metodos runge- kutta

8/19/2019 metodos runge- kutta

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Publ RIMS, Kyoto Univ.23 (1987), 583-611

Pseudo Runge-Kutta

By

Masaharu NAKASHIMA*

§ 0. Introduction

In this paper we shall stud y numerical methods for ordina rydifferential equ ations of the initial value problem :

y ' — (#9 y )( O . l ) r J * J(y (*o) =yo .W e shall assume that the function f(x 9 y ) satisfies the following

conditions :(A ) f ( x9 y ) is defined and continuous in the strip

(B) Th ere exists a constant L such that for any x with a


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5 8 4 M A S A H A R U N A K A S H I M A

(0. 2) y n+l = yn+h@ (xn,yn; h),

J C n ,

This method was first proposed by Runge [20] and subsequentlydeveloped by H eun [12] and K utta [13]. A ccord ing to the choice

of the stage number r and the parameters ah bih w,:, we shall derivevarious methods. In [1], [2], [3] and [4], B utch er has proved thefollowing results for Run ge-K utta method (0. 2) :

A(0=r (r = l , 2 , 3 , 4 ) ,A (5) =4,

Pi (6 ) = 5,A (7) =6,A 8) -6,

A 0) =7,A ( 1 0 ) = 7 ,A(H)=8,A(0=r-2 ( 1 2 < r ) ,

w h e re A (r) denotes the highest order that can be attained by ther-stage m ethod (0. 2) above.

A s we can see, th e r-stage Runge-K ut ta method (0. 2) requires rfunctional evaluations per step. W e shall look for other Run ge-K uttatype methods which have th e same order as (0. 2), but which requiresfewer funct ional evaluat ions than (0 .2 ) .


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PSEUDO RUNGE-KUTTA PROCESSES 585

A H =/(*», JO,

*u=/(*.+«A y. + h gX*u) (J=2, 3, •», r),

« < = S*«, ( i=2,3, - , r ) .y = i

In [6], [7] It is shown that the two-step Runge-Kutta method(00 3) has order

Gostabile [8] has also proposed the following pseudo-Runge-Kuttamethod :

( 0 . 4 ) JWi=J^ + Ai>,.*,.,

This method has the same order as that of the two-step Runge-Kuttamethod (0. 3) in the same stage0 Therefore, the two-step Runge-

Kutta method (0. 3) and the pseudo-Runge-Kutta method (0. 4)require fewer functional evaluations than Runge-Kutta method (0. 2).However computational experiments indicate that the local accuracyof the two methods (0. 3) and (0. 4) is frequently Inferior to that ofthe Runge-Kutta method 0.2).

To improve this defect, we [14], [15] have proposed anotherpseudo-Runge-Kutta method. O ur method is defined by

(0. 5) y n+l = vl j;M_i + v2 yn + h® *B_b xm j; n_ l9 jvn; A ) ,r

® x u-i, x m y n -i, y^ h)=ĥ w tki9z = 0

^0 =/ (*»- , JVn-l) , ki =J (X m y n ) ,

j=Q

The method (0. 5) has the following order :

In comparing our method and other three methods (0. 2) , (0. 3) and(0. 4) in the same order 3 our method requires fewer functional


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586 M A S A H A R U N A K A S H I M A

evaluations than the other methods and has almost the same accuracy

as Runge-Kutta method (0. 2) . Therefore our method is more economi-

cal than the other three methods.

The methods discussed above are all explicit. The main advantage

of explicit methods is that they give numerica l solutions explicitly at

each step. However, they are inadequate for the stiff problem [10].

A drawback of the classical Runge-Kutta method (0, 2) for the stiff

problem can be overcome by introducing stable methods.

J. C. Butcher [1], [3] was the first who considered an implicit

Runge-Kutta method, which is ^4-stable. The general r-stage implicit

Runge-Kutta method is defined by :

(0.6) y n+ i=

k,=f(x.+a,h, y n+h bukj),i=2

^ = 2^(1 = 2,3,4, - - . , r ) .

It is possible to consider some other implicit pseudo-Runge-Kuttamethods in a way similar to that of the implicit Runge-Kutta method

(00 6) . Our r-stage implicit pseudo-Runge-Kutta method is :

(0. 7) J n +i=vij n -i+v 2 yn +h0(x n - l, *M, jV- i?j y« ;A) ,r

0 (*»-i, * w j; n _ l5 y n ; A) =Z Wiki9z= 0

£o=(*„- , J>»-i) , * =(*», J)O

It will be seen that it is equivalent to certain implicit Runge-Kutta

methods in some special cases. For example, if we take r=2, V I = WQ= Wi=b 2=b 2o= ^2i

