Convergence analysis of discretization methods for nonlinear first kind Volterra integral equations

Numer. Math. 49, 67-80 (1986) Numerische MathemaUk �9 Springer-Verlag 1986

Convergence Analysis of Discretization Methods for Nonlinear First Kind Volterra Integral Equations

Jennifer Dixon 1, Sean McKee 1 and Rolf Jeltsch 2

1 Oxford University Computing Laboratory, 8-11 Keble Road, Oxford, UK 2 Institut f'tir Geometric und Praktische Mathematik, RWTH Aachen,

Templergraben 55, D-5100 Aachen, Federal Republic of Germany

Summary. An existence and uniqueness result is given for nonlinear Vol- terra integral equations of the first kind. This permits, by means of analogous discrete manipulations, a general convergence analysis for a wide class of discretization methods for nonlinear first kind Volterra integral equations to be presented. A concept of optimal consistency allows two- sided error bounds to be derived.

Subject Classifications: AMS(MOS): 65R20; CR: G1.9

1. Introduction

Many authors, including Brunner [1], Gladwin [3], de Hoog and Weiss [61, van der Houwen and te Riele [7], McKee [9], Taylor [12] and Wolkenfelt [13], have considered discretization methods for solving the linear first kind Volterra integral equation

i G(t,s)y(s)ds=f(t), O<_t<_T. (1.1) o

Few authors, with the notable exception of Gladwin and Jeltsch [3, 4], have considered numerical schemes for the nonlinear Volterra integral equation of the first kind

i G(t, s, y(s)) ds =f(t), 0 _< t _< T. (1.2) 0

In this note a class of discretization methods for the numerical solution of (1.2) is introduced. This class of methods has previously been considered for the linear equation (1.1) by Scott [-11"1, and includes linear multistep methods, reducible quadrature, block-by-block methods and collocation.

It is shown that convergence of these methods when applied to the nonlinear problem may be proved under the same conditions on the quadrature

68 J. D ixon et al.

weights as those required when the methods are applied to the linear problem, although the kernel G(t,s, y) will be required to satisfy additional continuity and differentiability conditions to guarantee the existence of a unique continuous solution of (1.2).

A concept of optimal consistency is introduced and two-sided error bounds, which permit the exact order of convergence of a discretization method to be deduced, are derived.

2. Existence and Uniqueness of a Continuous Solution

An existence proof for the nonlinear first kind Volterra integral equation (1.2) is included for completeness since the proof which shall be given does not appear in the published literature (but see Jeltsch [8]). Moreover, the convergence analysis for the discretization methods will employ discrete manipu- lative steps analogous to those used in this existence result.

Theorem 2.1. Assume

(i) f e C 1 [0, T],

(ii) f(0) = 0, (iii) G(t, s, y)e C(D x ~), D: = {(t, s): 0 < s < t < T},

dG (iv) -~- (t, s, y)e C(D • lR),

gG (v) for t = s is bounded away from zero, that is,

t3~ G-G (t, t, y) =>M>O for all tE[O, T] cy

and all y e R ,

~G (vi) -~- (t,

Then the integral equation

i G(t, s, y(s)) ds =f(t), o

has a unique solution y defined on [0, T] satisfying

(a) G(t, s, y(s)) continuous in D,

and

3G (b) --~ (t, s, y(s)) continuous in D.

Moreover this solution is continuous.

s, y) is (uniformly) Lipschitz continuous with respect to y.

0_<t_< T, (2.1)

Nonlinear First Kind Volterra Integral Equations 69

Proof Differentiating (2.1)

G(t, t,y(t))+ i ~G o ~t- (t, s, y(s)) ds =f'(t) . (2.2)

Then provided (i)-(iv) are satisfied each function y(t) defined on [0, T ] satisfying (a) and (b) is a solution of (2.1) if and only if it is a solution of (2.2).

To prove existence of a solution of (2.2) define a sequence y.(t) as follows:

' 0 G(0, 0, yo(t)): = f ( ) , t

G(t, t, y,+ a (t)): = f ' ( t ) - ! OG ~- (t, s, y.(s)) ds, n>0 .

