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MATHEMATICS OF COMPUTATION, VOLUME 27, NUMBER 124, OCTOBER 1973

The Order of Numerical Methods for Ordinary

Differential Equations

By J. C. Butcher

Abstract. For a general class of methods, which includes linear multistep and Runge-

Kutta methods as special cases, a concept of order relative to a given starting procedure is

defined and an order of convergence theorem is proved. The definition is given an algebraic

interpretation and illustrated by the derivation of a particular fourth-order method.

1. Introduction. Detailed theories have been published for the study of error

propagation in linear multistep methods on the one hand ([1], [2], [3]) and for Runge-

Kutta methods on the other hand ([2], [4]). While it is possible to modify these theories

to include various methods which fall between these extreme types ([5], [6], [7], [8], [9]),

it is of interest to analyse the behaviour of error propagation for a class of method

of sufficient generality to include all these special cases in a natural and straight-

forward way. In fact, we will consider the type of general method formulated by

the author [10].In attempting to find a suitable meaning for the concept of "order" which general-

izes the standard definitions in the well-known special cases, a rather surprising fact

arises. This is that the definition of order which seems most natural for this general

type of method does not necessarily, in the standard cases, assign a value to the order

which coincides with that given by traditional definitions. In fact, this new concept

coincides in the case of Runge-Kutta methods with "effective order" [11] so that,

for example, there exist methods of order 5 in the sense of this paper but only 4 in

the usual sense.

In the present paper, some of the ideas and methods introduced in [12] will be

made use of to show how the analytic definition of order may be expressed in algebraic

terms. To illustrate these developments an example is given of a method whose

derivation is based on this new meaning of order.

2. Notation. Let V be an iV-dimensional real vector space and let A, B be the

matrices of two linear operators V —> V with components a^, bn respectively (i, /' =

1, • • • , TV). As in [10], we consider methods (A, B) characterized by such a pair of

matrices.

Let X be a finite-dimensional normed real vector space and let/: X —> X satisfy a

Lipschitz condition with constant L. We shall consider the differential equation

y'(x) = f(y(x)). In computing solutions using (A, B), we have, at the end of step number

n, a collection of N approximations y\n), y2n), • • • , y^ which are computed from

Received November 16, 1972.

AMS (MOS) subject classifications (1970). Primary 65L05; Secondary 34A50.Key words and phrases. Initial-value problems, multistep methods, Runge-Kutta methods, order,

convergence.

Copyright © 1973, American Mathematical Society

793

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

794 J. C. BUTCHER

corresponding approximations to the solution after n — 1 steps. We write h as the

stepsize. That is, we regard ■y,(n) (/' = 1,2, • • • , N) as approximations at points h

further ahead than the approximations y'""1' (i = 1, 2, • • • , N). The formula for the

computation of ■y,.n) (i = 1, 2, • ■ • , N) is

(2.1) ¿b) = ¿ aii/ru + h e mo;*0).¿-i ¿-i

To represent this equation more compactly, we make use of the vector space V(X)

defined as the direct product of V and X.1fv= v, © v2 © • • • © vN G V(X) where

Pi, v2, ■ ■ ■ , vN (E X, then we define ||u|| = maxIS,s¡v ||p,-|| . Given a linear operator

A : V -+ V, we define [/I] : V(X) -> K(Z) by

[¿](»i ©»« © • • ■ ®Vn) = Z auVi © E «2,»i ©■•■©£ «iri»i.i=i i=i í=i

We thus write (2.1) in the more compact form used in [10]

(2.2) Y{n) = [¿]y(n-1) + h[B]F(Yln))

where Tu> = y[n) © ?<"' © • • • © y™ and the function F : V(X) -» F(Z) is defined by

F(v,®v2® ■■■ ® vN) = /(Dl) © f(v2) © ■ • • © f(vN).

Throughout this paper, we will assume that the method (A, B) is stable and con-

sistent. That is, sup {\\An\\ : n = 1, 2, ■ • • j < co. As = s and A£ + Bs = £ + s

where s G V is defined by st. = 1 (i = 1, 2, ■ • • , N) and £ is some member of V.

3. Definitions of Order. In framing our definition, we are motivated by a number

of considerations. In the first place, we might consider the obvious generalization of

the definition of order for multistep methods. This would lead to the following

definition.Definition 3.1. (A, B) is of MS-order q if there is | G V such that, for k =

0, 1, • • • , q and i — I, 2, ■ • • , N,

£= ¿ «„■(?, - D" + k ¿MÎ"1-¿-i í -1

Note that this definition in the case g = 1 corresponds exactly to the definition of

consistency.

