applicable

togeneral secondkindVolterra equations. Sincethe stabilityDefinition 7.1 refersto one special test equation, it will be clear that numerical stability in the sense of

inthesense ofLinz andNoble.Todemonstratethis wehave determined (cf.Remark

5.2) the values of the growth parameters yi ofthe method (7.2) and obtained 3’1 1

and y.

=.

Therefore, in view of Theorem 5.1, the method (7,2) having repetition

factor twois numerically unstable inthe sense ofDefinitions 5.1 and 5.2.

In

{}8 we

shall demonstrate the unstable behavior of the method (7.2) when applied to an

equationdifferent from (7.1).

Keech

[5]

employsessentiallythe samestabilitydefinition as

McKee

andBrunner,

and gives the following example’

(7.3)

Wl--21-0 1 2 0 2 1 0 2 4 1 0 4 0 4 0 4 0 4 1 1 2 0 4 0 4 0 4 0 2 1 0 2

Hismethodis reducible to a 2-cycliclinearmultistep method. Ithasrepetition factor two, andKeech showsthatthe interval of absolutestabilityis (-2, 0). Therefore, the method (7.3) isnumericallystable in thesense ofDefinition7.1. Itturns out thatthe

growth parameters associated with (7.3) are both equal to one, so that the method

ofKeechisalso numericallystable in the sense ofLinz andNoble.Clearly, themethod isnot stronglystable.

Instead of lookingat absolute stabilityas wasdone by

McKee

and Brunner and

byKeech,wecan alsoadopttheconceptof relativestability ofamethodwithrespect

to

(7.1)

with

,

R.That is,werequireall rootsofP(h

’)

0(seeRemark5.2)tosatisfy

(7.4)

1(/)[

[’1(/)1

2, 3,’’ ",

where

srl(/)

correspondstothe principalroot(i.e.

’1(/)

exp

(/)

+

O(/

p+

₁₎

foramethod of orderp). For theweakly stable method (7.2), the interval ofrelative stability has

the form (0, fl),/3 >0, whereas for the stable method (7.3), this interval is approxi-mately

(-],

1/4).

It is known (see

[12])

that the existence of an interval of relative

stability of the form (-a, fl), a,/3>0, implies that all growth parameters associated withthe essential zeros areequaltoone. Thisyields

THEOREM 7.1.

A

reduciblequadraturemethodisnumerically stable

(for

small h)

if

thereexists aninterval (-a,

13),

a,

_fl

>

O,such that the methodis relatively stable

_for

allh (-a,

_fl

).

If a method is numerically stable and not relatively stable for h (-a,

fl),

then the violationof

(7.4)

for some iscaused only by

the/a

orhigher orderterms in the

expansion of the essentialzerosri(h).Therefore forsmallh,the existenceofan interval of relative stability of the form (-a,/3) is also "almost" necessary for numerical

stabilityin the senseofLinzandNoble.

8. Numerical illustration. In this section we present numerical results partly as an illustration of the various stability concepts discussed in 5 and partly as a verificationof ourtheoreticalresults.

For

our experiments we have constructed the following quadrature method parameterized by y

(3’

O, 1)’ (8.)

w()

=1/2

0 1

1+3’

1 l+y 1 1 0 2-2y l+y

2+3"

2-23,

1+3,

2-23"

2+2y 2-23,

1+3"

2+3"

2-23, 2+23, 2-2y

_1+3’

The quadrature weights in (8.1) are reducible tothe linear multistep method {p,

with

p(’)=sr2-1

and

tr(’)=r2(l+3")/2+r(1-3")+(l+3")/2.

In

view of Corollary

4.1 the weights have an exact repetition factor of two. Furthermore, the method is

secondorderconvergent and the growthprametersassociated with

_rl

1 and

_r2

=

-1

are 3"1 1 and 3’2 3", respectively. The method has an interval of absolute stability

(-c,0) if3’

>

0.

It

has an interval of relative stability (0, c) if 3’

<

1, and (-o,0) if 3">1. Since3" 1, themethod isweaklystable in view ofTheorem5.1.

We

havesolved the integral equation of the form

(5.4)

i0

(8.2)

f(x)

1

+

,

exp

(x

(x y))f(y dy,

whoseexactsolution is, in view of(5.5), givenby

(8.3)

f(x)

(x +I exp(,

+x)x)/(h +x).

