continuous volterra-runge-kutta methods for integral equations with pure delay

C o m p u t i n g 50, 213-227 (1993) C o m p u t i n g

�9 Springer-Verlag 1993 Printed in Austria

Continuous Volterra-Runge-Kutta Methods for Integral Equations with Pure Delay

N. Baddour and H. Brunner, St. John ' s

Received November 23, 1992

Abstract - - Zusammenfassung

Continuous Volterra-Runge-Kutta Methods for Integral Equations with Pure Delay. In the following we give an analysis of the local superconvergence properties of piecewise polynomial collocation methods and related continuous Runge-Kutta-type methods for Volterra integral equations with constant delay. We show in particular that (in contrast to delay differential equations) collocation at the Gauss points does not lead to higher-order convergence and thus m-stage Gauss-Runge-Kutta methods for delay Volterra equations do not possess the order p = 2m.

AMS Subject Classification: 65R20, 65L06, 45L10

Key words: Volterra integral equations with delay, collocation, continuous Runge-Kutta methods, superconvergence.

Stetige Volterra-Runge-Kutta-Methoden fiir Integraigleichungen mit VerzOgerung. Diese Arbeit befagt sich mit Fragen der (lokalen) Superkonvergenz bei Kollokationsverfahren und stetigen impliziten Runge-Kutta-Methoden ftir Volterrasche Integralgleichungen mit retardiertem Argument. Es wird insbesondere gezeigt, dab (im Gegensatz zu retardierten Differentialgleichungen) Kollokation an den Gauss-Punkten nicht zu einer hSheren Konvergenzordnung fiihrt and dab deshalb m-stufige Gauss- Runge-Kutta-Methoden nicht die Ordnung p = 2m besitzen.

I. Introduct ion

In this pape r we analyze the numer ica l d iscre t iza t ion of Vol te r ra in tegral equa t ions with pure (constant) de lay z > 0,

y( t ) = 9( t ) + k( t , s, y ( s ) ) d s , t e I := [0, T ] , (1.1)

where the so lu t ion y is subject to the init ial cond i t ion

y( t ) = (b(t), t e [ - z , 0 ) , (1.2)

by co l loca t ion and re la ted con t inuous V o l t e r r a - R u n g e - K u t t a me thods in cer ta in (nonsmooth) po lynomia l spline spaces.

I t will be assumed tha t the given functions, ~b: [ - z , 0 ] -~ R, 9: I -~ R, and k: S~ • R ~ R (with S~: I • [ - v , T -- ~]) are (at least) con t inuous on their domains ; add i - t ional condi t ions will be imposed la ter when needed. Existence and uniqueness

214 N. Baddourand H. Brunner

results for (1.1) and for related Volterra integral equations with finite delay (e.g.

y(t) = g(t) + J'i k(t ,s ,y(e(s)))ds, t E I , (1.3)

with bounded e(s) < s) can be found, for example, in [3, 6, 11, 12, 13].

Note that the delay integral equation

y(t) = 7(t) + j [ ~c(t, s, y(s - r))ds, t e I ,

can be rewritten in the form (1.1) by setting k( t , s ,y ) := ~(t,s + r,y) and

g(t) := ~(t) + J~[ ~(t, s + ~, r

In recent years, various aspects of numerical methods for delay integral equations have been studied. In [ ! 4] convergence properties of Euler's method, the trapezoidal and midpoint method for (1.1) are analyzed. The papers [12] and [1, 2] deal, respectively, with Hermite-type collocation and ODE Runge-Kutta methods for delay integral equations which are somewhat more general than (1.3). An even more general version (in which the delay is state-dependent) is considered in [11]: here, the numerical solution is based on direct quadrature and cubic spline interpolation.

The paper [5] presents an analysis of a very general class of Runge-Kutta-type methods for Volterra integral equations; in [16] these methods are extended to Volterra integral equations with (constant) delay. Here, we also find a detailed analysis of the (P-)stability properties of these methods (and of collocation methods) for

y'(t) = 1 + a y(s)ds + b y(s)ds, (1.4)

where a and b satisfy the stability condition Jbl < -Re(a) . Note that (1.4) can be reduced to a delay differential equation with constant coefficients,

y'(t) = ay(t) + by(t - z), t ~ I .

However, an analysis of the local superconvergence properties of collocation methods for (1.1) is essentially still lacking. It is the aim of this paper to show that O(h2")-convergence at the mesh points HN can be attained by using the iterated collocation solution corresponding to collocation in S ~ ( H s ) (piecewise polyno- mials of degree m - 1 possessing jump discontinuities on Hu); on the interval I, the global order of convergence is given by p = m. If restricted to Hu, the collocation solution itself has, for the Gauss (-Legendre) collocation points, an order which in general does not exceed the global order p.