(2.3)

By (iii) and (v) for each te[0, T] G(t, t,y) is a continuous strictly monotonic function of y defined on N with range ~ . Hence yo(t) is well-defined and continuous on E0, T] (see Courant [2, Vol. I, p. 68]).

Assume inductively that y,(t) is well-defined and continuous on [0, T].

By (i) and (iv) f ' ( t ) - ' i c? G t o ~ ( ,s,y,(s))ds is continuous on [0, T]. It can then

be shown using (iii) and (v) again and then following the proof of the Implicit Function Theorem (see, for example, Courant [2, Vol. II, p. 120]) that y,+ ~(t) is well-defined and continuous on [0, T].

It is necessary to show that the sequence {y,(t)} converges to a limit function and that this limit function is a solution of (2.2) satisfying (a) and (b).

Condition (v) implies

IG(t , t ,y)-G(t , t ,z) l>Mjy-zl>O, foratl te[-0, T]

and all y, zeR, y =t=z. Using this a standard inductive argument yields

ly,+l(t)-y,(t)l< ~. max ly~(s)-yo(s)l, n>O, O<_s<_t

~G where L is the Lipschitz constant for ~ - (t, s, y) with respect to y.

Therefore, assuming without loss of generality that m > n,

lym(t)-y.(t)l < Z lYJ+t(t)-yj(t)I<IlY,-Yoll ~ \ M I j!" j = n j = n

Since the series ?r J 1

j=0 ~M-! f i is convergent it follows that {y,(t)} is a Cauchy

sequence and y.(t) tends uniformly to a limit

Lira y,(t)=y(t), te[0, T]. (2.4)

Since all the y,(t) are continuous y(t) is continuous on [0, T].

70 J. D i x o n et al.

The Lipschitz continuity of 0G dG ~ - ( t , s, y) with respectoG to y then implies that

the functions ~ (t, s, y,(s)) converge uniformly to ~t-(t, s, y(s)). This permits

the interchange of limit and integral in (2.3). Therefore y(t) defined by (2.4) is a continuous solution of (2.2). Since y(t) is continuous (a) and (b) are satisfied.

It remains to demonstrate the uniqueness of the solution. Suppose that y*(t) is any other, not necessarily continuous, solution of (2.1)

satisfying (a) and (b). Then y*(t) cannot be unbounded. Assume the contrary. Then there exists a sequence t,~[0, T] with Limt,=t*~[O,T] and y*(t,) unbounded as n--* oo. "~ ~

Since G(t,s,y*(s)) is continuous in D it must be bounded and hence a subsequence of {G(t*,t,,y*(t,))}~o= converges. Let this subsequence be {G(t*, t,~, y* (t,~))}~= o. Then by (iii)

Lim G(t*, t,k, y*(t,~)) = Lim G(t*, t*, y*(t,~)), k~o9 k~oo

which is finite. Therefore

6(t*, t*, y*(t.~))-6(t*, t*, y(t*)) y*(t,~)-y(t*)

can be made arbitrarily small. This is a contradiction to (v) and hence y*(t) must be bounded. Since y(t), y*(t) are both bounded solutions of (2.1) and hence of (2.2) it

follows, using (iv) and (v) that

ly(t)-y*(t)l < L ily(s)-y*(s)lds, O<t<T. o

By induction, for all n > 1,

ly(t)-y*(OI < ~.v max ly(s)-y*(s)[ O ~ s ~ t

and since y(t), y*(t) are bounded, uniqueness follows by letting n--, oo.

3. A Class of Discretization Methods

In this section a general class of discretization methods for the numerical solution of the nonlinear Volterra integral equation

t

S G(t, s, y(s)) ds =f(t) , t s [0, T], (3.1) 0

is introduced. Let T, h o be given with 0 < h o < T and T/h o = N o, a positive integer. Define

J . '= {h: h= T/N, N~N, N>=No}.