If we applied Definition 3.1 as it stands to Runge-Kutta methods expressed in

the form (A, E), we would conclude that every explicit Runge-Kutta method has

AfS-order no greater than 1. This is because the first off-step approximation found

in a Runge-Kutta step uses only the approximation at the beginning of the step and,

in fact, computes this new value by an Euler step. Later stages may make use of this

first-order approximation to construct other approximations at various off-step

points,but the final value given at the end of the step is designed to allow the effects

of the lower order errors to cancel out. Of course, only this final approximation is

used by subsequent steps. These considerations lead to the following definition.

Definition 3.2. (A, B) is of RK-typz if there is / G {1, 2, • • • , N\ such that for

all t? G Fand / G {1, 2, • • • , N\ it holds that (Av){ = v,. In this case, we can reframe

the method as an equivalent Runge-Kutta method and we define the RK-orâzv as

the order of that method.


THE ORDER OF NUMERICAL METHODS 795

This definition, besides being applicable only to a limited class of methods, suffers

from the disadvantage that it is not symmetrical in the members of {1, 2, • • • , N].

One way of combining features of these two definitions is to note that multistep

methods in general require starting procedures. In the case of linear multistep methods,

for example, Runge-Kutta methods are often used to supply the starting vectors

y[0), ' ' " > y^- Furthermore, in the case of methods of RK-type, the internal computa-

tions in step n could be regarded as being themselves computed from Runge-Kutta

methods starting from Y\n~u.

Let r denote a Runge-Kutta method. For a fixed function /, we interpret ras a

function on the real numbers which for stepsize h gives a mapping r(h) : X —» X

which takes z G X into the value computed using the Runge-Kutta method r from

starting value z. If we have a collection of Runge-Kutta methods r„ r2, ■ ■ ■ , rN, then

we can construct a method which gives N results f,,f2, ■ ■ ■ , fN given by

N N

(3.1) -r,(h)(z) = £ a,jri(h)(z) + A^ 6„«ft(A)G0).i=i ;-i

Note that r„ f2, • ■ ■ , fN are Runge-Kutta methods also, so that we can interpret a

method (A, B) as a mapping on the class of N-tuples of Runge-Kutta methods to

this same class.

For given z£I, let Ph(z) denote the value of y(h) such that y satisfies y(0) = z,

y'(£) = f(y(0) for all £ in the open interval {£ : £(A — £) > 0} and such that y is

continuous on the closed interval { £ : £(A — £) ^ 0J.

Definition 3.3. (A, B) is of order q relative to r,, r2, ■ ■ ■ , rN if for all/with con-

tinuous partial derivatives of order q in a neighbourhood of z, it holds that

(3.2) \\ñ(h)(z) - rt(h)(Ph(z))\\ = 0(\h\«+1)

as |A|->0fori = 1, 2, • ■ • , N.

Definition 3.4. (A,B) is of order q if there exist Runge-Kutta methods

r„ r2, • • • , rN such that (A, B) is of order q relative to r„ r2, ■ ■ ■ , rM.

To make practical use of this definition, we use algorithms consisting of three

parts: (a) a starting procedure, (b) a continuing procedure, (c) a finishing procedure.

Suppose the differential equation y'(£) = f(y(0) is given together with the value

of X*o). It is required to compute y(x,) where x, = xa + vh. The three parts of the

method are as follows: (a) we compute T<0) by y\0) = r¡(h)(y(x0)) for i = 1, 2, • • • , N,

(b) we use the method (A, B) to compute in turn Yl ' ', Y( 2 ', • • • , Yl " ', (c) we choose

one of y\"\ y2'\ ■ ■ ■ , y¡¡\ say y\'\ and compute an approximation to y(xv) by the

Runge-Kutta method (in general an implicit method) r such that r(h) = r^h)'1 (this

inverse function exists for sufficiently small h) operating on y1/'.

There are, of course, certain generalizations possible. For example, the finishing

formula may be chosen in such a way that it not only reverses the effect of the per-

turbation to the initial value introduced by the starting formula but also carries the

solution forward for a further step or part of a step. It is also possible, in some cases, to

find simple finishing formulae which make use of more than one of y['\ y2\ ■ ■ ■ , y#\

4. Rate of Convergence. We consider a sequence of approximations to y(x)

given by the algorithm described in the previous section. That is, for v = v0, v0 + 1,

• • • , we use a value h = (x — xa)/v and apply that procedure. We obtain an approxi-


796 J. C. BUTCHER

mation for each such v and we are interested in the behaviour of the error in these

approximations as h —» 0 (that is, as v —» °o ).

Denote by F¿n) the value of Y(n) computed with stepsize h = (x — x0)/v and

write Y[n) = j£î©j£i© ■ • • © y<%. The initial values are given by y™ = r,(h)(y(xo))

for / = 1,2, • • • , N and the final approximation is given by j>„ = r^h)~l(y['/j).

