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Publ.RIMS, Kyoto Univ .18 (1982), 325-364

Runge-Kutta Type Integration FormulasIncluding the Evaluation of the

Second DerivativePart I

ByTaketomo MITSUI*

IntroductionW e are concerning with theoretical study of num erical integration pro-

cedure for the initial-value problem ofordinary differential equation:(E)( I V )M a n y numerical analysts have been investigating the discrete variable methodsfor the problem. Consequently everyo ne can enjoy to solve numerically or-dinary differential equation in almost all computing centers. I t seems as if wegot the numerical integrator th rough the use of the comp uter. B ut the studyisy etcontinued fo r"better" numerical procedure.

Among the one-step methods, Runge-Kutta methods (RK methods, inshort) are popular because of the high accuracy and the feasibility of changingstep-size. In general the m ethods are expressed as follow s. The solution of(E )at x0+hi sapproximated by

where

Received February 27, 1981.* Research Institutefor M athematical Sciences, Ky oto U niversity , Ky oto 606,Japan.

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326 TAKETOMO MITSUIhis thestep-sizeandy0 is theapproximated valueatx0. This typ eof themethodis called (explicit) p-stage Run ge- K utta algorithm . A ccording to the choice ofthe stage number p and the parameters af, / ? f j - , \i{ we have many variations,among which the classical Runge-Kutta method or the Runge-Kutta-Fehlbergmethod is famous. A distinguished contribution for the study of the Runge-Kutta methods has been made byJ.C.B UTCHER([1]~[4]). He determinedthe attainable order of the RK metho ds up to 10-stage form ula. O n the otherhand he introduced the semi-explicit (the summation is up to i instead of i1in (*)) or implicit (the summation is up to p in (*)) formula.

RK methods (and perhaps many other quadrature formulas for the initial-value problem) are constructed on the principle that the required functionevaluation is onlyforf(x, y), i.e. the first derivative of the solution . It is qu itenatural becausew e areacquainted withthefunctional formof the firstderivativein the ordinary d ifferential equation. Recently, however, some propositionshave been made to employ the function evaluation of the second derivative ofthe solution. Functional form of it isgivenby

g ( * > y ) = fx ( x , j > ) + / , ( * , y ) f ( x , y ) -M . URABE[15]made a first attempt to employ g(x9 y) by presenting an

implicit one-step method with step-size control strategy. Let y0 and y-1 beapproximations of y(x) at x0 and x0 h, respectively. H is algorithm employsthepredictorgivenby

and the corrector given by

y o ^ - i + ^ a o i / . - i + 1 2 8 / o + i i / o(13 -4000-300,240

where /,=f(x0 +ih,yj )9 gt=f(x0 +ih,yj )9ft =f(xQ + h, j > 0 and g1=g(x0+ h,j>0- S ucceeding his result, J. R. CASH [5] has considered this type of formulamo re generally and m ade some stability analy sis. O n theother hand H. SHIN-T A N I[12], [13]has proposed some formulas analogous to RK formula em -ploying one evaluation for/(x,j;) and some for g(x, y). He has given thevalues of the parameters appearing in the formu las up to the order 7. H isresults, closely related to the present work, will be mentioned afterward.

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RUNGE-KUTTA TYPE INTEGRATION FORMULAS 327

In this context another type of integration formula, for the origination ofwhich H. H. R O S E N B R O C K [11] is given credit, is now being developed. Itemploys thepartial derivativefy ( x 9 y) and is reported to have good stability forstiff systemso f ordinary d ifferential equa tion s ([7], [9]).

Here we shall examine an explicit (p , g)-stage Runge-Kutta type formulaincluding the second derivative. It requires p times evaluations for the firstderivative and q times for the second derivative in a similar manner for RKm ethods. W e are interested in the following problems.

(1) W hat is the attainable orde r of the (p, g)-stage form ula from theviewpoint of its local accuracy?

(2) H ow are the parameters in the form ula determined?(3 ) W hat formula isgood fo r practical use?These problem swillbesolvedin thefo llowing sectionsand the forthcoming

paper by the author. The present paper is especially devoted to investigate the(1,g)-stage formulas.

First, we shall define explicit (p , ^)-stage form ula. N ext, some algebraiccomputations are carried out to investigate (p , g)-stage formulas. H ere S A Msoftware is used as a powerfultool. Then, theattainable order of (1, g)-stageformula isdetermined up to q = 4.

Remark. In the case of very complicated functional form off(x9 y) inhigher dimension, the calculation of the second derivative g(x9 y) requires alaborious work. It is the main reasonw hy the methods employing g(x, y) havenot been considered. B ut the recent developmentof the symbolicand algebraicmanipulat ion (SA M ) software brings the change of the situation. S A M soft-ware, for example , RE DU CE -2 or M A C S Y M A ,is now ahelpful tool fo r mathe-matical sciences. In fact, some S A M program m ay print expressions in aFO RT RA N n o ta ti on sothatone cancarryout the calculationof thesecond deri-vativefrom/(x, y) in an automatic way . O nce after algebraic computation wemay call g(x9 y) as aFUNCTION subprogram.

M oreover, S A M softwareisvery useful for the theoretical study of the RKand its analogous methods. For example,H . T O D A [14] has considered 5-stageRK limiting formula oforder 5. H e has util ized M A CS YM A essentially. W eshall also attempt to apply SA M for our s tudy.

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328 TAKETOMO MITSUI

AcknowledgementThe present author would like to express sincere thanks to Prof. S .

H I T O T U M A T U for his constant encouragement an d helpful discussions during thepreparation of the work . H e is also greatly indebted to Prof. K. I C H I D A forcarrying out thecom putations by interval arithmetic.

1.1. Explicit jp,g)-StageFormulaW e shall discuss numerical integration procedure for the initial-value

problem of ordinary differential equation:(1-1.1) - =/ ( x , 3 0 ,(1.1.2) y(xI)=y I.Here/is sufficiently smooth with respect to x and y . Let usdefine an explicit(p , g)-stage Runge-Kutta type formula including the second derivative of thesolution. Let g stand for the second derivative ofv(x),(1.1.3) g(x, y)=fx(x, y)+fy(x, y)f(x, y) .Explicit (p, g)-stage formula is given as follows.(1-1.4) )>n + i=y n+ hwhere

(1.1.5) \ i-i i-i{ kt=f(xn +a, h, yn+h P i j k j + h2 V U K J > > = 2, . . . p,

(1.1.6) t 1-1

Remark 1. The parameters /^, v , - , a, j8 l7 - , y j - , pf,

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f x2 >H 3;2)where at andfe {-are constants, then

and9 x y) =

Similar to the R K fo rmulas, the determining equations and the parameters arepossible to be slightly different between the single differential equation and thesystems of equa tions. For convenience sake we shall investigate the single case.The attainable order of the formula is not depend on whether (1) is single orsys tem.