For a given positive integer m independent of h the approximating space X h is then defined to be the vector space given by

X h = (R,,)N+ 1 = {Xh: X" = (Xo, xl , ..., XN) r, x i = (Xi, , . . . . Xlm)e R " , 0 <= i <__ N}

with norm rlxhll~:= max Ixil,,

O<i<=N where

IXi[m:= max ]xi~ I. l< .~<ln

The concept of block isoclinal and differentiation matrices will be required (see, for example, McKee [10] and Scott [11]).

Definition 3.1. An ( N + l ) m x ( N + l ) m matrix A h is said to be a bordered m- block isoclinal matrix if for some positive integer q, independent of h, A h may be written in the form

Ah= t lqm a'o

Ai

A'N_ q

a o

A1 Ao : ~

A u _ q . . . A 1

\ 0

, ( 3 . 2 )

Ao

where Iqm is the qm x qm identity matrix, A'i, O < i < _ N - q , are m x qm matrices, and Ai, O < i < N - q , are m x m matrices with A o non-singular. Without loss of generality it will be assumed that A 0 is the m x m identity matrix.

Note that if q = 0 A h takes the form

Ah= A 1 Ao

A N . . . A 1 " Ao /

and is termed an m-block isoclinal matrix.

(3.2')

It will be convenient to use the notat ion [A~[,~ to denote the infinity norm of the m x m matrix A i.

Definition 3.2. A n ( N + 1)m x ( N + 1)m bordered m-block isoclinal matrix A h is said to have bandwidth ( z + l ) m if there exists z e N , independent of h, such that

Ai=Om, for i>=z+l. (3.3)

The border is said to have length km if there exists k e N , independent of h, such that A'~ is the m x qm null matrix for i>k.


Definition 3.3. A bordered m-block isoclinal matrix A h with bandwidth (z + 1)m and border length km is said to be an m-block z-differentiation matrix if

that is,

L Aje=O where e = ( 1 , 1 , . . . , 1 ) r, j = 0

(3.4)

x zm matrix given by

t~ m Im 0 )

Om lm GA = . . . . . (3.5)

Om Im

\ -A t . . - A #

The class of methods for the numerical solution of (3.1) which is to be considered may now be introduced.

Let U h denote an ( N + l ) m x ( N + l ) m bordered m-block isoclinal matrix. Assume that there exists an ( N + l ) m x ( N + l ) m m-block z-differentiation matrix D h such that D h U h is a bordered banded m-block isoclinal matrix E h with bandwidth (z+ 1)m. Consider the class of methods for (3.1) expressible in the form

h Oh(yh) = f h, (3.6) where

fh = (h ~o . . . . , h Yq_ l, fq . . . . . fN) r eXh,

with ~ 0 , ~ l , . . . , ~ q _ l e l l m vectors of precomputed starting values and f = (f(ti~))eN', q <___ i < N, and

[Yi?, l<=a<=m, O<=i<=q-1,

(l~h(yh))itr-~- I L ~ (gij)~v G(tia, tjv, Yjv), l~-~~ q<i<=N, I j = o v = l

where Uu=Ui_ s for i-j>=O, j>=q, are the entries of the bordered m-block isoclinal matrix U h. Here Yi~ denotes an approximation to y(t~) where {t~}e[0, T], 1 <_a<_m, O<=i<N, is a given set of points.

Note that if q --0 no starting values are required,

f h = ( f o , f 1, ... , f N ) r e x h, and

i

(O"(yh))io= ~ ~ (Uij)~vG(ti~,tj~,yj~), l<~<_m, O<_i<-N. j=O v = l

L ~ (Ai)~v=O, l_<~_<m, j = O v = l

where (Aj)~v denotes the (a, v)th element of the m x m matrix As.

Definition 3.4. The block companion matrix G A for a bordered m-block isoclinal matrix A h (with A o =I~) with bandwidth (z+ 1)m is defined to be the zm


To illustrate the above class of numerical methods consider the following so-called block-by-block method introduced by de Hoog and Weiss [6].