Theorem 4.1. If the method (A, B) is of order q relative to r,,r2, ■ ■ ■ , rN and if

all partial derivatives off of order up to q are continuous and bounded in an open set

in X containing Y = fX£) : (k — xa\x — £) ^ 0}, there exist constants C, v0 such

that, for v ^ v0,

\\9, - y(x)\\ =g Cv~\

Proof. Let //?> = i£i © nl% © ••• © %(% G V(X) be defined by,w = r,(hXy(xo + nh)) for i = 1, 2, • • • , N. Also define Z[n) = J5Ç" - y[n) and

Wln) = F(H[n)) - F(Yln)) so that \\W™\\ ^ L\\ Z<"'|| . Our first task is toestimate

„ „ Eln) = Zln) - [A]Zl'"u - h[B]rVln)(4.1)

= Hln) - [A]H'ru - h[B]F(H[n)).

We write £<"> = ej"j © <?<"> © • • • © e,("¿ and we find that

If N

<£!■ = ri(h)(y(x„ + nh)) - ¿2 aiir,(h)(y(xo + (n - 1)A)) - h ¿Z buf(rj(h)(x0 + nh))i=i 1=1

= \fi(h)(y(xo + (n - í)h)) - ¿ a,iri(h)(y(x„ + (n - 1)A))V 1-1

- A ¿ büKrWWxo + (n- l)h)))\¡-i )

+ \r,(h)(y(xo + nh)) - rt(h)(y(x0 + (n - 1)A))

- A ¿ b{i(f(ri(hXy(Xo + nh))) - f(fi(h)(y(xo + (n - l)h))))\i-i >

= r,(h)(Ph(y(xo + (n - l)h))) ~ ñ(h)(y(x0 + (« - 1)A))

N

+ h ¿2 bti(f(rj(h)(y(xo + (n - 1)A))) - )(r,(h)(Ph(y(xH + (n - 1)A)))).i = i

Because of the boundedness of the partial derivatives of/at all points sufficiently

close to the points in Y, we can, by requiring h to be sufficiently small, obtain a

uniform bound C, |A|0+1 for the difference occurring in (3.2) for all z in Y and all

i = \,2, ■ ■ ■ ,N. We therefore have, for some v, and all v 3: v,, the following bound

\\eln)\\ Ú C, |A|S+1 (1 + |A| £ ||*||) ¡S C2 H°+1

where C2 = C, (1 + L \\B\\-\x - x0\/Vl). Hence, ||^n)|| á C2 |A|"+1 for m ^ p..

From (4.1) and the fact that Z(0> = 0, we obtain the equation

ZÍ" - A ¿ [if-'AlIPÍ" + ¿ [A~']Éin.



Let a, ß be chosen so that H^H g a, ||4\B|| g ß for all i = 0,1,2, ■■■ .We see that||Z<n,|| ^ íí"'where e^0' = 0 and

¿n) = |A| £|8 ¿4" + «C2a |A|«+1.i-i

Hence,

(n) (n-1) l.t r/i (tl) I /-r I j- I <Z "»-1

e„ — e, = I«I ¿pe, + C2a \h\

If L/3 = 0, this leads to the equation

(4.2) Hz!" 11 è C3 \h\\

where C3 = C2a \x — x0\ for all v è v2 — v,. Otherwise, we have

(1 - |A| Lß)(4n) + |A|« C2a/Lß) - (e^ + |A|« C2a/Li3).

In this case, we choose v2 such that, if h = (x — xa)/v2, then |A| Lß < 1 and such

that v2 ̂ vj. We then find

««•' + |A|« C2a/Lß - (1 - |A| £0rV + |A|' C2a/Li3)

so that

ei" - ((1 - |A| £0)"' - 1) |*|« C2a/Lß.

In this case, we choose

r - Ç* Ln / I* - Xo\ Lß \ \Cs ~ Lß VeXp Vl - Lß \x - Xo\/v2) ~ 7

so that (4.2) holds in this case as well with v ^ v2. Having obtained this bound on

l!^"' - I?'11 . we must still estimate

9, - y(x) - rrVtrWÙ - rl(h)-1(ri(h)(y(x)))

= rI(h)'\yï''i) - rWWjY

Let the Runge-Kutta method o(A)""1 be one that computes i+l approximations

z0, z,, • • • , z, from an initial value z by the formula

a

(4.3) Zi=z + h¿2 c,,/(z,), i = 0, 1, • • , s,)-0

and gives a result r/(A)"1(z) = z,. Since r^h)'1 is, in general, implicit, we assume that

|A| L max¿ XXo ko I < 1to guarantee existence and uniqueness of z0, z,, ■ ■ ■ , z,. This

is achieved by requiring that v è£ v3, where |x — x0\ L max.- 52*,0 k.,1/»^ = k say,

satisfies k < 1. We now define v0 as the greater of v2 and ¡^ and, from now on, we

assume that ^ v0. If z G -cYand z0, z., • ■ • , z, are computed from z in the same way

as z0, z1( • • • , z, are computed from z, we will now estimate r¡(h)~l(z) — r¡(h)~l(z).