1.2. TheTaylorSeries Expansionof the SolutionTo investigate Run ge-K utta ty pe methods of higher order, we are requiredto represent the solution y(x0 + li)for (E ) and (IV ) in S ection 1.1 into the p ow er

seriesof the stepsize h. For our quadrature formulait ispreferableto representthe solution into the power series utilizingthe second derivative g.

Twice integration after differentiation for theequa t ion, ,

implies th ef o r m ulaCXQ+h If* /J2 t>(Y\ ) (XQ+h (Ct }(1.2.1) AgLdx\dtss\ g(x,y(x))dx\dt. XQ jXQ UX JJCQ (JjCO '

The left-hand sideof the abovefo rmula isequal toy(x0 +h)-y(xQ )-hf(x0, X*o))

H ereafter the subscript0standsfor the evaluationatx = x0 andy =The right-hand sideof (1) becomes

(h {CxQ+, } C h(/\ \ g(x,y(xy)dx\dt = \ \Jo U.xo ' ^0 IJO

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330 TAKETOMO MITSUI

Then, assume that for O ^ C ^ / i ,X*o +C)can be expanded in the powerseries of by

m(1.2.2)where KJ(j = l , 2,...,m) is the coefficient to be determined later. S ubstitution(2 )into the integrand implies

kn

Here D0 is a differential operator definedby

Therefore wehave the following equation:

/

which determines K7recurrently .Twice differentiationfor (3)with respect to himplies

m i- m Ijl ( I I I \ /m-1 ,- \ f e f / P \fc ~1 )(1.2.4) Z ' =x 4 r E , ( -TTlVA')^~*(-r-)S\ ./ = o / ' /= o / ' U = o \ ^ / \ r = i ( r+ 1 ) /L \^/ JoJThen, we have the following important result.

Theorem 1. The coefficient Kt(/ = 0,1,..., m) is determined by the follow-in g way:

/or / = , 2,...3m.

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RUNGE-KUTTA TYPE INTEGRATION FORMU LAS 331

Here BSt t(x l9 x2 , . . . 5xs) is the multivariable polynomial of E. T.Bell of theorder (s, f) and

m(/, s) = m in (s, / s) .Note. KJappeared in the summation of the right-hand side of (5) is for j

less than or equal to / 2 sothatwe can determineKtrecurrently by theformula .Proof. K Q= Q Qis clear. Let us introduce two functions G(h)and A(h\ I)

as follow s:m 1( L 2 - 6 )

(1.2.7) ^ 4 ( A ; / ) = Z r }{G(h)}k\ Dl0-k(-^-} g I ,/=0, 1,.. . , m .N ote that G(h) is of the order h and G(0) = 0 holds. Then, the right-hand sideof (4) is equal to Zg=o hs.s

H ence, we have the equation(1.2.8) f -rr *'/= o s = o S 1For an integer / such that 1^/^m, /-timesdifferentiation to (8) with respect to/ ? implies(1.2.9) ic^'s= 0 \ S

B y (7),theequat ion

is clear. For s 1 , we havefrom (7 )

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332 T A K E T O M O MITSUI

For higher derivatives of composite functions, the formula employing the Bellpolynomial is known (J. R I O R D A N [10]). H ence, apply ing the formula, we havefo r Jt= l, 2,..., l-s

which implies

, , = o l OFrom (6), the equation

nt~* 1

holds. H ence we have,

Then /4 (s )(0; / -s) isgivenbym(/,s) / lS

= ] 2

I-S) I K0

which implies from (9)l-l I I \ f m U . s ) (l

l-l m l,s) 7f / k- k- k-z { z J,(,_ 'J-,) B -(T L--T-....v?This is the desired result. D

The multivariable polynomial B8t t(x ly x2,...,xs) has a recurrence form ulato calculate it conveniently by the application of any S AM softwa re. That is,

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RUNGE-KUTTA TYPE I N T E G R A T I O N F O R M U L A S 333

Z xk+l\k=ls+ l

= Z ,+ , , (* , . - ,*s+lK-r = lH ere, the bo th sides of the formula is considered as a polynomial of a. Therecurrence formula fo r Bst will be employed during th e calculations of KIbyRE D UCE -2 .

Theorem 1 tells us a concrete method to determine the Taylor series ex-pansion (2 )employing the second derivative. For example,W e may also carry out the process by the S A M software up to the desired order.The result by R E D U C E -2 is shown in Table 1 .

Note that because ofRemark 2 in the preceding section we do not care th eorder of the higher partial derivatives.

Another important result from Theorem 1 is that any KJis a summation ofan integer multiple of some product of [ _ D ( d /d y )q g ~ ] 0 . Hence, we shall callany product of [D (d/dy)qg] 0 an elementary differential of g. J. C.B U T CH E R[1] has used the terminology of elementary differential in a different sense.H oweve r, our study of the expression of the coefficient of Taylor series in theelementary differentials is on the similarpointof view.

In fa ct, the w ay of the proof o f Theorem 1 is applicable to the T ay lorseries expansion of y(x0 + h ) employingthe first derivative.

Theorem 2. Assume that(1.2.10)Then, the coefficient A, (/ = 0,1,..., m 1) isdetermined by

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334 TAKETOMO MITSUI

( 1 . 2 . H ) U - r / > < n + y 1lmy s)I ^/]o+ 5L I fL

7 = 1,2,..., m.W e shall call (10) the first type power series expansion in contrast with

(2) which willbe mentioned as the second type expansion.

Table1.