= 0, 02 = 1, w 2 = O o 5 , a 2 = 0. 5, 6 2 2 = 0. 5 in ( 0B7 ) , thenthe method (0. 7) becomes the second order Gauss-type implicit

method, and if r = 3, v 2 =l, Vi = wQ = Q, w 2~ ^ 3 = 0 « 5 , a 2 = 0.5 — V3/6 ,

fla=0. 5- V3/6, £ 2 = 6 2 0 = 0 , i a =0. 25, 6 22 -0. 25- V3/6, 63= 63o = 0,

631 = 0 8 2 5 — V 3/6 and 6 5 2 = 0. 25, then the method (0.7) becomes an

implicit Runge-Kutta method of order 4. In contrast with explicitmethods, implicit methods increase their attainable orders. In [3],


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Butcher proved for his method (0 . 6) , which is ,4-stable, th e followingresult :

W hereas our method (0. 7) has

A 0=r + 3 r = 2 , 3 ) .

A s we have mentioned a li t t le about the stiff problem, it is importantto derive ^4-stable methods. In this paper w e derive ^4-stable methodsfo r our me thod (0 .7 ) when r = 2 and 3. In these cases, we get

A 2) =2,

A 3) = ,

where A 00 i g the order that can be a t ta ined by an r-stage ^.-stablemethod.

We now outline th e organizat ion of this paper0 I n section 1 , wediscuss the attainable order with 2-, 3- and 4-stage methods ( 005 )8In section 2, we discuss the attainable order with 2 and 3 stageimplici t methods (0. 7) . In section 3 , we ana lyze a stability of theimplici t method (0 .7 ) . Finally, in section 43 w e give some num ericalexamples, which show some useful propert ies of our method. W ehave n ot discussed the selection of param eters app earin g in the me thod(0. 7) and other related results. These results will appear soon.

§ 1. Explicit M ethod

§1.1 N on-Bxlstence of O rder 5 with r= 2

T o investigate th e a t ta inable order of pseudo-Runge-Kutta method(0 .5) , w e are requi red to expand the formula ( 0 .5) as i ts Taylorseries about th e point (xm jyn) , where we a ss u m e t h a t j y O O — V n > Detailscan be f ound in [18], so shall be omit ted.

W e check whether it is possible for the method ( 00 5) to haveorder 5 with 2-stage, by investigating order conditions in two caseswhen w2 is equa l to zero and otherwise.

Case 1 .1 : w2=£Q. W e need the following order conditions.

( 1 . 1 ) C- 1 .)^ + £̂ -1̂ =4- (j = l , 2 , . . - , 5 ) ,3 i=0 3

(1 .2 ) ai=(-iy+l{b0+jb2Q} ( . / = 2 , 3 , 4 ) ,


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5 8 8 M A S A H A R U N A K A S H IM A

with a0= — I , a i—Q.The equation (1.2) implies that

S o we have

(1.3)


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§ l o 28 Non-Exlstence of Order 6 r = 3

B y going through the same procedure as in the case when r = 2, wecheck whether it is possible for the method (0. 5) to have order 6with 3-stage. We consider two cases (A) : a2=£a3 and (B) a 2 = a^

(A) : a 2 ^a 3 .S o we consider order conditions in the following four cases0

Case 2. 1: w 2±Q, ŵQ,W e need the following order conditions :

(1.4) (

~iyVl

+Earl

w i = 4~ (; = 1,2 ,. -,6),J * = ° J

(1.5) fli=(-D' +1 {

(1.6) ai=(-iy +l bs+j{(-iy +l b» + iil- lbx} (7=2,3,4).

The equations (1.5) and (1.6) imply that

(1.7) a 2 =-l,Q and fl 3 =-l,0.

If we define

A

/ J L _ i2

-— 1 a2 a23 2 3

1 alal

1 1 4 4

\ "5"

\ 1 »

6 fl2^3

/ 1 \

/ „ \

, ut=

1 I \

W 0

\ wJ

and V $ =

~2\13

1415

1 /\ ~ 6 7

then the equation (1.4) can be expresses as

In the case (1.7), we have

Rank(D 3 (a 25 0 3 ))=2 and Rank (A) =3.

Thus the equations (1. 4), (1.5) and (1.6) have no solution,,Case 2 0 2 :


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In this case, we need the condition (1. 4) with w 3 = 0. Define

A t e =

/ 1 , \T i «i\

-— 1 a l14 2

1 4/

y flz / 1

/ 2

i31415 /

V5 =

1

31415

16 J

, Ate) =

1 2\-y 02

I - . « i_ 1 14

5 fl2

-i- -1 al\ 6 fl2 /

, £/«=

Bl \a > o

w2

-I/

^ 4 ) 5 and

Then we need the conditions det ( A (^2) ) = det ( A fe) ) = 0» Since

i-« 2 -3) -0,

w e have


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PSEUDO RUNGE-KUTTA PROCESSES 59 1

(B): a 2 =a 3 .