From (4.3) and the corresponding equation for z0, z,, ■ • • , 2„ we have

Zi - Zi = z - z + h ¿2 c,,.(/(z,) - /(z,))1-0


798 J. C. BUTCHER

so that if max; ||z, — z,|| = f then f ^ ||z — z|| + fcf. Hence

lli-K*)"1« - ^(A)_1(f)|| Í5 (1 - kT1 \\z - z\\

so that if we write C, = C3/(\ — k), and substitute z = ;/',, z = r)\'\ and make use

of the inequality \\y1;} - vl'/M Ú ||5?> - Hl'>\\ , then we'find

(4.4) \\y, - y(x)\\ £ Ct |A|«.

To obtain the result of the theorem, we write C = C4 \x — Xop and substitute h =

(x - *„)/« in (4.4). D

5. Algebraic Criterion for Order. In this section, we make use of the terminology

of [12]. The basic idea in that paper is that there is a group G whose elements may be

represented by real-valued functions on the set T of (rooted) trees and such that every

member of a certain class of method (which includes all Runge-Kutta methods)

can be characterised by a member of the group. In particular, p is the group element

associated with the exact increment function for a differential equation integrated

through a unit step. If m„ m2 are two methods with group elements g,, g2, then the

method formed by successive application of m, and m2 has corresponding group

element g,g2. If g is a group element and A is a real number, then g'h) denotes the

group element such that for any / G T with order r(t), g'h)(t) = hrU)g(t). If g corre-

sponds to a method m then glh) corresponds to a method m' say, which is related

to m in such a way that the increment function for m' applied to the differential equa-

tion y' = f o y is identical to the increment function for m applied to the equation

y' = hfoy. In particular, if A is an integer, then /?'*' = ph (where the last exponent

is the group-theoretic power).

We now introduce some further terminology.

Definition 5.1. The derivative g' of g G G is defined by the recurrence

g'(r) = 1,

g'(tu) = g'(t)g(u), t,uE T.

From this point onward, we regard G as a linear space over the real field R. If g„

g2 G G, c„ c2 G R then c¡g, + c2g2 is defined by

(c,g, + c2g2)(t) = c,g,(t) + c2g2(t)

for all t G T.

Lemma 5.2. For a, ß, y, ß,, ß2, ■ ■ ■ , ßn G G ande, c,, c2, • ■ • , cn G R the following

relations hold:

(5.1) a(cß) = caß + (1 - c)a,

(5.2) a(ß -\- y) = aß -\- ay — a,

(5.3) a ¿Zcißi = ¿Zc,aßi + (l - £ c, kt-l t-l V ¿_! /

(5.4) aß' = a + (aß)'.

Proof To prove (5.1), we make use of the function X introduced in [12]. We

have, for t G T,



(a(cß))(t) = a(t) + X(a, t)(cß)

= a(t) + cX(a, t)(ß)

= c(a(t) + \(a, t)(ß)) + (1 - c)a(t)

= c(aß)(t) + (1 - c)a(t).

Similarly, to prove (5.2),

<*(ß + y)(t) = a(t) + X(a, t)(ß + y)

= a(t) + \(a, t)(ß) + a(t) + \(a, t)(y) - a(t)

= (aß)(t) + («7X0 - a(t).

Making use of (5.1), we see that (5.3) holds with n = 1. We now use (5.1) and (5.2)

to complete the proof of (5.3) by induction. We have, assuming the result holds with n

replaced by n — 1,

n /n-1 \ n-1

« ¿2 dßi = <*\¿2 cißi + CA ) = « ¿2 Cißi + a(cnßn) — a

n-1 / n-1 \

= ¿2 cAaßi) + ( 1 - ¿2 c, )a + cn(aß«) + (1 - C.)a - a

= Xci«!8. + l1 — ¿2cija->-l v »-1 '

Finally, we note that (5.4) is equivalent to

(5.5) (aß')(t) = a(t) + (aß)'(t)

for all t G T and this will now be proved by recursion. For / = r, the result is clear.

We now assume (5.5) holds when t = u and when t = v and prove it when / = uv.