=[O00]o

]0'0*o 30o[^00 Jo+ [Wo 0*o +4[000]o[>o0Jo +6f fo i Jo+0o0*o

+20[D00]0+ 10[0+15g0+000y 0+ 1800

]0flf,,.0+70[D00o0j,]o+450o 0*o

+ 150o 0*o [J>o0*lo+210o [+2100o[^o0]o'[.Dfayjo +6600 [D00]0- 0 * 0 '0y*+ 840g[D0gy ] 0 0W (0 + 1200g 0y i

0 * o 0*o +56[DS0]0\ _ D3 0 g y - ] 0 +14[>g0]00y > 0[D0ff Jo

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RUNGE-KUTTATYPE INTEGRATION FORMULAS 335

3y , o + 35ED20]

2,0Wi0+56[D00]0192[D00]0 [D00,]o [J>g0,]o +76[D00]0 9y,o

+280[D00]0

o 9y,0 IDfoylo +00, 9ytO+16800-[Dg0]0.[Do6fy 3?]0 +420^0

+36000-[D00]0-[D00y]0- g yyt +50800-28000+2520+4650 - 0^0-[D0y),]o+ 810g02j0-0^0 + 2100g [Dg0]08400g+2250g00gyyy+1050yyy

1.3. Im plicit (1, |)-S tage Form ulaFor the study of the general (1, g)-stage formula, it is convenient to analyse

the corresponding implicit (1, g)-stage formula, because we gain an insight intoits algebraic relations by them.

Consider an implicit (1, g)-stage formula as follows:(1.3.1) yn+i=yn+ViMl+]

1.3.2)

We need to analyse one-step integration by (1) and (2),so we may substituten= 0,i.e.

(1.3.3)1+ Z

JFor k l3 wehaveth e following expansion:

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On the other hand, K{has the expansion

H ence, the equ ation(1.3.4) yi ,holds. The solution y(x0 + h) has the power series of h as fol lows:(1.3.5) y(xQ+ h) = y0+hf0+h*g0 + 0(h*).Comparison of the terms having th e same power of h in (4) and (5) yields th efollowing equations.

Thus, Ju1=l , a1 =0 and Z ? = iv i= l/2 must hold in the implicit (1,g)-stageformula.N ext, sinceKthas the expansion

the comparison of the third order termofyl and y(xQ+ h)impliesthe equationy [^o^]o=y(^fo+/o-^,o)= ZviKp^.o+ffn/o-^.o)]

From the viewpoint of homogenuity of differential operation, P i= ffu must holdfor / = , 2,..., g.

H ence, let us rewrite (3) by

(1.3.6) I Xf=g(x0 + pth, y0 + P / / I / O 4-We shall assumethe Taylor series expansionfor 1 by

(1.3.7)where Kf,willbe determined by the similar manner fo rTheorem 1.

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RUNGE-KUTTA TYPE INTEG RATION F ORMULAS 337

Theorem 3. For each /, KUin (1.3.7) isdetermined by the following:j *10= 00*

n 1 Q\ J(L3-8)**,.,(t lino,-,* I T.jKy...,)^-1-' C T H T ) '1} .j=l j i_ \ try J0>

/= , 2, . . .,m .A f o f e . B y the above formula,KU can be determined in the ascending order

of the second subscript, i.e., Ki2 ( / = 1 , . . . ,q) after Kn , /c /3( / = , . . . , ^f)after Ki2 ,and so on.Proof. Tw o-variable Tay lor series expan sion for K{gives

h- +

= / = o/ m - 1 / Tir \ \ Fi- (S?)4*)[*

Thus, putting G,(/i)as(1.3.9)

r=0 \ j = i rand noting that G/(/i) is of the order h and Gj(0)= 0, wehave theequation

t *'- f -which is similar to (1.2.4). H ence, follow ing after the proo f of Theorem 1, wehave

From (9), for fc=l, 2,...,s

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338 TAKETOMO MITSUIholds, which implies

Thus wehave th e desired result.B y Theorem 3, KU isgiven asfollows:

=P?[^o0]o+ 2(Z *ij)0o9y ,o >jVS'O*2Z *ljKj

P) [0o0j jThe result with the help of RE DU CE -2 is shown in Table 2. H ere we employthe notation

Tik= >,%, i = l, 2,...,q, /c= 0, 1,2,....From (6) and (7), the Taylor series expansion fo r yis givenby

(1.3.10) y,=yo+o+ Z v=y0+V 0+ _

o+f (Sr=2 i=lO n the other hand, from (1.2.2), the (second type) Taylor series expansion

of the solution y(x0 +h)is given by(1.3.11) y(where KJis represented by (1.2.5). The comparison of (10) and (11) brings thedetermining equation of the implicit (1 ,g)-stage form ula,fo r whichwehave thefollowing

Theorem 4, Each KU in (10) is constructed with the all elementary differ-entials included in the corresponding ?cz in (11). Hence the determin ing equa-tion of the implicit (1, q)-stage formula is of theform such as

(an integef)x(a polyn omial ofvt, p iy iij ) = (an integer).

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Proof. From(10) and (11), K:/ an d TCMmust satisfy

which implies

i

From(1.2.5)and(1.3.8)theconclusionfollows. D

KiO=Table2.

g0 gy,0o -g0 f l[D 0 f f ] 0 [D0ff J0

[DgffJo +24(Z T y 7 j0)ff0

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'[030,Jo+5040(ZTy(rror

+2520(E(p,rjo+2 pJ.T i0)T lJTJO)g gy t

+6720(E,]o+20160(E p,pjitJ Tj+ 10080(Z(p? + pfru

+20160(ZZTyTJtTu

340 TAKETOMO MITSUI

+ 5040(Z Z T+2520p/T?,[D00]g-[D00Jo +42pfTi0g0[D& J0+2520(1p.-p/Pi+py .T o[D00Jo [080Jo+840(Z)P?+pfriJTJO)g0 gy,0 [080 J o+ 5 0 4 0 (2 ; z 0 ,+ P J+p*)T r*o)0o 0 i o [000 J o

0w.o+252QPiTi0Ti2g0 [Dg0]0[000,Jo

+20160(Ep0 i+pfa]Tn) [D of if] 0+ 6720(Z (P?+ 40320(Z Z (p, + pj +p)TuTJJ krH )[jDo^o-fll.0' C ^ o f fJo+6720rar,3[D0fli]0 [D8+20160(ZtyTj r,, +T^))U >0g-ti gy,o gyy ,0

o +10080(Z rfpjrtJTJO)go Wig J+6720(ZPtPtpi+pfruTjo-tio [DoffJ0 [Dgff J0

1680(Z(pf+P>UTJ.0+40320(Z Z (p0j + PJ P k +plpdtijiflCT l[0)go 9y,o[D0ff JS+20160(Z Z (P? +PJ+pD J'^^o)0off5,o'[020Jo+40320(ZZ Z Ty T7tTH TJO )ffo'0*,o+1680TIO Ti4g0[DjJ K t

