C o m p u t i n g 50, 2 1 3 - 2 2 7 (1993)
Computing 9 Springer-Verlag1993 Printed in Austria
Continuous Volterra-Runge-Kutta Methods for Integral Equations with Pure Delay N. Baddour a n d H. Brunner, St. J o h n ' 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 GaussRunge-Kutta-Methoden nicht die Ordnung p = 2m besitzen.
I. Introduction In this p a p e r we a n a l y z e the n u m e r i c a l discretization of V o l t e r r a integral e q u a t i o n s with p u r e (constant) d e l a y z > 0, y(t) = 9(t) +
k(t, s, y ( s ) ) d s ,
t e I := [0, T ] ,
(1.1)
where the s o l u t i o n y is subject to the initial c o n d i t i o n y(t) = (b(t),
t e [-z,0),
(1.2)
by c o l l o c a t i o n a n d related c o n t i n u o u s V o l t e r r a - R u n g e - K u t t a m e t h o d s in certain ( n o n s m o o t h ) p o l y n o m i a l spline spaces. It will be a s s u m e d t h a t the given functions, ~b: [ - z , 0 ] -~ R, 9: I -~ R, a n d k: S~ • R ~ R (with S~: I • [ - v , T -- ~]) are (at least) c o n t i n u o u s o n their d o m a i n s ; a d d i tional c o n d i t i o n s will be i m p o s e d later when needed. Existence a n d uniqueness
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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 O D E 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 polynomials 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. C o l l o c a t i o n and Iterated C o l l o c a t i o n Let t n := n h ( n = 0 . . . . . N - 1; t N = T ) define a u n i f o r m partition for I = [0, T], and set HN := {to . . . . . tu}, Io := [to, t l ] , I. := ( t . , t . + l ] (1 _< n < N -- 1). Since the solu-
Continuous Volterra-Runge-Kutta Methods for Integral Equations with Pure Delay
215
tion y of (1.1) has p r i m a r y discontinuities at the points ~ :=/~r, p = 0 . . . . . M (we suppose, without 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 .
r
(2.1)
F o r given integers d > - 1 and m > 1 the piecewise p o l y n o m i a l space S~)+a(HN) is defined by Sin+d(N) ca) H := {u: I ~ R;ull, =: U.
~ 7 ~ m + d ,. U n~v) -l(tn)
=
U(.v)(t,),V
=
0 . . . . . d}
.
(2.2)
The dimension of this vector space is obviously given by dim S~)+a(HN) = N m + d + 1. This shows, in the context of collocation, that the natural choice of d in (2.2) will be governed by the nature of the functional equation to be solved: if the equation under consideration is a (delay) differential or integro-differential equation of order K, then d = K - 1; when solving (delay) integral equations like (1.1) we choose d=-l. F o r given real n u m b e r s {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 collocation solution u ~ S~-I_~(HN) to (1.1) is then given by the equation u(t) = g(t) +
k(t, s, u(s))ds,
t ~ XN,
(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 ) (j = 1 , . . . , m ; n = 0 . . . . . r -
cI)(t),
t = t,,~ ~ X N ,
(2.6)
1), where
q5(0 :=
k(t, s, ~(s))ds,
t - r < 0.
(2.7)
The delay integral ~(t) represents a further potential 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 integration formulas (compare T h e o r e m 3.4 for details). The iterated collocation solution, ui,, corresponding to the collocation solution 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)
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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) Foreachn 0, . . . ~ N - 1, (2.11) defines a unique Un := (Un, x. . . . . . U. . ) T 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, tkl(tnj, 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(tnj ) + Dn(tn.j)
(j = 1 , . . . , m ) ,
(2.14)
Continuous Volterra-Runge-Kutta Methods for Integral Equations with Pure Delay
217
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 ' ~= 1w l k ( t n ' j ' t nl - r - t - c j c l h ' ~ gkk=(1c j c l ) g n - r ' k )
(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
~ i=n-r+l
k(t,,j, ti + vh, d~(tl + vh))dv
(2.16)
(in applications, the integral ~(t) of (2.7) can frequently be found analytically), or by a suitable quadrature approximation 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)), i=n-r+l / = 1
(2.17)
where
~j,t:=cj+(1-cj)cl,
#j,t:=(1-cj)wl
( j , l = 1..... m).