Let \(a, u) = ¿2w£t c(w)w, \(a, v) = ^jEr d(x)x where c, dm&p all but a finite set

of trees to 0. Using results proved in [12], we find

(aß')(uv) = a(uv) + X(a, uv)(ß')

= a(uv) + (X(a, u)\(a, ü))(ß') + a(v)\(a, u)(ß')

= a(uv) + ¿2 c(w)d(x)ß'(wx) + a(v)\(a, u)(ß')w,x€T

= a(uv) + ¿2 c(w)d(x)ß'(w)ß(x) + a(v)\(a, u)(ß')v> ,xET

= a(uv) + \(a, u)(ß')(a(v) + \(a, v)(ß))

= a(uv) + ((aß')(u) - a(u))(aß)(v)

= a(uv) + (aß)'(u)(aß)(v)

= a(uv) + (aß)'(uv).

This completes the proof of (5.5) and therefore of (5.4). D

Let the Runge-Kutta methods r,,r2, ■ ■ • ,rN have group elements g,,g2, ••• ,gw

associated with them when A = 1. Thus, for a general stepsize A, the group elements


800 J. C. BUTCHER

are g¡M, g2h\ ■■■ , g"'. We wish to study the group elements associated with f,,

f2, ■ • • , fN given by (3.1).

Theorem 5.3. The group elements associated with r„ r2, ■ ■ ■ , fN are given by

«¡"."Ii", ••• ,W »here

ff N

(5.6) g, = ¿2 aitg, + ¿2 biig'i, i = 1,2, ■■■ , N.i-l i-l

Proof. We first note that (5.6) does indeed define g,(r) for i = 1,2, ■ ■ ■ , N and

all t G T, since in the equation

If N

(5.7) =#,(*) = J2 aii8i(t) + ¿2 biig'M, i = 1, 2, • • ■ , N,i-l 1-1

g'^t) on the right-hand side can be written as the product gi(t,)g¡(t2) ■ ■ ■ g¡(ts) say,

where f., r2, • • • , t, have lower orders than t.

By absorbing A into the function /, we can, without loss of generality, restrict

ourselves to the case A = 1 so that g,"0 = g and g¡h) = g4 for / = 1, 2, • • • , N.

Suppose the Runge-Kutta method r, (i = 1, 2, ■ • • , N) is defined by (S,, d, s/) where

Si is a (finite) set, c, is a linear operator on the space of real-valued functions on S,

and Si is a particular member of S¿. The result computed by the method ru starting

from an initial value z G X, is Xs.) where, for all íG5¡,

Xi) - z + (c, X (/ o y))(S).

Suppose, without loss of generality, that Si, S2, ■ ■ ■ , SN are disjoint and that none

of them contain any of the numbers 1, 2, • ■ • , N. Form a new set S = S, U S2 W • • ■

KJ SNKJ ¡1,2, • • • , N\ and define a function y : S —> X satisfying

(5.8) y(s) = z + (c X (f ° y))(s)

for all s G S, where c : *(S) —> *(5) is defined by c(x) | S, = c,(x | Sv) for

i = 1, 2, • •• , Nand

M N

c(x)(i) = ¿2 ai,Ci(x | S,)(s,) + £ **#*(/). / = 1, 2, • • • , iV.i-i i-i

Note that (5.8) defines y for all / with sufficiently small Lipschitz constant, and

that y(Si) = ri(\Xz), X0 = ^OX2) f°r all i — 1,2, • • ■ , N. Let the functions p, v bedefined for S and c as described in [12] and, for i — 1, 2, • • • , N, let p{, vt be the

corresponding functions computed for 5,, c,, so that for all t G T, p¡(t) = ix(0 I $

and v¿(í) = KO | St. Also, for all t G r and all / = 1,2, • • • , N, v(tXsi) = g,(0,K'XO = £•(')> and m(?X0 = c?X0- We now use the relation v(t) — c(ß(t)) to compute,

for i = 1,2, ■ ■ ■ , N,

gt(t) = K»X0 = c(p(t))(i)N N If If

= ¿2 a«cMt) | Si)(Si) + ¿2 bup(t)(j) = ¿2 a„c,(M,(0)(i,) + ¿2 baß(t)(j)i-l i-l i=l i-l

ff ff !f If

= ¿2 atMtXsi) + ¿2 bup(t)(j) = ¿2 aiigi(t) + ¿2 bug'iU).i=i i=i i=i i-i

Hence, (5.6) holds. D



We now use the result of Theorem 5.3 to express a condition for order. Let

Ta denote the set of trees with up to q nodes.