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+20160(5tij(T>2 T j0 + Ti0T J 2 + Tj0 T J 2))g0

+40320(Z+ 40320(E rlJ (p iT i0 T jl+201607 00-[D00J8

+20160(Ep,pj-ctfiTm +TJO )TJO )g% +10080(Z rrfpjTjo + 2p?Ti0)T JO )g20+ 20160(Z E Ty(2T7llr,oTk0

+20160(1

1.4. Determ ining Equation for 1, g)-Stage Form ulaA n algebraic computation according to Theorem 1 gives the number of

elementary differentials includingineach /c,. Let m(be thenumberofelementarydifferentials in K , and define th e integer M , b y

Each ml and Ml are given in Table 3 up to / = 8.Thus, we have Ml restrictions for the parameters v f - , pi9 xu of the formula .In the implicit (1, ^f)-stage formula the number of the parameters to be de-termined, say N(q\ can be given as a simple function ofq

It implies that the implicit (1, g)-stage formula can attain at least the order(1+ 2) whereI is the largest integer satisfyingthe inequalityM N . Theserelations are showninTable4.

H owever, the above argument based on merely counting the number ofthe equations that m us t be satisfied, ignores the relationship between them.

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342 T A K E T O M O MITSUIIn fact, it may happen that Ml restrictions are satisfied with fewer than Mlvariables. B ut, since we are interested in the explicit for m ula rather thanimplicit one, we shall not com e into m ore investigation for the attainable orderof the implicit formula.

An explicit (1 , g)-stage formula, which is defined by the parameters v^ ,ptandru (^ = 0 for;^i) in(1.3.6),hasN(qE) parameters to bedetermined. H ere

isgiven by

H ence, similar consideration for the implicit case gives the largest integer /*satisfying the inequalityM N . Table4 includes the relations betweenq,the number of stages, and /*.

Table3./ m, M,012345678

112369

172646

1247

13223965

111

Table4.

12345678910

38

1 5243548638099120

345566778

25914202735445465

234455667

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RUNGE-KUTTATYPE I N T E G RA T IO N F O R M U L AS 343

S H I N T A N I has given some explicit one-step methods utilizing the secondderivative [12], who se form ulation coincides w ith our explicit (1 , g)-stageformula. H e has determined the parameters fo r q = l, 2, 3, 4 and 5which giveth e formula obtaining th e order 3, 4, 53 6 and 7 respectively. H is resultsattainthe orders that we have argued as the least number /* for each stage formula.H ence, it is a question whether S hintani's results can be improved .

W e shall consider the d eterm ining eq uation for the explicit (1, g)-stageformula. Tables 1 and 2givetheequation asfollows. W eemployth enotationfor summation symbol such that th e upper limit of summation can not exceedth evariableof the preceding summation symbols,i.e.2r"(Sj"O

mearss X ? = i "( J= i"" ) - M oreover , the symbol Ti kmeans

Tlk =Q andDetermining equations./= 0: (E-0) 22v , =l/= : (E-l) 62>,p,=l1 = 2: (E-21) 1 2 E v i j 0 ?= l

(E-22) 24 2 ^7,0= 17 = 3: (E-31) 2oi>(pf= l

(E-32) HOSv.r,(E-33) 12oiv^T/0

1=4: (E-41)(E-42)(E-43)(E-44) 360i 7,0 =6(E-45)(E-46)

(E-52)(E-53)(E-54)(E-55) 504022:^^ =(E-56)

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344 TAKETOM O MITSUI

(E-57) 5040E E vt(p t+p^jTjo =8(E-58)(E-59)

1 = 6: (E-601)(E-602)(E-603)(E-604) 10080E ViplTi2 =1 5(E-605) 20160E E vT0.T,2= 1(E-606) 6720(E-607) 40320E E vf(p f+ p^j(E-608) 20160 =10(E-609) 1680 z vipfr,0= 15(E-610) 40320Z S viPip^jTJO =18(E-611) 20160 Z v,.(p?+^)T/J.TJ.0=21(E-612) 40320ESS v.TyT^T*,= 1> j k(E-613) 20160E v(7;.07;.2= 15(E-614) 40320E viPiTi0Tn = 60(E-615) 10080Evip?r?0=45(E-616) 20160E E v . T y C T j o+2T0)r;o= 18(E-617) 6720EVir?0= 15

1 = 1: (E-701) 72Ev ip/= l(E-702) 3024Evir/5= l(E-703) 15120Evfp i7;.4= 7(E-704) 30240E v;p?T i3=21i(E-705) 60480E E V , . T (J .r/3 = 1(E-706) 30240E v,pfrj2= 35(E-707) 181440E E v^ + TyT =12( j(E-708) 15120 EvipfTu= 35(E-709) 362880E Ev^^tyT,. =28(E-710) 181440E Z v((p?+p5)TijTJ.1= 31(E-711) 362880E Z EW T*, = 1

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RUNGE KUTTA TYPE INTEG RATION F ORMULAS 345

E-712) 181440Zv TMT 2= 5E-71 3) 181440 Z v,p,.T?i = 70E-714)(E-715) 1 8 1 440 Z Z v,p,p/p,+pj*tJTlo = 1 05(E-71 6) 60480Z Z v ; ( p ? +pJ)TyTy o= 45E-717)362880 Z Zv,. pi+/+t)tiyTJ.tTk0=5(E-718) 60480zv , ,or ,3= 21(E-719) 181440Z VipiT io Ti2=W 5( E-720) 1 8 1 440 Z v f p ? 7]07) , = 2 10(E-722) 362880Z Zv^(Ti0Tn+(E-723) 30240 z viP?T? 0= 105(E-724) 181440Z Z(E-725) 181440Z Z(E-726) 1 8 1 440 Z v,T?0T,, = 1 05(E-727) 60480Z v,p,T?0= 1 05(E-801) 90Zv ( pf=l(E-802) 5040Zv ;T i6=l(E-803)(E-804) 75600Z V i p ? T j 4 = 28(E-805) 1 5 1 200 Z Z v , t y 7} 4= 1' y( E-806) 100800Z v , p ? Ti3= 56(E-807) 604800Z Z v ; ( p f+P7)tyT73= 14(E-808) 75600Z v f p f 7;2= 70(E-809) 1 8 14400Z Z WiPj^jTj i =40(E-810) 907200Zz'v/P?+P?)tyTy2=43(E-811 ) 1814400Z Z Z v,TI7t;trt2= 1i j fc(E-812) 453600Z v (r?2= 35(E-813) 30240 z v/P?rn =56(E-814) 1814400Z Z W0fy>i +pfaTj =192(E-815) 604800Z Z v^p?+pJ^T,,.=76(E-816) 3628800Z Z Z vfoi +Pj+l > k ) i i j* j k Tk l = 18