2. Co l loca t ion and I terated Co l loca t ion

Let t n := n h ( n = 0 . . . . . N - 1; t N = T ) define a u n i form part i t ion for I = [0, T] , and set HN := {to . . . . . tu}, Io := [to, t l ] , I . := ( t . , t . + l ] (1 _< n < N -- 1). Since the so lu-

Continuous Volterra-Runge-Kutta Methods for Integral Equations with Pure Delay 215

t ion y of (1.1) has p r imary discontinuities at the points ~ :=/~r, p = 0 . . . . . M (we suppose, wi thout loss of generality, that T is such that T = M r for some positive integer M), the mesh HN is assumed to be constrained, i.e.

h = - z for some r ~ N . (2.1) r

For given integers d > - 1 and m > 1 the piecewise po lynomia l space S~)+a(HN) is defined by

ca) H {u: I R;ul l , ~ 7~m+d, U n - l ( t n ) = = . Sin+d(N) := ~ =: U. . ~v) U(.v)(t,),V 0 . . . . . d} (2.2)

The dimension of this vector space is obviously given by

d im S~)+a(HN) = N m + d + 1.

This shows, in the context of collocation, that the natura l choice of d in (2.2) will be governed by the nature of the functional equat ion to be solved: if the equat ion under considera t ion is a (delay) differential or integro-differential equat ion of order K, then d = K - 1; when solving (delay) integral equat ions like (1.1) we choose d = - l .

Fo r given real numbers {cj} with 0 < c I < . . . < c,, < 1, define the set XN := {t.,j} of collocation points by

t , , j : = t , + cjh ( j = 1 . . . . , m ; n = O . . . . . N - l ) . (2.3)

The collocat ion solution u ~ S~-I_~(HN) to (1.1) is then given by the equa t ion

u(t) = g(t) + k(t, s, u(s))ds, t ~ X N , (2.4)

with

u(t) = r on [ - r, 0). (2.5)

If t = tn,2 is such that tn, j - T ( = tn_r,j) < 0 (recall that, by (2.1), z = rh = tr), then (2.4) becomes

u(t) -- g ( t ) - cI)(t), t = t,,~ ~ X N , (2.6)

( j = 1 , . . . , m ; n = 0 . . . . . r - 1), where

q5(0 := k(t, s, ~(s))ds, t - r < 0. (2.7)

The delay integral ~(t) represents a further potent ial source of error since in general one will not be able to evaluate this integral analytically; instead, one will have to resort to suitable numerical integrat ion formulas (compare T h e o r e m 3.4 for details).

The i terated collocation solution, ui,, corresponding to the col locat ion solut ion u determined by (2.4), (2.5) is defined by

uit(t ) := g(t) + k(t, s, u(s))ds, t e I . (2.8)

216 N. Baddourand H. Brunner

It has the property that

uit(t) = u(t) whenever t e XN.

Moreover, if the delay integral introduced in (2.7) can be found analytically, then we have uit(t) = y(t) for t e [0, z] (compare also the proof of Theorem 3.2).

In order to put (2.4) and (2.8) into a form amenable to numerical computation, let t ~ I,, and define the lag term Dn(t ) by

Dn(t ) := k(t, s, u(s))ds (2.9)

(with Dn(t ) = - ~ ( t ) if t < z). Since u ~ ~,,-i on I n, we may write

u(t n + vh) = ~ Lk(v)Un, *, t n + vh E In, (2.10) k = l

where U,,, := u(tn,,), and where L k denotes the kth Lagrange fundamental polynomial for the set {cj}. Thus, (2.4) assumes the form

U.,j = g(tn,j) + Dn(tn,j) , j = 1 , . . . , m, (2.11)

where, for t = t n + zh ~ I.,,

D.(t) = h ~ k(t, ti + vh, u(ti + vh))dv + h k(t, t._~ + vh, U(tn-~ + vh))dv. i = 0

(2.i2)

. . . ~ U T F o r e a c h n 0, N - 1, (2.11) defines a unique Un := (Un, x . . . . . . . . ) ERm, and hence a unique representation of the collocation solution u on In. The corresponding iterated collocation solution at t = t, + zh E I~ is then given by

u,,(t) = g(t) + Dn(t), (2.13)

with D,(t) as in (2.12).