Let 0 < q l < r / 2 < . . . < r i m = l

be a given fixed set of points and define

ti~=(i+~l~)h , l<_a<m, O<_i<n-1,

where (n - 1) h = T. The approximations {Yi~} to {y(ti~)} are determined from the equations

h ~ ~ a~G(ti~,tj~,yj~)+h a~G(ti~,t,~,yi~)=f(t~ ), j=O v - 1 v = l

q* where a,~= ~ Lv(s)ds, l <a, v<m,

0

av=a,.~, 1 <v<m,

l<a<m, O<i<n-1,

(3.7)

To express this method in the form (3.6) set n - 1 = N, q = 0 and let U h be the (N+ 1)m • (N+ 1)m m-block isoclinal matrix of the form (3.2') with

a l l a12 . . . a i m \

g o ~ a21. a22 ''- a2m). ,

\ a l a2 am ,, mxm

(i Ui a l a2 m ,

1 a2 am/mxm

l <_i< N.

(3.8)

Noting the block repetition in U h choose T= 1 and take D h to be the m-block r-differentiation matrix with no border and Do=l,.,

D1 = 0 . . . 0 11

0 ... 0 il •

Then E h= D h U h is an m-block isoclinal matrix with

E0= U0, El=O,. , l < i<N.

74 J. Dixon et al.

It follows that the block-by-block method (3.7) of de Hoog and Weiss may be included in the class (3.6) of numerical methods for the nonlinear Volterra equation (3.1).

Further examples illustrating how appropriate U h and D h matrices may be constructed for other numerical methods may be found in Scott [11]. The class of discretizations of (3.1) which are expressible in the form (3.6) is wide and includes, in addition to block-by-block methods, direct linear multistep methods (McKee [9]), generalized linear multistep methods (van der Houwen and te Riele [7]), reducible quadrature (Wolkenfelt [13]) and collocation (Brunner [1]).

4. A Stability Condition

In this section a stability condition which will be required to be satisfied by the matrices U h and D h involved in the discretization (3.6) will be derived. This condition, together with consistency, will be shown in Sect. 5 to be sufficient for the convergence of a numerical method belonging to the class (3.6).

Using the identity

G(ti~, tj~, yj~) = G(tj~, tj~, yj~) + h(G(ti~, tj~, Y j r ) - G(tjv, tj~, yj~))/h,

~h(yh) is decomposed into the sum

~kh (yh) = U h G h (yh) + h V h (yh),

where Gh(y h) is a vector with entries yj~ for l < v < _ m , O < = j < q - 1 , and G(tj~, tj~, Yi~) for 1 <-v<m, q < j < N .

Equation (3.6) may then be rewritten as

h U h Gh(y h) = f h _ h E Vh(yh),

and premultiplying by the inverse (h uh) - 1

G h (yh) = (h U h ) - l f h _ h ( U h) -1 V h (yh). (4.1)

Equation (4.1) is the discrete analogue of Eq. (2.2). Let [yh[m denote the vector

lYhl~ = (lYolm, lY l I . . . . . . lYNlm)T ~ x N + 1

Condition (v) of the existence Theorem 2.1 then implies

([ G n (yh) _ G n ( zh) lm)i ~ M' ([ yh _ Znlm)i, (4.2)

for each i, O < i < N , and all yh, zn~X n, y h ~ z h ' where M ' = m i n (1, M). Using En= D n U h,

I(( v ~) - ~ ( v n (y~) - v n (zn))), I .

= I((Eh)- 1 (O n Vn(yh) _ On Vn(zh)))iim

N o n l i n e a r F i r s t K i n d V o l t e r r a I n t e g r a l E q u a t i o n s 75

i =< ~ ]((Eh) - 1),~1 m I(D h Vh(y h)-D h Vh(zh))j]m

j = 0 k

=< max ~ I((Eh) - 1)kjl,, max I(D h Vh(y h) -- D h Vh(zh))j[,, k j=O j < i

k i <max ~ I((Eh) - 1)kdlmE ~, I(yh--zh)jl,,,

k j = 0 j = 0

for some constant E, independent of h, using condition (vi) of Theorem 2.1 and (3.4).