The fully discretized collocation scheme generates a collocation solution fi E S~,~_I(HN), given by
~(t, + vh) = ~ L k ( V ) O n , k ,
tn
+ vh e I,
(0 < n <_ N - 1),
(2.18)
k=l
with ~ := u - f i r 0 in general. Equations {(2.14), (2.17), (2.18)} represent a continuous implicit m-stage VolterraRunge-Kutta (CDVRK) m e t h o d for the delay integral equation (1.1). This class of collocation-based Runge-Kutta methods forms an important subset of the general D V R K methods discussed in [16]. The corresponding discretized version of the iterated collocation equation (2.13) will be referred to as the iterated C D V R K method. If t = t,+ 1 it yields n--r
fiit(t,+l)=g(t,+l)+h
~ i=0
wlk(t,+l,ti+czh, Ui, l),
r<_n<_N-- 1. (2.19)
l=1
F o r 0 < n < r (i.e. for t,+ 1 _< t, = z) we obtain the approximation -1
fii,(t,+l) = g(t,+l) - h
~ ~ wlk(t,+l,ti + clh, d~(tl + czh)). i=n+l-r l=l
(2.20)
We conclude this section with the following obvious but important observation concerning the solution of (1.1) at t = 0. Assume that the given functions g, k and
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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
HN
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)),
t9I
(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
y(t) = Yo +
;
ay(s)ds +
f7
by(s)ds,
t 9 I,
(3.3)
Continuous Volterra-Runge-KuttaMethodsfor Integral Equations with Pure Delay
219
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 of 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~~ v(t, + zh) = v(t,) + h ~ flt(z)v'(t. + c~h),
for the DDE (3.2), and set 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 - a h A ) - l e . v ( t , ) + bh(I - ahA)-IAV~_~,
(3.6)
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N. Baddour and H. Brunner
where the meaning of the vector V. and the matrix 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 product in R". F o r 0 < n < r we have V._r =
([}n -r := ((fi(tn -r, 1)..... (#(tn -r,m) ) T" We n o w turn to the delay integral equation (3.3). On the subinterval I. the colloca-(-1J(/7N) is given by tion solution u e a;,-1
u(t. + zh) = ~ L~(z)g.,~,
U.,l := u(t. + qh).
l=1
The collocation equation defining u is
.1
U.,j = F. + O.-r,j + h ~ 0c " au(t. + zh)dz
(3.8)
(j = 1,...,m),
with F. := Yo +
ftOn au(s)ds + ftOnr bu(s)ds
(3.9)
and
O.-r,j := h CoJbu(t._r + zh)dz.
(3.10)
U. = (I - a h A ) - l e . F. + (I - ahA)-l~g._ r,
(3.11)
This leads to
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. F o r this choice Of Cmit follows from (3.10) that
q?._r = bh" AU._r which in turn implies
U, = (I - a h A ) - l e . u ( t . ) + bh(l - a h A ) - l A U . _ r
(r < n < N - 1).
(3.12)
Since c,. = 1 we m a y 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 . ) Let now 0 < n < r, i.e. - r
+ bh(w,U._,)
(r < n < N -
_< t._~ < 0. Observe first that, due to (3.4),
u(O) = U(to) = Yo -
I ~ b(~(s)ds = ~b(O) = V(to). d-*
1).
(3.13)
Continuous Volterra-Runge-Kutta Methods for Integral Equations with Pure Delay
Also, setting q,-r : = (qn-rm, 1. . . . .
221
q,_,.,,)r, the collocation equation (3.8) yields
U, = (I - a h A ) - 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 - a h A ) - l e . v ( t , ) + bh(1 - a h A ) - l A @ , _ , (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 T h e o r e m 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 counterpart 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 ~bis a polynomial whose degree does not exceed the degree of precision of the m-point interpolatory quadrature 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 interpolatory quadrature formulas (cf. (2.20)), may be written as Uit(tn+l)
=
Yo +
f?
au(s)ds +
fl
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 equation (3.3). Theorem 3.2. Let the assumptions (i), (ii) and the notation of assume that the collocation parameters {c;} are such that Cm < q,_,,j (j = 1.... ,m;O <_ n < r) are approximated by the quadrature formulas Q,_r,j, then the iterated collocation corresponding to its collocation solution u satisfies ui~(t.) = v(t,)
Theorem 3.1 hold, and 1. I f the delay integrals m-point interpolatory solution uit for (3.3)
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,.
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Proof." It follows from the definition o f u , and from (3.9) that F n = ui,(t,). I f 0 < n < r, then the a s s u m p t i o n on the a p p r o x i m a t i o n s 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-
ah(w,U,)
+ bh(w, ~ , _ r ) ,
with (cf. (3.11)) U , = (I - a h A ) - l e . u i d t , ) + bh(l - a h A ) - t A ~ , _ r
(0 < n < r).