Theorem 5.4. The method (A, B) is of order q iff there exist g,, g2, ■ ■ • , gN G G

such that, for all t G TQ and i = 1, 2, • • • , N,

If N

(5.9) (pgi)(t) = ¿2 atigi(t) + ¿2 bi,(pgi)'(t).¿-i i-i

Proof. Given Runge-Kutta methods r,,'r2, ■•■ ,rN, let g[h), g¡h), ■■ ■ , g(Nh) denote

the group elements associated with r,(h), r2(h), • • • , rN(h). By Theorem 5.3, the group

elements associated with f,(h), f2(h), ••• , rN(h) are g[h), g¡h), • • ■ , g^1' where g,,

g2, • ■ ■ , gw are given by (5.6). However, the group element associated with the

operation that takes z G *to ri(h\Ph(z)) is/j^g'"' = Og,)"". Hence, (5.9) will hold

if((pgiXt) = gt(0 for / = 1,2, •■• ,N and all t G Ta. This holds iff, for eachi = 1,2, • • • , N, the Runge-Kutta method which, for stepsize h, takes z G X to

(r^hy1 o fi(h)Xz) is of order q. This is equivalent to (3.2). D

Corollary 5.5. The method (A, B) is of order q iff there exist g,, g2, ■ ■ ■ , gN G G

such that, for all t G T„ and i — 1, 2, • • • , N,

N ff

(5.10) gi(t) = ¿2 «,,(p-^,xo + ¿2 bug'id).i=i ¿-i

Proof This result follows directly from Theorem 5.4 by substituting p~1gi for

gi(i = \,2, ■ ■ ■ ,N). However, we will give an alternative proof based on the group-

theoretic interpretations of (5.9) and (5,10). Let G0 denote the normal subgroup of G

such that, if g G Gq and / G T„, then g(t) = 0. We will prove that

(5.11) grVMZ aug, + ¿ ¿„Og,)') = g:\¿2 a^p-'g,) + ¿2 bug') ,\,_1 i-l / \,_i ,=! /

which will establish the result since (5.9) is equivalent to the statement that the left-

hand side of (5.11) is in G„ and (5.10) is equivalent to the statement that the right-hand

side of (5.11) is in G„. We have, using Lemma 5.2 and the consistency of (A, B),

(N N \

¿2 aa8i + ¿2 bii(pgi)')1-1 1-1 '

= *:'(£ au(p-lgi) + ¿ biiP-l(pgi)' + (l - ¿ au - ¿2 bt,)p-1)\ 1 = 1 i-l \ ,_1 1 = 1 / /

(!f N \

¿2 aij(p~lgj) + ¿2 bijg'X D1-1 1-1 '

In the next corollary, which we state without proof, we consider a method (A, B, C)

in which (2.2) is replaced by

F<»> = [^jyt"-» + h[B]F(Y{n)) + h[C]F(Y(n'l)).

Corollary 5.6. The method (A, B, C) is of order q iff there exist g,, g2, • ■ ■ ,gN

G G such that, for all t G TQ and i = 1,2, ■ ■■ , N,

N N !f

(pgd(t) = Z "t,g,0) + E bute,)'«) + ¿2 cug'i(t).i-l i-l i-l


802 j. c. butcher

Table 6.1

(TT.T>I

P<t)

l(b42+»43>

-H\3b32+ï«\2+b43)(1-»42-V T(b42+b43-})2-Ít43b32

4<l-b42-b43>2+!h43b32

-1|+îh53b32+î(b42+b43' ib53b32+T£(b42+b43)

6. An Example. By contrast with the classical Runge-Kutta method (A, B)

where

0 0 0 0 1

0 0 0 0 1

(6.1) A = 0 0 0 0 1

0 0 0 0 1

J) 0 0 0 1.

we consider a method (A, B) where

0 0 0 10

0 0 0 0 1

(6.2) A = 0 0 0 0 1

0 0 0 0 1

LO 0 0 0 1.

B =

0 0 0 0 0

I 0 0 0 0

0 | 0 0 0

0 0 10 0

6 3 3 6

B =

i1

è - b3

2 — bt2 ¿>43

6-

0

0

b32

bi2

0 0 0

0 0 0

0 0 0

¿43 0 0

ba3 6

and A32, A42, bi3, b53 are to be chosen. The advantage the method (6.2) would have over

(6.1) is that it requires three rather than four derivative calculations per step. This


THE ORDER OF NUMERICAL METHODS

Table 6.1 (continued)

803

t(tt.t)

H*«*«*

>*,*■-.,>

12 12 42 43 i, 42 43 i, 43 32

l-i(b,,+b,,)3 8 42 43

(J-b )(l-?(b.,+b,,))2 J2 3 i, 42 43

2"T2(b42+b41>+i<bi9+b/ l)2+ii, *bv>3 12 42 43 i, 42 43 4 43 32

-Tï(b42+b43>ib53b32

t(t.tt)