Hence i ([(uh) - l(Vh(yh)-- Vn(zh))l,,)i< C ~, (ly h - zhl,,)~, (4.3)

j = o

for some constant C, independent of h, provided there exists C1, independent of h, such that

i

m a x ~, ] ( ( E h ) - 1)i j] m <_ C 1. ( 4 . 4 a ) i j = 0

It can be shown (see McKee [10]) that if m = l (4.4a) is satisfied if and only if the eigenvalues of the companion matrix G E for En lie strictly inside the unit circle.

Using the further identity

G(ti, , tj~, yj~)- G(tj,, tj,, y3~)

= ( t . , - tj~) ~ - (tj~, t~., yj~) + h G(ti~, t~, y~),

_G(tj , , tj,, yjv)_h ( ~ ) ~BG (tjv, tjv, yj~))/h),

to decompose Vh(y h) into the matrix sum

V h (yh) = uh(a) G*(1) (yh) + h W h (yh)

where U h(1) is the matrix such that ~/h(yh)= Uh(l)yh when G(t, s, y)=l t -sJy , and Gh(1)(y h) contains the derivatives c~G(tj~, tj,, y~)/~3t, it can be shown by employ-

2 2 ing a similar argument that, provided ~ G(t, s, y)/c3t is assumed Lipschitz in y, (4.3) is also satisfied if for some constant C 2, independent of h,

max I((D h Eh) - 1)i j[ m ~ C 2 . (4.4b) l, 3

The matrix DhE h is a banded bordered m-block isoclinal matrix with bandwidth (2z+ 1)m. Denote the block companion matrix for DhE h by GOE. It can be deduced from the results of McKee [10] (see also Scott [11]) that (4.4b) holds provided the eigenvalues of GoE lie on or within the unit circle with those on the unit circle simple eigenvalues.

The discretization (3.6) will be said to be stable if either (4.4a) or (4.4b) holds.


The stability conditions (4.4a), (4.4b) depend only upon the quadrature weights and not upon the kernel G(t,s,y). Consequently the same stability conditions apply to both linear and nonlinear Volterra integral equations of the first kind (compare Scott [111; see also McKee [91).

For the block-by-block method (3.7) the condition (4.4a) is satisfied since in this case

i

max ~ }((Eh) - 1)ij],, = [(U0)- 11,,, i j = 0

with U o given by (3.8). The a~v's are independent of h, therefore it follows that

I(Co)-~l~< C1,

for some C~, independent of h. Consequently the block-by-block method is stable.

5. Convergence

In this section convergence of the discretization method (3.6) is proved assuming the stability condition (4.4) is satisfied and the method is consistent.

Let rh: C[0, T 1 ~ X h be the usual grid restriction operator, that is,

(rhy(t)) i~=y(tJ , l<tr<_m, O<i<_N. (5.1)

A concept of optimal consistency may then be defined as follows.

Definition 5.1. Let yeC[O, T 1 be the unique solution of the integral equation (3.1). The discretization (3.6), which may be rewritten in the form (4.1), is said to be optimally consistent of order s > l if there exist constants Ci~, l < a < m , O < i < N , some of which (but not all) may be zero and all of which are bounded independently of h, such that

l(oh)ic,[: = [Gh(rhy)-- (h Ua) - l f h + h(Uh)- 1 (Vh(rhy))io [ = Ci~h s + O(M+ 1). (5.2)

Optimal consistency permits two-sided bounds for the error rhy--y h to be derived.