Wetook the values(,,

x)

(1,-2) and(I,

x)

(-2, 1). Sevendifferentmethods were

used, tobe specific: the method of

McKee

and Brunner given by

(7.2)

and denoted

by W0; the method of Keech given by (7.3) and denoted by

W;

the method

(8.1)

with 3"

=

and 3’ 3 and denoted by

W()

and W2(3), respectively; the methods

employing quadrature weights which are reducible to the second and third order

backward differentiation method andtothethirdorder Adams-Moultonmethodand

denoted by

BD,

BD3

and

AM,

respectively. (Notethat the method

_AM

isidentical

to the quadrature method employing the third order

Newton-Gregory

quadrature

rules.)

Sincethe polynomial0 associated with the methods

BD,

BD

and

AM

satisfies

the strong rootcondition, these methods have an asymptotic repetition factorofone

(cf. Corollary 4.2) and are strongly stable (cf.Theorem

6.4).

The method

_W

has an

exact repetition factor of two and is stable but not strongly stable. The remaining methods also havean exactrepetition factor oftwobut areweakly stable(seeTheorem

5.1).

In

viewof the values of thegrowthparameters,weexpect thatfor(i,

x)

(1,

-2)

allmethods except

W2(3)

yield stable results, whereas for(,,

x)

(-2,

1)

themethods

Wo

and

W2()

areexpectedtobehaveunstably.

To

demonstrate clearly this unstable behavior of some of the methods, it is

necessarytointegrateover arather longtime interval.

We

have integrated theproblem

(8.2) onthe interval

[0,

25]

withstepsizesh 0.1 andh 0.05.

In

Tables8.1 and8.2

wehavelisted the true erroronlyforh 0.05 (the results forh 0.1 show the same behavior).

From

thesetablesweconcludethat, dependentonthe values I and x,themethods

with agrowth parameterdifferentfromone(thatis, W0,

W()

and

W(3))

areunstable,

TABLE8.1

Trueerror]:or(8.2)withh 1, =-2 and h 0.05.

x Wo W9_() W9(3) W1 BD2 BD3 AM3

5.0 _{-3.01o -2.51o -6.71o-2 -6.61o-3 -6.21o-3 -4.41o-4 -8.01o-5} 15.0 _{-3.01o -2.51o -1.31o -6.71o-3 -6.21o-3 -4.41o-4 -8.01o-5}

25.0 _{-3.01o -2.51o} -3.1lO -6.71o -6.21o -4.41o-4 --8.010

TABLE 8.2

Trueerrorfor(8.2)withh=-2, andh 0.05.

x _Wo W2() W2(3) W1 BD2 BD3 AM3

5.0 1.51o 5.31o -3.01o-3 -1.21o-3 -1.21o-3 2.91o-5 5.71o-6

15.0 _{1.21o 7.71o-1 -4.11o-3 -1.71o-3 -1.71o-3 6.61o-5 1.11o-5}

25.0 _{9.510 6.010 --4.110 --1.71o 1.710 6.610-5 1.110}

Keech (W1) yield stable results for both problems. Although it was notincluded in

the tables of results, we also noticed that for the stable method of Keech the true error changes sign at everymesh point, whereas the strongly stable methods yield a

smooth global error. Clearly, the numerical results are in full agreement with the

theory.

Inorder to eliminate the effectof thequadratureerror(which maybe quitelarge

whensolving(8.2) with h =-2 andtx 1), we have alsoinvestigated the effect ofan

isolated perturbation (see

[2]).

The methods were applied to (8.2) yielding values nextthe valueof

_fl

wasperturbedbyanamount of0.01,andthemethod was applied

once again yielding perturbed values

]rn.

In

Tables 8.3 and 8.4 we have listed the

difference

[-f[

at some meshpoints. The tables show that for both problems the

perturbation is damped by the strongly stable and stable methods, whereas it is

amplified by the weakly stable methods (dependent on the values of h and

).

We

remark that for the method of Keech wehaveperturbed

_f2

instead of

fl,

sinceit can

be seen from (7.3) that a perturbation of

_fl

has no effect on the even-numbered gridpoints whicharedisplayed in the tables.