Consider now (2.11) and (2.13): in general, the integrals in Dn(t) cannot be evaluated analytically but have to be approximated by suitable quadrature formulas. We choose interpolatory m-point quadrature formulas whose abscissas are given by the collocation points. Specifically, if z = cj,

J

k(t., j, t ._ ~ + vh, u(t._~ + vh))dv

will be replaced by

wj, t k l ( tn j , t ,_r + qczh, u(tn_r + cjcth)), / = 0

with wj, l := cjwt, wt = S~ Ll(v)dv; u(t._r + cjqh) is given by (2.10), with v = cjct and with n - r replacing n. Thus, the resulting fully discretized collocation equation corresponding to (2.11) is given by

On, j = g( tn j ) + Dn(tn.j) ( j = 1 , . . . , m ) , (2.14)


with n - r - 1 m

D(t.,j) := h E ~. k(t . j , ti, l, Oi,l) i = 0 1 =1

+ h c j f ' ~ w l k ( t n ' j ' t n - r - t - c j c l h ' ~ g k ( c j c l ) g n - r ' k ) ' l = 1 k =1 (2.15)

provided that n - r _> 0 (i.e. t, - r = t,_ r > 0). If n - r < 0, then/)( t~, j ) is given either by the exact value of - ~(t. , j) ,

D,(t.,j) = - h k(t,,j, t,_r + vh, ~(t._, + vh))dv d

- h ~ k(t,,j, ti + vh, d~(tl + vh))dv (2.16) i=n-r+l

(in applications, the integral ~(t) of (2.7) can frequently be found analytically), or by a suitable quadra ture approximat ion to - ~ ( t , , j ) , e.g. by

On(tn, j ) = - h ~ l~j, lk(tn,j, tn_ r -1- ~j, th, qb(t._. + ~j, zh)) l = 1

- 1

- - h ~ ~ Wlk(tn,j, ti, l,q~(ti,l)), (2.17) i=n-r+l / = 1

where

~ j , t : = c j + ( 1 - c j ) c l , # j , t : = ( 1 - c j ) w l ( j , l= 1 . . . . . m).

fully discretized collocation scheme generates a collocation solution fi E The S~,~_I(HN), given by

~(t, + vh) = ~ L k ( V ) O n , k , t n + vh e I, (0 < n <_ N - 1), (2.18) k = l

with ~ := u - f i r 0 in general.

Equat ions {(2.14), (2.17), (2.18)} represent a continuous implicit m-stage Volterra- Runge-Kutta (CDVRK) method for the delay integral equat ion (1.1). This class of col locat ion-based Runge-Kut ta methods forms an impor tan t subset of the general D V R K methods discussed in [16].

The corresponding discretized version of the i terated collocation equat ion (2.13) will be referred to as the iterated CDVRK method. If t = t,+ 1 it yields

n - - r

f i i t ( t ,+l)=g(t ,+l)+h ~ wlk(t,+l,ti+czh, Ui, l), r<_n<_N-- 1. (2.19) i = 0 l = 1

For 0 < n < r (i.e. for t,+ 1 _< t, = z) we obtain the approximat ion

- 1

fii,(t,+l) = g(t,+l) - h ~ ~ wlk(t,+l,ti + clh, d~(tl + czh)). (2.20) i=n+l-r l=l

We conclude this section with the following obvious but impor tan t observat ion concerning the solution of (1.1) at t = 0. Assume that the given functions g, k and

218 N. Baddour and H. Brunner

q~ in (1.1) and (1.2) are continuous, and let lim,.~o_ ~b(t) =: ~b(0) exist and be finite. Then the solution y of(1.1), (1.2) is continuous at t = 0 if, and only if, g satisfies the condition

g(O) = ~(0) + f ~ k(O, s, fb(s))ds. (2.21)

Recall also that, in contrast to the above situation, the solution of the delay differential equation y ' ( y ) = f ( t , y ( t - z)), t �9 I, with initial condition (1.2), is (for continuous data) always continuous at t = 0 (while y' in general is discontinuous).

3. Local Superconvergence on H N

It was shown in [4, 15] that if the initial-value problem for a delay differential equation,

y'(t) = f ( t , y ( t ) , y ( t - r)), t �9 I (z > 0),

is solved by collocation in S~~ with constrained mesh Hs, and if the collocation points are given by the Gauss (-Legendre) points (i.e. if the {cj} are the zeros of the Legendre polynomial P,,(2s - 1)), then

max [y(t,) - u(t,)J < Ch 2", (3.1) l <n<_N

provided the exact solution y has continuous derivatives of order 2m on each subinterval (/~z, (p + 1)~) (# = 0, . . . , M - 1). In other words, the Gauss collocation solution for constrained meshes exhibits local superconvergence of order p* = 2m at the mesh points t = t, (while the global convergence order is p = m; compare [-4]).

This superconvergence result does not carry over to the Gauss collocation solution u �9 S ~ ( I I u ) for the delay Volterra integral equation (1.1): instead of (3.1) we only have

max [y(t,) - u(t.)] = O(hm). l <_n<_N

However, for the corresponding iterated Gauss collocation solution u u, the result (3.1) is again true.