-^b43b32+|(b42+b43)<1-b42-b43>

y(i-b32><b42+b43-1>

b43b32<l-b42-b43>+í!Í<b42+b43>1<b42+b43)2tb"-b«)

-,tÍb43b32+t(b42+b43)(l-b42-b43>

(|-b32)(-7+ib43b32> + <1-b42-b43)(t(b42+b43>-b32<t+7(b42+b43)»

<1-b42-b43)(-í+i<b42+b43>Í(,>42+b43»2+b43b32>

■?¡Íl(b42+b43>-i<b42+b43'2+ib43b32+i1'53b32(1-b42-b43)

is because f(y\n)) is identical to f(yl4n~1)) and need not be recomputed and because

f(yin)) is not made use of and therefore need not be computed at all.

What we shall do is to choose b32, bi2, bl3, b53, so that the method (6.2) is of order 4

relative to r,, r2, r3, rit r5 where r5 is the trivial Runge-Kutta method defined by

r5(h)(z) = z. This will mean that the method will not require any special finishing

procedure. Let p~lg„ p~xg2, p~lg3, p~lgt, p~lgs (where g5 = p) denote the group

elements corresponding to r,(\), r2(\), r3(l), r4(l), r5(l) so that the conditions for the

method to be of order 4 are that, for t G Tt,

(6.3)

(6.4)

(6.5)

(6.6)

(6.7)

gi(t) = (p-'gJO),

g2(t) = hg',(t),

gÁt) = (h - b32)g',(t) + b32g'2(t),

gi(t) = (1 - ¿>42 - bi3)g',(t) + bi2g'2(t) + bi3g'3(t),

P(t) = gs(t) = ±g',(t) + (f - b,3)g'2(t) + b53g'3(t) + \gi(t).

To write these as polynomial conditions on A32, bi2, A43, b53, we substitute in turn

t = T,TT, TT-T, T-TT, (tT-t)t, TT'TT, t(tT-t), t(t-Tt) HltO (6.3)—(6.7). FrOIU (6.4),

(6.5), (6.6), we find, for each t, the values of g2(t), g3(t), gt(t); from (6.3), we find the

value of g,(t); and, from (6.7), we obtain a condition on A32, • • • , A53. Because of the

form chosen for B, the equations for t = t, tt, tt-t, (ttt)t will be automatically

satisfied. However, for completeness, the computations are shown for all the 8

members of Tt in Table 6.1.


804 J. C. BUTCHER

Equating g5(r) with p(t) in the cases t = r• tt, tt- tt, t(tt■ t) and t(t■ tt), we

obtain the following conditions

12 — 6 '(6.8) \(bi2 + 6«) - A + ÍA5363:

(6.9) Mbi2 + bi3) + \bb3b32 = \,

(6.10) S — T2(bi2 + *43) + J*53*32 = T2,

(6.11) -A + Mb*2 + bi3) - \(b,2 + bi3f

+ 6*43632 + 2*53*32 (1 — 642 — 643) = ^4-.

Equations (6.8), (6.9), (6.10) are satisfied if and only if

(6.12) A42 + bi3 = 11/12,

(6.13) ô53*32 = 7/36,

and, with these values, (6.11) simplifies to

(6.14) ¿43*32 = 5/12.

The general solution to (6.12), (6.13), (6.14) is

A32 = 6, *42 = 11/12 - 5/12(9, ¿>43 = 5/126, bb3 = 7/360

where 6 is any nonzero number. There do not seem to be compelling reasons for

choosing any particular value for 8; an analysis of the region of absolute stability,

for example, shows this to be independent of 6. Hence, we select 6 = \ on grounds

of simplicity. The method (6.2) becomes in this case

A =

0 0 0 1

0 0 0 0

0 0 0 0

0 0 0 0 1

.00001

B =

_i_12

1T2

5 7T¥ TS"

0 0

0 0

0 0

0 0

i 0.

To use this method in practice, we could find Runge-Kutta methods r,, r2, r3, rt

which satisfy the requirements that r{(l) has group element lying in p~1giGi and use

these methods to compute starting values for Y¡0) (i = 1, 2, 3, 4). However, since

only A/(y40)) and Y(50) are used in subsequent steps, r„ r2, r3 can be omitted from con-

sideration and Y[0) need be computed only to third-order accuracy. The starting

procedure that will now be suggested takes the result forward one step (so that we

may think of T"', Y¡1) as being the starting values computed) by the classical fourth-

order method (6.1) and then computes Y{1) by using an increment function that

requires no further derivative calculations. If Yu Y2, Y3, Y4, T5 are the approximations

computed using (6.1) with Y, equal to a given initial value X*o), we compute 74u, Y¡1}

as follows

Y2 = Y, + UK Y,),

Y3 = Y, + \hf( Y2),



Yt = Y, + hf( Y3),

if' = Y5= Y,+ \h(f( Y,) + 2/( Y2) + 2}( Y3) + /( Y<)),

Y\l) = Y,+ h(c,f( Y,) + c2f( Y2) + c3f( Y3) + c4/( F4)),

where c„ c2, c3, c4 are real numbers. If g{h) is the group element associated with