I J K

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346 TAKETOMO MITSUI

(E-817) 604800E vtTnTi3=56(E-818) 1814400(E-819) 907200EV^^I=280(E-820) 1814400E Ev;T0T;i(2T;i+ 7 ) =66(E-821)(E-822) 907200E v^fpfa, },,=168(E-823) 604800E Ev /p?+pj)tyT}0=248(E-824) 151200Z S v r f p f+P^IIJTJO = 85(E-825) 3628800EZ L(E-826) 1814400i E(E-827) 3628800E S E

i j k I(E-828) 1 51200E vtTi0T;4= 28(E-829) 604800E v{PiTmTi3= 168(E-830) 907200E v(p?T;oTi2=420(E-831) 1814400E E vfit{TuTj0 +Ti0Tj2+ TJOTjJ=113(E-832) 604800E v ;p?Ti0T; t= 560(E-833) 3628800S E tAPj(E-834) 3628800E E vt-r,j(p tT loT(E-835) 1814400Ev,rror?i=280(E-836) 75600E v;pfT?0=210(E-837) 907200E Evi(E-838) 1814400(E-839) 907200E ZvfitfrjTj^rf Tio)TJO =465(E-840) 1814400EE Z v,T (X2T Jk T f0 Ti j t +tikTj0Tko +Ty t(E-841) 907200E vtTJ0Tt2=210(E-842) 1814400E ViPiT?0T n= 840(E-843) 302400E v ;p?r?0=420(E-844) 1814400E E v,tyTi0TJ0= 84i i(E-845) 604800E E VjTy (3rf0+ T2JO)TJO=225(E-846) 151200E v frf0= 105

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1.5. A ttainable O rder of 1, 1)-and 1,2)-Stage FormulasThe explicit (1 , g)-stage formula is said to have th e attainable order m if

m is the largest integer fo r which

among all combinations of the parameters of the formula, where y(x) is theanalytical solution an d y is given by (1.3.6). The definition of the attainableorder will be extended, ifnecessary , to general (p , g)-stage formula.

It is obvious that a (1, g)-stage formula has the attainable order m if andonly if the determining equations corresponding up to I have at least one so-lution, but they have no solution up to / + , where l = m2 .

Theorem 5, The attainable order of (I, \)-stageformula is3.Proof. The left-hand side of the equation (E-22) is equal to

which vanishes for (1, l)=stage formula because y T j-= 0. This means thatth e parameters v a nd pl can satisfy merely the equations (E-0)and(E-l). D

Theorem 6B Theattainable ordero/(l, 2)-stageformula is equal to4.Proof. Assume that the formula attains order 5, that is, the parameters

v i > V2 5Pi? P 2a nd ^21satisfy the equations (E-0)-(E-33).(1.5.1) v, +v2=(1.5.2)(1.5.3)

(1.5.4) ' i r i - i - * 2 f 2 1 2(1.5.5) v2p2T21=^

(1.5.6) v2p1T21=12Q

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(1.5.7)The equations(1),(2) and (4) yield a m atrix equation

I I 1 / 2 \P i P21/6 | v2) = 0 ,P\ P22

which impliesI 1 1/2

(1.5.8) Pi p2 1/6 = 0

fo r the existence of nontrivial solution[vt,v2, l]f. O n the other hand,(3),(5)and (6) give the values

Pi = ~y > P i~ "y 5which specify the determinant of (8) by 1/250. This con tradiction implies thestatement. D

Note. S H I N T A N I gives (1, l)-stage formula with parameters v1= i/2 andPi= 1/3. H e also gives (1 ,2)-stage form ula with v1=(9-hv /6)/36, v2=(9-VQ/36, p1=(4-N/6)/10, p2= (4 +x/6)/10, T21=(9 +v'6)/50. These parametersare not unique solution(l.S.l)-(l.5.4). The solutio n of them is represented w ithon e parameter p by the following:

Pi=PP2=(2p-l)/2(3p-l),v , = l / 6 ( 6 / 9 2 - 4 p+ l),

1.6. Attain able Orderof(1,3)-StageForm ulaThe determining equation for the explicit (1, 3)-stage formula are given as

follows:

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(E-0) i(E-l) ViPi+v2p 2+v3p3=-~

(E-22) V2T21+V3(T31+T32)=

(E-32) v2p2T 21+v3p 3(t3 1+T3 2)=(E-33) v2 p i i 2 1+v3 (p jT 3 1+ p2 T 3 2 )= L-

30(E-42) v2pit21+v3pi(T 3 1+T32)= gL(E-43) v2 p 2p 1T 2 14 -v3p 3(p 1T 3 1+p2T32)= -~(E-44) v2 p ?T 2 1+ v 3 (p ?T 3 1+piT 3 2 ) :=-jig-(E-45) v2 T |1 + v 3 (T 3 1 + T 3 2 ) 2 = L_(E-46) v3T 21T 32=-720"

It is remarkable that none of the factors on the left of(E-46) can vanish.Assume that two of pl5 p2, p3 are equal, say P/ = PJ. Then, from (E-0),

(E-l),(E-21),(E-31)and(E-41),we seethatIP ip }\P iP i

}

P KP 2 KI

PKP 2 K

1 ~216

JLl ~6112

J_

"v +v ~ jVK

- }= 0,

- +v -p vP K ^ K-1

= 0,

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350 TAKETOMO MITSUI

1P ip f

1P KP 2 K

12120

J_

"p fCv .+ v/P K

= 0.