Before making these statements more precise (in Theorem 3.3), we will discuss an important and relevant relationship between collocation (in S~)(HN)) for the delay differential equaton (DDE)

y'(t) = ay(t) + by(t - ~) (a, b = const.), t �9 I , (3.2)

with initial condition (1.2), and collocation (in S~-_1~ (/-/N)) for the corresponding delay integral equation

; f7 y(t) = Yo + ay(s)ds + by(s)ds, t �9 I , (3.3)


with the same initial condition (1.2). According to (2.21), these two initial-value problems define the same (unique) solution y on I if, and only if,

Yo = ~b(0) + b f ~ (b(s)ds. (3.4)

Although in this paper we focus on integral equations with pure delay, the following theorem deals with the more general delay equaton (3.3); it will also be of interest in a wider context.

Theorem 3.1. Assume:

(i) v ~ S(,,~ is the collocation solution to the DDE (3.2), with v(t) = O(t) if t e [--z, 0]. The underlying mesh H s is assumed to be constrained, i.e. h = z/r for some r ~ N.

(ii) u e S~-I_~(HN) is the collocation solution to the delay integral equation (3.3) (subject to the continuity condition (3.4)), based on the same set o f collocation points as v.

Then

u(tn) = v(tn) for n = 1 . . . . . S

if, and only if, c,~ = 1 and the approximations to the delay integrals

J

qn-r,j:= ~b(t~_,+zh)dz ( j = 1 . . . . . m ; O < _ n < r )

are given by the m-point interpolatory quadrature formulas

Q,-~,j := aj, lO(t,-~ + clh) aj, l := Ll(z)dz �9 1=1

This result gives a first indication on why the local superconvergence results for (general) delay differential equatons ([4, 15]) corresponding to collocation at the Gauss points {t, + cjh} (for which Cm < 1) will not carry over to collocation for Volterra integral equatons with delay.

Proof: Consider first the collocation solution v ~ S~~ for the DDE (3.2), and set

v(t, + zh) = v(t,) + h ~ flt(z)v'(t. + c~h), z e [0, 1], (3.5) /=1

with flz(z) := Sf~ Lz(s)ds. If follows from the collocation equation for (3.2),

v'(t~ + cih) = av(t n + cih) + bv(t~_r + cjh), j = 1 . . . . , m,

(where tn_, = tn - r) that

V~,j = v(t,) + h ~ aj,~(aV,, l + bV~_,,l), / = 1

with aj, z : - fl~(cj) and Vn,j := v(t~ + cjh). Setting e := (1 . . . . . 1) r e R ~ this yields

V, = (I - ahA) - l e . v ( t , ) + bh(I - ahA)-IAV~_~, (3.6)


where the meaning of the vector V. and the matr ix A is obvious. Thus, setting z = 1 in (3.5) and using wj := flj(1),

v(t.+l) = v(t.) + a h ( w , V . ) + b h ( w , V . _ r ) , (3.7)

where ( . , - ) denotes the usual inner p roduc t in R". For 0 < n < r we have V._r =

([}n -r := ((fi(tn -r, 1 ) . . . . . (#(tn -r,m) ) T"

We now turn to the delay integral equat ion (3.3). On the subinterval I . the colloca- -(-1J(/7 tion solut ion u e a;,-1 N) is given by

u(t. + zh) = ~ L~(z)g.,~, l = 1

The col locat ion equat ion defining u is

with

and

This leads to

U.,l := u(t. + qh).

~ c 0 " .1

U.,j = F. + O.-r,j + h au(t. + zh)dz ( j = 1 , . . . ,m) ,

ftOn ftOnr F. := Yo + au(s)ds + bu(s)ds

CoJ O.-r,j := h bu(t._r + zh)dz.

U. = (I - ahA) - l e . F. + (I - ahA)-l~g._ r,

(3.8)

(3.9)

(3.10)

where ~ . - r := (0.-r,~ . . . . . 0._r,,.) r. C o m p a r i n g (3.8) and (3.9) we observe that F. +1 = u(t. +1) if, and only if, Cm = 1. For this choice Of Cm it follows f rom (3.10) that

q?._r = bh" AU._r

which in turn implies

U, = (I - ahA) - l e .u ( t . ) + bh(l - ahA) - lAU._r (r < n < N - 1). (3.12)

Since c,. = 1 we may write

u(t.+l) = U.,m = u(t.) + h f / au(t. + zh)dz + h f ] bu(t._~ + zh)dz,

and so we find

u(t.+l) = u(t.) + a h ( w , U . ) + b h ( w , U . _ , ) (r < n < N - 1). (3.13)

Let now 0 < n < r, i.e. - r _< t._~ < 0. Observe first that, due to (3.4),

u(O) = U(to) = Yo - I ~ b(~(s)ds = ~b(O) = V(to). d-*

(3.11)


Also, setting q,-r := (qn-rm, 1 . . . . . q,_,. , ,)r, the collocation equat ion (3.8) yields

U, = (I - ahA) - l e . u ( t , ) + bh(I - ahA)-lq,_~ (0 < n < r),

and hence (using again c,, = 1)

u(t.+l) = u(t,) + a h ( w , U , ) + bh. f ~ q~(t,_~ + zh)dz.