Y[u computed this way, we find

g(r) = c, + c2 + c3 + Ct,

g(rr) = 5(C2 + c3) + cit

g(TT-T) = \(C2 + C3) + C4,

g(r-Tr) = \c3 + 5^4,

so that, equating these to the appropriate g4(i) given in Table 6.1, we obtain the values

c, = 1/12, c2 = 7/72, c3 = 59/72, c4 = 0.

Although this new method is put forward to illustrate the definition of order

rather than as a practical alternative to existing methods, it does seem appropriate

to see how well it performs in at least one example. Accordingly, the four-dimensional

system of equations y{(0 = yi+,(0 (i = 1, 3), tf«) = -yi-ÁO/iyAüY+yÁÜT2(i = 2,4) withXO) = (1,0,0, 1) was integrated through v steps with stepsize A = ir/2v

and with v = 10, 20, 40, 80 using the classical fourth-order Runge-Kutta method and

the method derived in this section. At each value of v, for each method, the value of

"*(Á'!r/2) — y,) was printed out. For each method, this vector was almost independent

of v and had the values

Runge-Kutta: (.03, .10, .10, .13), New Method: (.22, .04, .05, .27).

Thus, the fourth-order error behaviour for the new method is experimentally

confirmed in this example. Although a comparison of these error vectors favours the

Runge-Kutta method, a comparison based on numbers of derivative calculations

(rather than numbers of steps) would favour the new method.

7. Acknowledgments. This paper was prepared at the University of Dundee

where the author was participating in the North British Symposium on Differential

Equations. He expresses his thanks to the Science Research Council of Great Britain

which sponsored the Symposium and to the University of Dundee which made its

facilities available to him.

Department of Mathematics

University of Auckland

Auckland, New Zealand

1. G. Dahlquist, "Stability and error bounds in the numerical integration of ordinarydifferential equations," Kungl. Tekn. Hogsk. Handl. Stockholm, No. 130, 1959, 87 pp. MR21 #1706.

2. P. Henrici, Discrete Variable Methods in Ordinary Differential Equations, Wiley,New York, 1962. MR 24 #B1772.

3. M. Urabe, "Theory of errors in numerical integration of ordinary differential equa-tions," J. Sei. Hiroshima Univ. Ser. A-l Math., v. 25, 1961, pp. 3-62. MR 25 #759.


806 J. C. BUTCHER

4. H. Shintani, "Approximate computation of errors in numerical integration of ordi-nary differential equations by one-step methods," /. 5c/. Hiroshima Univ. Ser. A-I Math.,v. 29, 1965, pp. 97-120. MR 31 #6383.

5. M. Urabe, H. Yanagiwara & Y. Shinohara, "Periodic solutions of van der Pol'sequation with damping coefficient X = 2 ~ 10," /. Sei. Hiroshima Univ. Ser. A-I Math., v. 23,1960, pp. 325-366. MR 23 #A1889.

6. W. B. Gragg & H. J. Stetter, "Generalized multistep predictor-corrector methods,"/. Assoc. Comput. Mach., v. 11, 1964, pp. 188-209. MR 28 #4680.

7. J. C. Butcher, "A modified multistep method for the numerical integration ofordinary differential equations," /. Assoc. Comput. Mach., v. 12, 1965, pp. 124-135. MR 31#2830.

8. C. W. Gear, "Hybrid methods for initial value problems in ordinary differential equa-tions," /. Soc. Indust. Appl. Math. Ser. B. Numer. Anal., v. 2, 1965, pp. 69-86 MR 31 #3738.

9. J. C. Butcher, "A multistep generalization of Runge-Kutta methods with four orfive stages," /. Assoc Comput Mach., v. 14, 1967. pp. 84-99. MR 35 #3896.

10. J. C. Butcher, "On the convergence of numerical solutions to ordinary differentialequations," Math. Comp., v. 20, 1966, pp. 1-10. MR 32 #6678.

11. J. C. Butcher, "The effective order of Range-Kutta methods," Conf. on NumericalSolution of Differential Equations (Dundee 1969), Springer, Berlin, 1969, pp. 133-139. MR43 #1432.

12. J. C. Butcher, "An algebraic theory of integration methods," Math. Comp., v. 26,1972, pp. 79-106.


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