The condition that the above three equations have non-trivial solutions, impliesthe equations with respectto p/5 pKas follows:

PiPK- -^(PI +PK)+ 12 =5P / P K - - 7 2 ( P /+PK )+ 20] =0

and

, P Ian d pK must satisfy th e equationsyPiP*:-y (P I+PK)+ 12= ~6~PiPK~ ^(PI +PK )+ 20

= i2PiPK~ 2o(P/+ Px) + 30=0 which isimpossible. The caseof pa=p2 = Ps leadsto a contradiction becauseof

V i + v 2+v3=-i-,P i ( v i+ v2+v3)=-i-,

inducedby(E-0), (E-l),(E-21), respectively.Henceno two ofpl9 p2, p3 areequal. It isconvenientto define

Since we consider th e explicit formula, Al=pH2 holds. Then, we havesimultaneous linear equations

r 1 llnPi P2 PSP I pi pi

= 0

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by (E-21), (E-22), (E-31),(E-32), (E-41)and(E-42). Sinceno two of pt, p2,p3 are equal, all ofvlAl9 v2A2) v3A3 vanish. B y virtueof the above mentionedremark, wedistinguishthe following four cases.

Case1. A1=A2= A3=Q.TheequationsA1=0 andA2= 0implypt=0 and T21=pi/2. Then,(E-43)

and (E-44) bring the equations v3p 2p 3T 3 2=1/180 and v3pT 32=1/360, wh ichgivetheidentityp3=2p2. Thus(E-l),(E-21)and(E-31) implythe equation

= 0,

which yieldsa quadratic equation of p220pi-15p2+ 3 = 0

because ofp2^0 . B ut the above quadratic equation has no real roots.Case 2 . v=Qan d A2= A3=0.In such case, wehave the equations

- 2p 2p 4 p ipi 8p|

16112120 _

r V2V3

-1

=and

Substitution of T 2 1 = p 2 / 2 and T3 1= (p 3 /2 ) T32into(E-44) impliesPf+24v3T32(pi-p?)= i.

Employing th e equation

induced by(E-46), we obtainp1(l-24v3T32)= 0.

H ence, th e equation pL=Q or v3T 3 2= l/24 holds. The case of P = 0 is equiva-lent to Case 1. The equation v3T 32=1/24 yields

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352 T A K E T O M O M I T S U I

Pi+24vyr32(p2-p1)=p2=y-by (E-33). The equation (*),how ever, implies p^ = l/15, which leads to acontradiction.

Case3. v2= 0 andAl = y 3= 0.Al=0 impliespl=0. Then,(E-l)and (E-21) bringtheequations

= -Land

H ence we see p3= l/2 and v3= l /3. I t contradicts th e equat ion v3p|= l/20induced by (E-31).

Case4. v1=0, v2= 0 and ^43= 0.The equations(E-0) and (E-l) yield v3= l/2 and p3= l /3. Ag ain, it

contradicts the equationv3p|= l/20 inducedby(E-31).Thus, we can conclude thatthedetermining equations (E-0)-(E-46) have

no solutions. The proof of the following theorem is now accomplished.Theorem 7. The explicit (1, 3)-stageformula can notattain order 6. Its

attainable order is5.Note. S H I N T A N I gives(1 , 3)-stage formula with parameters

4 , v 3 =(5-V5)/24, P l=0, p2= (5-V5)/10,), T 3 1 = 0 and T32= (3 +x/5)/20.

These parameters arealsonotunique solution of(E-0)-(E-33).

1.7. Attainable Orderof(1,4)-StageFormulaThe determining equation for the explicit (1 , 4)-stage formulaare given by

th e following:(E-0) V i-h V2+ V 3 - h V 4 = \ 1 2 3 * 2(E-l)

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E-22) V2T21-fV3 T31+T32)4-V4T41-fT42+T43)=

(E-31 ) v pf+v2pl+v3pl+v4pt= ^(E-32)

(E-33) V2pT2 +V3(p1T3t+P2T32)+V4(p1T41+p2T42+P3T43)

(E-4i) V1p t+v2pf+v3pt+v4pt=^E-42) V2piT2j+V3pi(T3j+T32)+V4p| T41+ T42+T43)=

(E-43)

(E-44) v2p ? T 2 1 + v 3 ( p ? T 3 ](E-45)

(E-46)

(E-52)(E-53)(E-54) v2jo2^T21+v3p3(p|T31+piT32)+v4jo4O?i(E-55) V2/3|T21+V3(pjT31+/)iT32)+V4(p?T41+pil42-r 3,43j,--g --

(E-56) v3(E-57)(E-58)

+V4p2(T4T42+T42T43)+V4p3(T31T43+T42T43)=

(E-59)

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354 TAKETOMO MITSUIThe question is whether an y parameters vt, p t an d rtj exist to satisfy these

22 equations simultaneously. It is helpful for investigation to introduce thefollowing notations:

T i0= Z* ty, Tfl= Z* Pjty (i = 1, 2, 3, 4) ,(T i0 andT11 meanzero.)

( 1 = 1 , 2 , 3 , 4 ) .Then, from (E-21), (E-22), (-31), (-32), (-33), (-41), (-42), (E-43),

(-51), (E-52)and(-53),weeasilyseethat(1.7.1) ZM i= Z V i P i d i =S ^ pM i= Zi i i iand(1.7.2) ZvA=ZVjp^:Theequations (1)means

/ 1 1 1 1P i P2 Pa P4 v2J2 = 0.V3^3

I P I p\ pi pi/p f pi pi PW e now distinguish two cases according as tw o of p l5 p2 , p3, p4 are equal or

otherwise.Case1. Two of pl5 p2, p3, p4are equal.Assumethat two of them are equal, say p/ = pj. Equat ions(E-0),(E-l),(E-21), (E-31), (E-41)and (E-51) give three simultaneous linear equations as

follows: / IPipu

1P KP 2 K

Pi

iPLPP

_ 1 _ \216112120 /

\V xV L-1

= o

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/

P ip }\ p }

/P iP ?u

1P KPiP K

P KPiPi

1

PLPiPi11

PLP 2 L

P 3 L

112120

30 /1 \12]20

30142/

c + )PKTKP,v,\ ~1 // \P\vl VJ)

P K2

\ ~~1 /

= 0,

= 0.