The analogous relations for the D D E (3.2) are

V. = (I - ahA) - l e . v ( t , ) + bh(1 - ahA) - lA@,_ ,

(cf. (3.6)) and

/)(tn+l) = D(tn) -'~ a h ( w , V , ) + bh . (w,~O,_r) .

Clearly, U, = V, (and hence u(t.+O = v(t.+l)), n = 0 . . . . . r - 1, if, and only if,

q . _ , = A ~ . _ ~ .

For r < n _< N - 1 the assertion of Theorem 3.1 then follows from (3.12), (3.13) and (3.6), (3.7). [ ]

The above proof also reveals that, for the choice c,, = 1, the collocation solution v for the D D E (3.2) and its counterpar t u for (3.3) do not yield the same value at t = t. if the delay integrals q , _ , j (0 <_ n < r) are given analytically (unless ~b is a polynomial whose degree does not exceed the degree of precision of the m-point in terpola tory quadra ture formula based on the collocation parameters {cj}).

The iterated collocation solution to (3.3), with the delay integrals (2.7) (t = t,, with n - r < 0) approximated by m-point in terpolatory quadra ture formulas (cf. (2.20)), may be written as

f? fl . . . .

U i t ( t n + l ) = Yo + au(s)ds + bu(s)ds

= uu(t,) + ah(w, U , ) + b h ( w , U , _ , ) ,

where, for 0 < n < r, U ,_ , = ~ ._ r . The following result reveals that collocation for the D D E (3.2) is equivalent (in a sense made precise below) to iterated collocation for the delay integral equat ion (3.3).

Theorem 3.2. Let the assumptions (i), (ii) and the notation of Theorem 3.1 hold, and assume that the collocation parameters {c;} are such that Cm < 1. I f the delay integrals q,_,,j ( j = 1 . . . . ,m;O <_ n < r) are approximated by the m-point interpolatory quadrature formulas Q,_r,j, then the iterated collocation solution uit for (3.3) corresponding to its collocation solution u satisfies

ui~(t.) = v(t,) for n = 1 , . . . , N .

Thus, if the {cj} are the Gauss points in (0, 1), the above result together with the superconvergence result of [4] implies that u~t is superconvergent of order p* = 2m at the mesh points t = t,.


Proof." It follows f rom the definition o f u , and f rom (3.9) that F n = ui,(t,). I f 0 < n < r, then the assumpt ion on the approx imat ions to the delay integrals q n - , j and (3.10) lead to ~ , _ , = bh. A ~ , - r . Hence,

U i t ( t n + l ) = Uit(tn) -1- a h ( w , U , ) + bh(w, ~ , _ r ) ,

with (cf. (3.11))

U , = (I - ahA) - l e . u id t , ) + bh(l - a h A ) - t A ~ , _ r (0 < n < r).

If we now compare these last equat ions with (3.6) and (3.7) we readily find that the assert ion u , ( t ,+ l ) = v(t,+l) is true for 0 < n < r, since u/,(0) = Yo = v(0).

Fo r r _< n < N - 1 the propos i t ion of Theorem 3.2 is a direct consequence of(3.11), which now can be writ ten as

U , = (I - ahA) - l e . ui,(t,) + bh(I - a h A ) - I A U , _ , ;

of (3.6), (3.7), and the expression for uidt.+ 1) preceding Theorem 3.2. [ ]

We continue by stating the main result of the paper for the linear version of (1.1),

y(t) = g(t) + K(t , s)y(s)ds, t ~ I , (3.14)

with initial condi t ion (1.2). The extension of Theorem 3.3 to the nonl inear case (1.1) will be discussed at the end of this section.

Theorem 3.3. Assume that the given functions in (3.14) and (1.2) are smooth: g C m +d(I), K e C" +d(S~), and 0 ~ C" + e l - r , 0], for some (given) integer d with 0 <_ d <_ m. Suppose that the delay integral ~ (t) (cf . (2.7 or (2.16)) can be evaluated analytically.