TABLE 8.3

Effect ofanisolated perturbation (h 1,/x=-2; h 0.1).

x Wo

W2(61-)

W2(3) W _{BD2 BD3 AM3}

5.0 4.710 5.610 2.710 1.010-5 4.41o-6 8.810 6.31o-6

15.0 2.21o 2.61o-lO 9.41o+3 4.81o 2.1lO 4.01o 2.91o

25.0 1.41o 1.41o 3.31o+8 1.41o-14 1.41o-14 1.41o-14 1.41o-14

TABLE 8.4

Effect ofanisolated perturbation(h =-2,tz 1; h 0.1).

x Wo Wz() W2(3) Wl BD2 BD3 AM3

5.0 8,510 8.310 --6.710 4.410-5 1.310-5 3.710-5 1.910-5

15.0 _{6.710 6.510 --1.510-1o 1.510-9 4.110-1o 1.810-9 8.910-1,’}

Allcalculations wereperformedon aCDC-CYBER750 computer system using singleprecision (60-bit wordlengthwith a 48-bitmantissa).

9. Concluding remarks. Motivatedbyaconjecture ofLinz,wehave investigated for a special class ofquadrature methods the relationship between the (asymptotic)

repetition factor and numerical stability.

We

have shown that the methods with an

(asymptotic) repetition factor of one are strongly stable, which implies numerical stability in thesenseofLinzeand Noble.However,ifamethod hasarepetition factor greater than one, weneed additional

_information

in ordertodetermine whether that methodis numericallystable or not.

To

be specific, wehavetocheck that thevalues

of the growth parameters associatedwiththe essentialzeros ofthe polynomialp are

equaltoone.

In

general,these values can be determinedfrom the stability polynomial

associated with the method when applied to the test equation (7.1), and in this connection the analysis of Baker and Keech

[2]

can be used, although that analysis

wasdeveloped foradifferent type of

stability.

Onthe otherhand,it isthe rulerather thanthe exception that foranonartificially

constructed method, the growth parameters associated withthe essential zeros ( 1)

are different from one (see e.g.

[13,

p.

247]),

so that we share the general opinion

that methods with an (asymptotic) repetition factor greater than oneshould not be

generally employedfor the solution ofsecond kindVolterraintegral equations.

REFERENCES

[1] C.T.H.BAKER,TheNumericalTreatment_ofIntegralEquations, ClarendonPress,Oxford,1977.

[2] C.T. H. BAKERANDM.S. KEECH,Stability regions inthenumericaltreatmentofVolterraintegral equations, thisJournal, 15 (1978),pp.394-417.

[3] P. HENRICI,Discrete VariableMethodsin OrdinaryDifferentialEquations, John Wiley,NewYork, 1962.

[4] W. H. HOCK, Asymptotic expansions _for multistep methods applied to nonlinear Volterra integral

equationsofthe second kind,Numer.Math., 33 (1979),pp. 77-100.

[5] M. S. KEECH,Stability inthenumerical solutionofinitialvalue problemsinintegralequations, Ph.D. Thesis, Univ.Manchester, England,1977.

[6] M. KOBAYASI, On thenumericalsolution_ofVolterra integralequationsofthe secondkindby linear

multistepmethods,Rep.Statist.Appl.Res. Un. Japan.Sci.Engrs.,13 (1966),pp.1-21.

[7] J.D. LAMBERT, Computational Methodsin OrdinaryDifferentialEquations, John Wiley,London, 1973.

[8] P.LINZ, Numerical methods]:orVolterra integralequations with applicationstocertainboundary value problems,Ph.D. Thesis, Univ.Wisconsin,Madison,1968.

[9] J. MATTHYS, A-stable linear multistep methodsforVolterra integrodifferentialequations,Numer.Math., 27(1976),pp. 85-94.

[10] S. MCKEE AND H. BRUNNER, The repetition factorand numerical stability of Volterra integral

equations,Comput.Math.Appl.,6(1980),pp.339-347.

[11] B.NOBLE,Instability when solving Volterra integralequationsofthe secondkindby multistep methods,

in Conferenceon the Numerical Solution of DifferentialEquations, J. L. Morris,ed., Lecture NotesinMathematics109,Springer-Verlag, Berlin,1969.

[12] H.J.STETTER,Astudy_ofstrongand weak stabilityindiscretizationalgorithms,thisJournal, 2(1965),

pp.265-280.

[13]

,

Analysis _ofDiscretization Methodsfor Ordinary DifferentialEquations, Springer, Berlin-Heidelberg-NewYork, 1973.

[14] P. H. M. WOLKENFELT, Linearmultistep methods and theconstruction _ofquadrature_{formulae for} Volterra integral and integro-differentia!equations, ReportNW76/79, Mathematisch Centrum, Amsterdam, 1979.

[15]

.,

The construction of reducible quadrature rulesfor Volterra integral and integro-differential