Theorem 5.1. Let yE C[0, T] be the unique solution of the integral equation (3.1) and let yh~Xa be defined by (3.6). I f the discretization (3.6) is optimally consistent of order s> 1 and if the stability condition (4.4) holds then for all h e J sufficiently small

CL 110hi[ o~ =< [[rhY -Yh]l o0 <= Cv II 0hll ~ (5.3)

convergence is of order exactly s. For some non-zero constants C L, C v, independent of h,

Proof. Upper bound. By Definition 5.1 and (4.1)

G h (r h y) _ G h (yh) = _ h ( U h) - 1 ( V h (r h y) _ V h (yh)) + O h" (5.4)


Using (4.2), (4.3) together with the stability condition (4.4),

i m'( lrhy--yhl , , ) i~hC ~ ([rhy--yhlm)j+(lohlm)i, O<i<--N.

j = o

Let Xi=([rhy--yh[m)i and 6= ]lOh[[~o. For h sufficiently small there exists C1, Cz, independent of h, such that

i--1

xi~hC1 2 xJ~C2t~, O<i<_N. j = 0

Invoking the standard discrete Gronwall inequality yields (see, for example, Henrici [5, p. 244])

x i~CzcSexp (Cx ih ) , O<_i<_N, which implies

Ilrh y-- yhN ~ <= Cu Ilohll ~.

The upper bound implies convergence of order at least s.

Lower bound. Rearranging (5.4) and taking norms

(10hlm)i <= (I Gh( rh Y) -- Gh(yh)lm)i + h(l(Uh) - ~ (Vh(r h y) - Vh(yh))l,.),

for O<_i<N.

Since O G (t, s, y) exists there exists L >0 such that (?y

([Gh(yh)--Gh(zh)lm)i~L(lyh--Zhlm)i , fo r all i, O<_i<_N, (5.5)

and all yh, z h E X h. Using (5.5) and (4.3) together with (4.4)

i ([ OhIm)i ~ L(I rh Y - - yh [m)i + h C ~ ([ r h y - yhl,.)j

j =0

~(L+ CC)[[rhy--yh[[ ~, where C = Sh.

This holds for each i, 0 < i < N; consequently

II0hl[| < ( L + CC) l[rh y-- yhl[~,

and this yields the lower bound. It is straightforward to check from the definition of optimal consistency

that the lower bound implies convergence of order greater than s is not possible. Convergence of order exactly s may now be deduced.

The optimal consistency error may be related to the quadrature error, and it may be convenient to do this as follows to verify the exact order of convergence of a numerical method for (1.2).


Define (Th)i =J'y(ti~)--yi, , l<0_<m, O < i < q - 1 ,

(5.6) ) (h~kh(rhy)--fh)i~, l < 0 _ < m , q<i<_N.

Definition 5.2. The starting values are said to be of order s if for some constants Ci,, 1 _< a < m, 0 < i < q - 1, bounded independently of h,

Yi~ - y(ti,) = Ci, hs + O( hs+ x). (5.7)

Definition 5.3. The quadra ture error is said to be of order s if for some constants Ci,, 1 <_ a <_ m, q <=i< N, some of which (but not all) may be zero, and all of which are bounded independently of h,

(Th)i, = C~ h s + O(h ~+ 1). (5.8)

The quadra ture error is said to be of order s with a ( p + 1)-term asymptot ic expansion if, for l_<a_<m, q<=i<=N, there exist functions Ck(t)eck[o,T], 0 <= k <= p, ( independent of h) with %(0) = 0 (cp(t) ~ 0) such that

p

(Th)~ = h ~ ~ hZ(r h c o _ t(t))i~ + O (h s+ p+ i). (5.9) l = O

F r o m Section (5.2) and (5.6), for 1 < o-_< m, q < i <= N,

( r h ) i . = (h U h oh)i~,, and for l < a < m , O < i < q - 1 ,

(Th),, = (0h)i, = Y (ti ~) 2 Y, ,. (5.10)

For i>=q i

I(Th)i[m <= h max ~ I(uh)ij],, max l(oh)j[m i j = 0 s

<=M llOhlI~o,

for some M >0 , independent of h, since the elements of U h are independent of h.

It follows f rom (5.3) that

CL M - 1 II Thll o0 < CL II 0all ~ < [LrhY --Yh[I ~o. (5.11)

Using (5.7), (5.8) it may be shown that (5.11) implies the order of convergence cannot exceed s.

To obtain an upper bound for II0hll | t w o cases will be considered separate- ly.