I f h = z/r is sufficiently small, i f the collocation parameters {cj} are chosen so that the orthogonality conditions

Jk := 11 sk ~] (S -- e f t& = O, k = 0 . . . . . d - 1; with Jd r 0, (3.15) do j = l

hold, and i f % = 1, then the collocation solution u ~ S ~ ~ ( Hu ) defined by {(2.4), (2.5), (2.10)} is locally superconvergent at the mesh points t = t, (1 <_ n <_ N) whenever d > 0 :

max ly(t,) - u(t,)l < Ch m+a (3.16) l <_n<_N

for some f ini te constant C.

I f d = m, i.e. i f the { cj} are the Gauss (-Legendre) points (for which c,. < 1), then there exist constants C and C* so that

max ly(t,) - u(t.)[ < Ch =, (3.17) l <_n<_N

but

max [y(tn) - Ui,(tn)[ ~ C*h 2'n. (3.18) l<_n<N


Here, the iterated collocation solution u~t is determined by (2.8).

Remark: For "classical" Vol terra integral equat ions (i.e. for (1.1) with z = 0), local superconvergence results for the cases d = m, d = m - 1 (Radau H points), and d = m - 2 (Loba t to points) were derived in [7, 9].

Proof: The col locat ion equat ion (2.4) (applied to the linear delay equat ion (3.14)) m a y be writ ten in the "cont inuous" form

u(t) = -($(t) + g(t) + K(t, s)u(s)ds, t �9 I ,

where the defect ~ vanishes on XN:

~(tn,~) = 0, j = 1 . . . . . m; n = 0 , . . . , N - 1; (3.19)

moreover , for 0 < v _< m + d, 8~(t) is piecewise continuous, with finite j umps at t = ~, := #~ (# = 0 . . . . . M - 1). The collocat ion error e := y - u satisfies the delay integral equat ion

e(t) = 5(0 + D(t), t �9 I , (3.20)

where

If D(t) := K(t, s)e(s)ds. (3.21)

If t �9 [0, r], then the assumpt ion that the delay integral ~o_~ K(t, s)c~(s)ds is known analytically implies that D(t) = 0 on [0, ~], and this yields e(t) = ~(t) for t �9 [0, z-l.

Since the col locat ion solut ion u and the i terated col locat ion solution u , are related by

uit(t) = u(t) + 6(t), t �9 I ,

the error cor responding to u , m a y be writ ten as

ea(t ) = e(t) - ~(t), t ~ I . (3.22)

In part icular , if t �9 [0, z] ( = [0, t ,]) then we have u,(t) -- y(t) (recall the r emark following (2.8)), and hence

eit(t ) = 0 for t ~ [0 ,z] .

The following l emma on the representat ion of the solution to the delay integral equa t ion (3.20) contains the key to the p roof of Theo rem 3.3.

L e m m a 3.4. On the interval [~,, ~,+1] (0 <_ # <_ M - 1; Mz = T) the solution to the delay integral equation (3.20) is given by

# f l - i t e(t) = 6(t) + ~ K,(t,s)6(s)ds, (3.23)

i=1

where

~ t ~ T Ki(t ,s ) := K(t ,v)Ki_l(v ,s)dv (2 < i __ #),

+(i -l)z

with K l ( t , s ) := K(t,s).


Proof o f Lemma 3.4: Consider first the interval [z, 2z]. Since e(t) = 6(0 on [0, z], (3,20) and (3.21) imply

e(t) = 3(t) + K~(t,s)g)(s)ds.

In order to establish (3.23) in general, a straightforward induction argument is used. Here, the main tool is a variant of the standard Dirichlet formula, namely

Details are left to the reader. []

The result of Lemma 3.4 and its companion expression for the iterated collocation error (which follows from (3.22)),

I i - i~ e~t(t) = ~ Ki(t,s)8(s)ds, t e [~,, ~,+1], (3.24)

i = l

form the basis for establishing the local superconvergence results in Theorem 3.3. Consider first (3.23) with t = t, in the interval [~,,~,+~] (with/~ > 1). Setting

Iltn,i(t v + vh) := Ki( tn , t v + vh)3(t~ + vh),

and observing that t, - iz = t, - irh = t,_~r we may write (3.23) as

n --ir - 1

e(tn) = 6(t,) + h ~ i=1 v=O

t~ n - J r - 1

= 3 ( t , ) + h ~ ~, i =1 v =0

f ~ T~,i(t~ + vh)dv

( ~t=a wt~U"'i(tv + czh) + El')~} "

Here, we have replaced each integral over [0, 1] by the sum of its m-point interpolatory quadrature formula and the corresponding quadrature error F,(.") - - l ,v"

(Note that, by our assumption on the delay integral ~(t), we have E(") = 0 if L,V