If we n o w c o m p a r e these last equations with (3.6) and (3.7) we readily find that the assertion u,(t,+l) = v(t,+l) is true for 0 < n < r, since u/,(0) = Yo = v(0). F o r r _< n < N - 1 the proposition of T h e o r e m 3.2 is a direct consequence of(3.11), which now can be written as U , = (I - a h A ) - l e . ui,(t,) + bh(I - a h A ) - I A U , _ , ; of (3.6), (3.7), and the expression for uidt.+ 1) preceding T h e o r e m 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 condition (1.2). T h e extension of T h e o r e m 3.3 to the nonlinear case (1.1) will be discussed at the end of this section. T h e o r e m 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, if the collocation parameters {cj} are chosen so that the orthogonality conditions Jk := 11 sk ~] (S -- eft& = O, k = 0 . . . . . d - 1; with Jd r 0, j=l do
(3.15)
hold, and if % = 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: m a x ly(t,) - u(t,)l < Ch m+a
(3.16)
l <_n<_N
for some finite constant C. I f d = m, i.e. if the { cj} are the Gauss (-Legendre) points (for which c,. < 1), then there exist constants C and C* so that m a x ly(t,) - u(t.)[ < Ch =,
(3.17)
l <_n<_N
but m a x [y(tn) - Ui,(tn)[ ~ C*h 2'n. l<_n<N
(3.18)
Continuous Volterra-Runge-Kutta Methods for Integral Equations with Pure Delay
223
Here, the iterated collocation solution u~t is determined by (2.8). Remark: F o r "classical" Volterra integral equations (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 ( L o b a t t o points) were derived in [7, 9]. Proof: The collocation equation (2.4) (applied to the linear delay equation (3.14)) m a y be written in the "continuous" 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 u m p s at t = ~, := #~ (# = 0 . . . . . M - 1). The collocation error e := y - u satisfies the delay integral equation
e(t) = 5(0 + D(t),
t 9 I,
(3.20)
where
D(t) :=
If
K(t, s)e(s)ds.
(3.21)
If t 9 [0, r], then the a s s u m p t i o n that the delay integral ~o_~K(t, s)c~(s)ds is k n o w n analytically implies that D(t) = 0 on [0, ~], and this yields e(t) = ~(t) for t 9 [0, z-l. Since the collocation solution u and the iterated collocation solution u, are related by
uit(t) = u(t) + 6(t),
t 9 I,
the error corresponding to u, m a y be written as
ea(t ) = e(t) - ~(t),
t ~ I.
(3.22)
In particular, if t 9 [0, z] ( = [0, t,]) then we have u,(t) -- y(t) (recall the r e m a r k following (2.8)), and hence
eit(t) = 0
for t ~ [0,z].
The following l e m m a on the representation of the solution to the delay integral equation (3.20) contains the key to the p r o o f of T h e o r e m 3.3. L e m m a 3.4. On the interval [~,, ~,+1] (0 <_ # <_ M - 1; M z = T) the solution to the delay integral equation (3.20) is given by # f l -it
e(t) = 6(t) + ~
K,(t,s)6(s)ds,
i=1
where
~
t~T
Ki(t,s ) :=
+(i -l)z
with K l ( t , s ) := K(t,s).
K(t,v)Ki_l(v,s)dv
(2 < i __ #),
(3.23)
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N. Baddour and H. Brunner
Proof of 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)),
e~t(t) = ~
I i -i~
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
=3(t,)+h
f
~ T~,i(t~ + vh)dv
e(tn) = 6(t,) + h ~ i=1
v=O
t~
n-Jr-1
~
~,
i =1
v =0
( ~t=awt~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 --l,v" F,(.") (Note that, by our assumption on the delay integral ~(t), we have E(") L,V = 0 if 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
e(t,)=g(t,)+h
n-it-1
~
~
i=1
v=O
-~,F'(")~ (l_<n_<N).
(3.25)
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) (re [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
Continuous Volterra-Runge-Kutta Methods for Integral Equations with Pure Delay
225
see that the n u m b e r of error terms occurring on the right-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 + Cmh = t, ~ X N, and thus 6(tn) = 0, 1 _< n _ N. This proves (3.16). N o t e that under this constraint on Cmwe 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). F o r 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 attained if the {c~} are the Lobatto points (zeros of s ( s - 1 ) P ' _ l ( 2 s - 1)). The m a x i m u m value of d in the orthogonality condition (3.15), d = m occurs if, and only if, the collocation parameters are the Gauss (-Legendre) points in (0, 1). F o 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 continuous of degree m, vanishing on XN 75 Hs). This yields (3.17). The estimate (3.18) for the iterated Gauss collocation solution follows from (3.22). [] The above proof readily suggests that the local superconvergence results of T h e o r e m 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 quadrature approximation to the delay integrals
r
--
f+o
k(t., s, ~(s))ds
n
r
=h ~
k(t,, t i + vh, 4(t; + vh))dv
i mtl --r
are given by
~(t,) = h ~ i=n-r
wtk(t.,ti, z,q~(ti,l)) (0 < n < r).
(3.26)
1=1
Due to the orthogonality condition (3.15) satisfied by the parameters cz defining the quadrature abscissas, the corresponding quadrature 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.
If 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
N. Baddourand H. Brunner
the property that
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,) - â&#x20AC;˘(t,)l < Ch m. J. <n<_N
The approximation furnished by the iterated C D V R K correspondin 9 to the Gauss points satisfies
method {(2.19), (2.20)}
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).
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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).
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