The condition that these equations have non-trivial solutions, implies the de-terminantsofmatricesto bevanishing. Hence,we seethat the equations

(P/- PK)(PK~PL)(P L- Pi){~

^Q (Pi ~ PK ) ( P K ~ PL)(PL~Pi){~

and

-4^0"(pl~PK)(pK~PL)(pL~Pi){~ 35PiPzPL+21(pjpK+ P K P L +Pi.Pi)

hold. We candistinguish three cases.( i) Atleast three ofpt areequal.(ii) p^pj andpK= pL.(iii) pj,pK, pLaredistinct, and the above equations hold.

But,the case(i) can not holdby thesimilar reason mentionedat the firstpartofSection 1.6. In the case(ii), (E-0),(E-l),(E-21), (E-31)and(E-41)implytheequations

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3 5 6 T A K E T O M O M I T S U IJL21

1 ' 2P, PK 6P2, Pi Ty = 0,

JL\ ;6Pi PK J2

Pi PK in x0 -1= o

P/ Px ^P? Pi

120.

-1

Thus, by the same reason as for case (i), these equations lead to a contradiction.In thecase (iii),weeasilysee theequations

PiPK+PxPL+PLP/= yand

PlPKPL= 35 >

which implythat p/9 pK 9 pL aredistinct roots of thecubic equation(1.7.3) *3-yJC2+y;c--^=0.

The cubic equation is irreducible and has three distinct real roots given asfollows. Let 9be anangle such that(1.7.4) cos30= (o< y)Then,therootsare

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*0= y (3 + 2^2cos 0),= -L(3- J2cosO+ ^6 sin6),1.7.5)

Some algebraic properties on the equation (3) are the followings:Lemma1.7.1. TherootsR0, Ri9 j R _ areequal ton o n e0/0, 1/7, 1/3.Proof. Substitution of 0, 1/7 and 1/3 into the cubic polynomial of (3)

gives -1/35, 16/1715and8/945, respectively. DLemma1.7.2. The equation (3 ) has no common roots wi th the cubicequation(1.7.6) *3--|*2-{

Proof. Put

and

The Sylvester's determinantD(/l9/2)isequalto -3437/32768000. DLemma1.7,3. The equation (3) has no roots, one of which is the triple of

another.Proof. Assumethat one of roots is equal to the triple of another, then

th eroot satisfy another cubic equation

That is ,^3 x2 + x = 0

Putfi(x)=x -y*2+y*-35

an d

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358 TAKETOMO MITSUI

f2 \x) = x ~-j"x +2 *~945TheS y lvester 's determinantD (fl9 /2) isequalto 80384/41351522625. Q

Case 1.1. Pi=p2.From (E-45) and (E-57), the assumption gives Pi=p2=1/7. Due to

Lemma1.7.1,weleadto a contradiction.Case 1.2. Pi=p3.W e may assume that pl5 p2, p4 are distinct. From (E-0), (E-l) and

(E-21),wehave the equation

(1.7.9)

The solutionof (9) isgivenby

/ I 1 1 \Pi P2 P4

P I P i p i i

i \V2\ ^ 1 =

1 \2Ti

\ 1 2 /

V4 /

/ IP i\ p ll

P2

Pi

1VP A .* i

1 l\2161\ 1 2 /

12(p1-p2)(p4-p1)-6p4p +2(p4-12(p1-p2)(p2-p4)

12(p2~p4)(P4-Pi)N ote that vi+ v3,v2, v4 can not vanish. The reason is as follows: For ex-ample, assume that 6p2p4+ 2(p2+ p4) 1 = 0. Then

* 6p2-2holds. S ubstituting this intothe cubic equation

we see that

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RUNGE-KUTTA TYPE INTEGRATION FORMU LAS 359

64pj-24pj-12p2+ 3 _n35(6p2-2)3But , by Lemma 1.7.2, there is no common roots for the cubic equations

an d

Then, from the equations(1) and (2) wehavev 1A l +v3A3=v2A2=v4A4=0

and

Since v2^0,A2= B2=Q holds. This means the equations 2i=p2/2 andP i T 2 i = P 2 / 6 3 which imply 3p =p2. H ence, we lead to a contradiction b yvirtue of Lemma 1.7.3.

Case1.3. p1=p4.This is equivalent to Case1.2.Case 1 A. p2= p3.W e can assume that pl9 p2, p4 are distinct. From (E-0), (E-l), (E-21),

wehave th e solution

12(p1-p2)(p4-p1)12(Pl-p2)(p2-p4)

v-v+v -

v-4 12(p2-p4)(p4-p1)N o t ethat each numerator on the right can not vanishby the same reason as inCase1.2. Then, (1 ) implies v1^L1=v2^4 2+ v 3 y 4 3 =v4^4 4= 0. Since v1^0,wese e^i=03 which meansPI =0. Thus,weleadto a contradiction.

Case1.5. p2= p4and

Case1.6. p3= p4.

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360 TAKETOMO MITSUI

Both of them are equivalent to Case 1.4.Case 2. N o two ofpl9 p2, p3, p4 are equal.The equation (1) implies vlAl=v2A2=v3A3=v4A4=Q. Thus, we dis-

tinguish cases according as vfor At vanishes. W e have, however,the followingresults.

Lemma1.7.4. Thecase ofvj = vK= vL=Qcan not occur.Proof. In this case, v7= 1/2 by (E-0). Then , (E-J ) implies pr=1/3.But,

they do not satisfy (E-21). DLemma1.7.5. Thecasefo r vK= vL= 0 leads to acontradiction.Proof. The equation vK= vL=Qy ields a linear sy stem

y ,~

by (E-0),(E-l). SincePi^pj, this systemhas thesolutionl-3pj = l-3p,V/

6(Pl-Pj) ' Vj

6(Pj-Pl) 'Substitution of this into(E-21) an d(E-31)implies

Pu t X PI+ PJ, Y=pfpj9 we have

Thus, we easily seethat X = 4/5, 7=1/10. That is , pL an d PJare equal to theroots of the quadratic equation(1.7.10) X2_+ ss0fwhich has real distinct roots. O n the other hand, we seethat

the lefton(E-4l) =vIpj + Vjpj

19600"

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RUNGE-KUTTA TYPE INTEGRATION F O R M U L A S 361

which is a contradic tion. DDue to the above Lemmas, wedist inguish five cases.Case 2. 1. v ,=0an d A2= A2= A4 =Q. B y a similar consideration as in

Case 1. we seethat p2, p3, p4 satisfy

P2P3+P3P4+ P4P2= ~f

P2P3P4-J5

Hence, they are equal to the distinct rootsof the cubic equation (3). N ote that ,contrary to Case 1,pi isequal to none of them.