0 < n < r.) Since the defect 6 vanishes on XN (recall (3.19)), the values of T,,~(t~ + qh) are zero, and thus the above expression reduces to

,u n - i t - 1

e ( t , ) = g ( t , ) + h ~ ~ -~,F'(")~ ( l_<n_<N). (3.25) i=1 v=O

The orthogonality condition (3.15) implies that the (interpolatory) m-point quadrature formulas with abscissas {t~,~} and weights {wz} all possess the degree of precision m + d. Thus, since the integrands T,,~(t~ + vh) are smooth for v e [0, 1], the quadrature errors in (3.25) can be bounded by

IEI~)~[ <_ Ch re+d,

where, by definition of T,,i(t ~ + vh), the constant (~ depends on the derivatives (of order up to m + d) of Ki(' , t~ + vh) and 6(t~ + vh) ( r e [0, 1]) (as seen from the definition of the K~ in Lemma 3.4, the "iterated" kernels K~ inherit the smoothness of the given kernel K). Noting that we have # < M - 1, Mz = M . rh = T = Nh, we


see that the number of error terms occurring on the r ight-hand side of (3.25) is at most equal to (M - 1)N/2 = O(N), we finally arrive at the estimates

le(tn)l < [O(t.)l + Ch re+a,

and, by (3.24),

lei,(t.)l < Ch m+a (n = 1 . . . . ,N)

for some finite constant C.

If c m = 1, then t,_l + Cm h = t, ~ X N, and thus 6(tn) = 0, 1 _< n _ N. This proves (3.16). Note that under this constraint on Cm we have d _< m - 1, with d = m - 1 if, and only if, the {cj} are the Radau II points, i.e. the zeros of Pm(2S- 1 ) - Pm_l(2S -- 1). For continuous u (corresponding to the choice c 1 = 0, c m = 1 (m >_ 2)), the optimal value of d in (3.16) is d = m - 2: it is a t tained if the {c~} are the Lobatto points (zeros of s ( s - 1 ) P ' _ l ( 2 s - 1)).

The maximum value of d in the or thogonal i ty condit ion (3.15), d = m occurs if, and only if, the collocation parameters are the Gauss (-Legendre) points in (0, 1). Fo r these points we have 0 < cl < "" < Cm < 1, and hence 3(t~) ~ 0. It is easily seen from the definition of the defect ~ that, in general, we can do no better than 6(t) = O(h '+) for t ~ XN. (Consider (3.14) with Ki = const., ~ = eonst.; in this case

~ S(m~ i.e. piecewise cont inuous of degree m, vanishing on XN 7 5 Hs). This yields (3.17). The estimate (3.18) for the i terated Gauss collocation solution follows from (3.22). [ ]

The above proof readily suggests that the local superconvergence results of Theorem 3.3 are also true for the discretized collocation solution t~ e S~-I__~(Hs) defined by (2.14), (2.17), (2.18), and for the corresponding iterated collocation solution ai, (cf. (2.20)), provided the quadra ture approximat ion to the delay integrals f+o

r -- k ( t . , s, ~(s ) )ds n r

= h ~ k(t,, t i + vh, 4(t; + vh))dv i m t l - - r

are given by

~(t,) = h ~ wtk(t.,ti, z,q~(ti, l)) (0 < n < r). (3.26) i = n - r 1=1

Due to the or thogonal i ty condit ion (3.15) satisfied by the parameters cz defining the quadra ture abscissas, the corresponding quadra ture errors a r e O(hm+a). We leave the remaining details of the proof to the reader but summarize these results in

Theorem 3.5. Let the assumptions of Theorem 3.3 hold, and assume that the approximations to the delay integrals ~(t,,j) and qb(tn) (where 0 <_ n < r) are given by the quadrature processes (2.17), (2.20) and (3.26), respectively.

I f h = z/r is sufficiently small and if the orthogonality conditions (3.15) hold, then the solution ~ given by the continuous implicit D V R K method {(2.14), (2.17), (2.18)} has

226

the property that

N. Baddourand H. Brunner

max ly(t,) - a(t,)l ~ C*h re+a, l <_n<_N

provided that c,, = 1 (and thus d ~ m - 1).

I f the {ei} are the Gauss points in (0, 1), then

max ly(t,) - •(t,)l < Ch m. J. <n<_N

The approximation furnished by the iterated C D V R K method {(2.19), (2.20)} correspondin 9 to the Gauss points satisfies

max ly(t,) - i,t(t,)l < C*h am. [] l <n<_N

Finally, we remark that Theorems 3.3 and 3.5 can be extended to the nonlinear Volterra equation (1.1), as follows. Instead of (3.20), the equation for the collocation error e now has the form

e(t) = 6(t) + (k(t, s, y(s)) - k(t, s, u(s)))ds, t e I .