(a) Assume condit ion (4.4a) is satisfied, and that the quadra ture error is of order s with a 2- term asympto t ic expansion. For i > q

l(Oh)ilm=h- i [((Uh )- i Th)ilm

= h - 1 i((Eh)- 1 O h Th)il,.


q-1

j=O i

+ h - 1 max ~ I((En) - 1)ijl m max I(D h Th)jlm. i j = 0 q<j<i

Using the asymptotic expansion (5.9) to consider I(DhTh)jl, , for j>q , and recalling that D h is a differentiation matrix satisfying (3.4), it follows that, provided the starting values are of order s,

](Oh)il,,<CihS+O(hS+l), i>q.

Combined with (5.10) this implies

jlOh]l | < Cvh~ + O(hS+ ~), (5.12)

for some C v > 0, independent of h. (b) Assume (4.4b) is satisfied, and that the quadrature error is of order s

with a 3-term asymptotic expansion. For i>q

I(0h)il,, = h - ' [((O h Eh) - 1 (Dh)2 Th)il,, q--1

< h - ' ~ ](Zh)jlm j = 0

i

+ h - 1 max I((D h Eh) - 1)ij[,, ~, [((oh) 2 Th)jl,n. J j=q

It can again be shown using (5.9), with p = 2 , and (3.4) that as long as the starting values are of order s the bound (5.12) holds.

This leads to the following corollary to Theorem 5.1.

Corollary 5.1. Let y~C[O, T] and yh~Xh be as in Theorem 5.1. Assume the stability condition (4.4a) ((4.4b)) is satisfied. I f the quadrature error is of order s with a 2-term (3-term) asymptotic expansion and the starting values are of order s then for all h~J sufficiently small convergence is of order exactly s.

For the block-by-block method (3.7) the quadrature error can be shown to be of order m with a 2-term asymptotic expansion. Therefore, since condit ion (4.4a) is satisfied and the scheme requires no starting values, it follows from Corol lary 5.1 that the method is convergent of order exactly m.

R e f e r e n c e s

1. Brunner, H.: The approximate solution of integral equations by projection methods based on collocation. Math. Comput. No. 1/78. Dept. Math., University of Trondheim 1978

2. Courant, R.: Differential and Integral Calculus. Vol. I and II. London: Blackie and Sons 1945 3. Gladwin, C.H.: Numerical Solution of Volterra Equations of the First Kind. Ph.D. Thesis,

Dalhousie University 1975 4. Gladwin, C.H., Jeltsch, R.: Stability of quadrature rules for first kind Volterra integral equa-

tions. BIT 14, 144-151 (1974)

80 J. Dixon et al.

5. Henrici, P.: Discrete Variable Methods in Ordinary Differential Equations. London: John Wiley 1962

6. de Hoog, F., Weiss, R.: On the numerical solution of Volterra integral equations of the first kind. Numer. Math. 21, 22-32 (1973)

7. van der Houwen, P.J., te Riele, H.: Linear multistep methods for Volterra integral and integro- differential equations. Report NW 151/83. Amsterdam: Mathematisch Centrum 1983

8. Jettsch, R.: Existence and uniqueness of a continuous solution of a nonlinear first kind Volterra integral equation. Notices Am. Math. Soc. 23, A580 (1973)

9. McKee, S.: Best convergence rates of linear multistep methods for Volterra first kind equations. Computing 21, 343-358 (1979)

10. McKee, S.: Discretization methods and block isoclinal matrices. IMA J. Numer. Anal. 3, 467- 491 (1983)

11. Scott, J.A. (n6e Dixon): A Unified Analysis of Discretization Methods. D. Phil. Thesis, University of Oxford 1984

12. Taylor, P.J.: The solution of Volterra integral equations of the first kind using inverted differentiation formulae. BIT 16, 416-425 (1976)

13. Wolkenfelt, P.H.M.: Reducible quadrature methods for Volterra integral equations of the first kind. BIT 21, 232-241 (1981)

Received February 14, 1985/September 10, 1985

Documents

Convergence analysis of discretization methods for nonlinear first kind Volterra integral equations