On the other hand, th e equation (2 )yields v2B2=v3B3=v4B 4= 0. TakingLemma 1.7.5 into account, we are sufficient to consider th e case B2= B3= B4=Q.

A 2= B2=Q implies th e equations t2\p\\1 and P^2\=P\I6, which givePi= =P2/3. ^3= B3= 0 implies a linear system

which has the solution

1-7.11) T

31= rX~ TL LL~> T32 =

Since p2, p3, p4 are distinct,(E -0), ( E - l ) , (E-21) give th e solution for v2, v3, v4as

(1.7.12) 12(p2-p3)(p3-p4) 9

V4= I2(p3-p4)(p4-p2)(E-44)gives the equation

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362 TAKETOMO MITSUI

v 4 (pfT 4 1+pii42+p2T43)= 4n -360-l1-n4_;-/>2//-L36 0 T31

= 360~T8V2/?2- Isv3^2Pi (4p3- 3p2)by (9). Hence,we may represent the left on (E-54) as the polynomialof p2and p3. By (9) and (10),

31+piT32+V4p4(pfT41 ~

v3p2pi (4p3- 3p2)

378QQ(p2-p3)

Let usdenotethenumeratorof theaboveby (p(p2, p3). Thequestioniswhether

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RUNGE-KUTTA TYPE INTEGRATION FO RMULAS 363

Computation by interval arithmetic shows the following:i)e[-11.6391, -11.6390],-i)e[-4.70784, -4.70783],

(p(Rl9 R0) 6 [3.20479, 3.20480],< p ( R l9 R _ i ) e [0.821791, 0.821792],

_ ,*o)e[12.1750, 12.1751],,Ki)e[-0.956777, -0.956776].

(O n the interval ari thm etic, see[8]. A bo ve calculation is carried out by theprogram m ade by K. Ichida on H ITA C VO S 3 at the E ducational Center forInformation P rocessing, Ky oto U niv.) N one of them vanishes under thecondition (4 ) because every interval given above is away from zero. Thus, wehave a contradiction.

Case 2.2. v2= 0 and Ai = A3= A4= . A1=Q impl ies P t= 0. Then,P 2 > Pa*P4can nt vanish. From (E-l),(E-21),wehave

3 12p3(p3-p4) ' K4 12p4(p4-p3) 'S ubstituting these into(E-31),(E-41),w e seethat

35(P 3+P4)2-2p3p4(P3+P4)-P3p4= y .

Put J = p3+ p4, 7=p3p4, then wehave X = l, 7=1/5. The left on (E-51)isequalto

which is a contradiction.Case 2.3. v3= 0 and Ai=A2= A4.Q, E qui vale nt to Case 2.2.Case 2.4. v4=0andA1=A2= A3=Q. Equivalent to Case 2.2.Case 2.5. Ai=A2= A3= A4=Q. A1=Q implies Pi=0, which means

B 1=0. Then, theequations S vA = v^Bj= vp?B r.= 0 yield v2B2=v3B3=v4B4=Q. Since v2s v3, v4 are assumed to be non-zero, wehave B2= B3= B4

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364 T A K E T O M O M I T S U I

= 0. # 2 = P i T 2i -P2/6 = 0 impliesp2=Q because p1 = 0 . This contradicts theassumption that no two of p{ are equal.N ow , we have accomplished to investigate the whole cases. In con clusio n,we have

Theorem 8. The explicit (1 ,4)-stageformula can no t attain order 1. Itsattainable order is6.

Note. S H I N T A N I gives (1, 4)-stage fo rm ula w ith param eters v1= l/20, v2v3-8/45, v4=7(7-N /2T)/360, ^=0. p2=(7-v/2l)/14,T2,=1/2, T 3 1 = ( 3 ~ X / 2 T ) / 1 9 2 ,T32=( 2 l - fv / 2 T ) / J 9 2 , p 4= (7 +v/2l)/

14 , T 4 1 = ( 2 1+5x/2l)/294, T42=(N/2l-3)/84, T43=( 21+v ' 2f) /147. These pa-rameters arealso not unique solutionof(E-0) -(E-46).

References[ 1 ] B utcher,J. C ., Coefficients for the s tudy of Ru nge -K utta integration processes, J .Austral. Math. Soc., 3 (1963), 185-201.[2] , O n Run ge-K utta processes of high order, J. Austral. Math. Soc., 4(1964),

179-194.[3] , Im plicit Ru nge -K utta processes, Math. Comp ut. , 18 (1964),50-64.[4] __ _ 5 Qn tjie attainable order of Runge-Kutta methods, Math. Comp ut. , 19(1965),408-417.[ 5 ] Cash, J. R., H igh order methods for the num erical integration of ordina ry differential

equations, Numer. Math., 30 (1978),385-409.[6 ] Hearn , A. C., R E D U C E 2 User's Man ual, second ed., Un iv . U tah, M arch 1973.[ 7 ] Kaps, P. and Rentrop. P., Generalized R ung e-K utta methods of order four with

stepsize control for stiff ordinary differential equations, Numer. Math.,33 (1979),55-68.[8] M oore , R. E. , Interval Analysis, Prentice-H all Inc., N ew Jersey, 1966.[ 9 ] N 0rsett, S. P. and W aifbr and t, A., O rder con ditions for R osenbrook-ty pe method,Numer. Math., 32 (1979), 1-15.[ 1 0 ] Riordan, J., Derivatives of composite functions, Bull. A. M. S., 52 (1946),664-667.[ 1 1 ] Rosenbrock, H . H ., Some general implicit processes for the num erical solutio n of

differential equations, ComputerJ ., 5 (1963), 329-330.[ 1 2 ] S hintani, H ., O n one-step me thod s utilizing the second derivative, Hiroshima Math.J., I (1971),349-372.[ 1 3 ] - , O n explicit one-step methods utilizing the second derivative, Hiroshima

Math.J., 2(1972),353-368.[ 1 4 ] Toda, H ., O n the truncation error of a limiting formula of Runge-Kutta methods,Res. Electrotech. Lab., N o. 772, 1977.[ 1 5 ] Urabe, M ., An implicit one-step method of high order accuracy for the numerical

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