Under appropriate differentiability and boundedness conditions for k we then find, setting u ( s ) = y ( s ) - e ( s ) ,

Ok 1 c32k 2 k(t, s, y(s)) - k(t, s, u(s)) = ~ (t, s, y(s))" e(s) + ~ ~y2y2 (t, s, z(s))" e (s),

where z(s) is between y(s) and u(s). The role of K(t,s) in (3.21) is now assumed by K(t, s):= (Ok/#y)(t, s, y(s)), and the analogue of (3.21) contains a perturbation term involving e2(s). Since the global convergence of the collocation solution u is given by Hy - u[[| = He/~ = O(h") (compare [8]), it follows that

[leZHo~ = O(h TM) for any set {cJ in (2.3).

The remaining parts of the proofs are then readily adapted to deal with these modications.

4. Concluding Remarks

It should be possible to combine the techniques used in [4, 15] for delay differential equations with the ones employed in this paper in order to extend the above results to Volterra integral equations with variable delays (e.g. to (1.3)) or with multiple delays (I-6]). The analysis of collocation and continuous Volterra-Runge-Kutta methods for Volterra integral equations with state-dependent delays apprears to be much more complex to deal with (compare [11] where a class of numerical methods for such delay equations is studied).


Acknowledgements

The work of the first author was carried out while she held an NSERC Undergraduate Student Research Award (Summer 1992). The second author is supported by an NSERC Operating Grant (A9406).

References

[1] Arndt, H., Baker, C. T. H.: Runge-Kutta formulae applied to Volterra functional equations with fixed delay. In: K. Strehmel (ed) Numerical treatment of differential equations: 4th lnternat. Seminar NUMDIFF, Halle-Wittenberg, 1987, 19-30. Leipzig: B. G. Teubner Verlagsgesellschaft 1988 (Teubner-Texte zur Mathematik, 104).

[2] Baker, C. T. H., Derakhshan, M. S.: R-K formulae applied to Volterra equations with delay, J. Comput. Appl. Math. 29, 293-310 (1990).

[3] Banag, J.: An existence theorem for nonlinear Volterra integral equation with deviating argument. Rend. Circ. Mat. Palermo (2) 35, 82-89 (1986).

[4] Bellen, A.: One-step collocaton for delay differential equations. J. Comput. Appl. Math. 10, 275-283 (1984).

[5] Bellen, A., Jackiewicz, Z., Vermiglio, R., Zennaro, M.: Natural continuous extensions of Runge- Kutta methods for Volterra integral equations of the second kind and their applications. Math. Comp. 52, 49-63 (1989).

[6] Bownds, J. M., Cushing, J. M., Schutte, R.: Existence, uniqueness, and extendibility of solutions of Volterra integral systems with multiple, variable lags. Funkcial. Ekvac. 19, 101-111 (1976).

[7] Brunner, H.: Iterated collocation methods and their discretizations for Volterra integral equations. SIAM J. Numer. Anal. 21, 1132-1145 (1984).

[8] Brunner, H.: Collocation and continuous implicit Runge-Kutta methods for a class of delay Volterra integral equations. To appear in J. Comput. Appl. Math.

[9] Brunner, H., Van der Houwen, P. J.: The numerical solution of Volterra equations. Amsterdam: North-Holland 1986 (CWI Monograph, 3).

[10] Cahlon, B.: On the numerical stability of Volterra integral equations with delay argument. J. Comput. Appl. Math. 33, 97-104 (1990).

[11] Cahlon, B., Nachman, L. J.: Numerical solutions of Volterra integral equations with a solution dependent delay. J. Math. Anal. Appl. 112, 541-562 (1985).

[12] Esser, R.: Numerische Behandlung einer Volterraschen Integralgleichung. Computing 19, 269- 284 (1978).

[13] Sugiyama, S.: On functional integral equations. Mem. School Sci. Engrg. Waseda Univ. 41, 135-153 (1977).

[14] V5.I5., P.: Convergence theorems of some numerical approximation schemes for the class of non-linear integral equations. Bul. Univ. Gala~i Fasc. II Mat. Fiz. Mec. Teoret. 1, 25-33 (1978).

[15] Vermiglio, R.: A one-step subregion method for delay differential equations. Calcolo 22, 429-455 (1985).

[16] Vermiglio, R.: On the stability of Runge-Kutta methods for delay integral equations. Numer. Math. 61,561-577 (1992).

N. Baddour and H. Brunner Dept. of Mathematics and Statistics Memorial University of Newfoundland St. John's, Newfoundland Canada A1C 5S7

continuous volterra-runge-kutta methods for integral equations with pure delay

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