In this case, we need the order condition (L 4) with a 2 =a^ If wedef ine

C/5WQ

_ I

then the equation (1.4) can be expressed as

5 = 0 and A(

Therefore we get a 2 = — l, 0 0 As we have seen In Case 2 0 2, the equa-

tion (1.4) has no solution.

§1.3. Non-Existence of Order 7 with r=4

Proceeding as before, we check whether it is possible for the method

(0. 5) to have order 7 with 4-stage 0 We consider the two cases (A) :

a* = £f l,( i =£.;') and (B) : f l , -= f ly ( i=£ / ) -(A): ai *aj (i*j).

Let us consider the order conditions in the following eight cases,

Case 3 B 1 : w 2 ^0 9 w^O and w±=£Qa In this case, we need the

following order conditions:

(1.8)J i= Q J

(1.9) 0*=(

(1. 10) flH(

(1. 11) fli=(

The equations (1.9) , (L 10) and (1.11) imply that

(L 12) 02= -1,0, fl 3 = -1,0 and a,= -l,Q e

Define


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« =H^0


— —1 U 2U3U±\ ui -y-

--Q- 1 "2"3"4 «XO -Q-

— — 1 a2alal , C/6 = w2 and V Q -4

- i -i -•

Then the equation (1.8) can be expressed as

In the case (1. 12), we have

Rank (A) =2, Rank (A) =3,

where A=(A, F 6 ).Therefore, the equations (1.8), (1.9), (1.10) and (1.11) have no

solution.

Case 3 . 2 : w 2 = Q, w^O and z e >4= £ 0 . In the case b & W z + b&Wt^Q,we need the same conditions as that of case 3. 1. And in the case^32^3 + ^ 4 2 ^ 4 — 0 , we need the conditions (1.8), (1.10) and (1.11)with b&=b42 = Q9 which imply a 3= — 1,0 and a 4= — 1 ? 0. Thus we

can prove the non-existence of the solution of (1. 8) in a way similar

to that of Case 2. 2,

Case 3 .3: z e ^ ^ O , w 3 = 0 and w^Q, In this case, we need thefol lowing condition, in addition to (1.8):

(1.13) flj=(-l)' +1 02+iU (f=2,3,4,5).

The equation (1. 13) implies that a 2 = Q. Therefore, as we haveconsidered in Case 2 .2 , there are no solutions satisfying (1.8).

Case 3 . 4 : w 2 = 0, w 3 = 0 and w^Q. In this case, we need thesame order conditions as that of Case 2. 2. Therefore we can prove

the non-existence of solutions of (1.8).

Case 3.5: ^ 2 ^0, w 3 ^0 and z e >4 = 0. In this case, we need thesame conditions as that of the case r = 3 (see Section 1 .2 ) . Thus wecannot get order 7.

Case 3. 6: z#2 = 0, le^O and ^ 4 = 0. This is equivalent to Case 2. 2 eCase 3 . 7 : w23=Q9 w 3 = Q and w 4 = Q. This is equivalent to Case


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2,2,

Case 3, 8: w 2 = Q, w 3 = 0 and w± = Qa In this case9 as we haveconsidered in Case 1.2, the equation (1.8) has no solution,, Nowlet us go on to

(B): fl,-=fly (i*j).Let us consider the order conditions in the six cases.

Case 3.9: b b w^Q. In this case, we need the conditions (1.8)

and (1.9) without assuming a i= â j(i^j) B Therefore we can prove

the non-existence of solutions of (1.8) and (1.9) in a way similar

to that of Case 2, 2,

Case 3. 10: w 4 = 0 e This is equivalent to Case 3 0 6 0Case 3 0 11 : £ 4 3 = 0,, 632^3 + 6 4 2 ^ 4 ̂ 0 , In this case9 w e need th e

condition (1. 13) without assuming 0. = 0 , - ( z = £ ; ) > and we get < 22 = 00Therefore we can prove the non-existence of solutions of (L 8) in a

w ay similar to that of Case 20 28Case 3 , 1 2 : 6 4 3 = 0 5 , 6 3 2 ^ 3 + 6 4 2 ^ 4 = 0. In this case9 if we assume a 2 = a

then w e need the condition (1.11) with 6 4 2 = 6 4 3 = 03 which implies< z4= — 1, O o If we assume a 2 ~a^ then we need th e condition (1. 10)

with 632 = 0 which implies < 23= — 1909 and if we assume < 23 = < 24 9 then weneed the condition (1 ,9 ) which implies a 2 = — 1.0. Therefore we can

prove the non-existence of solutions of (L 8) in a way similar to that

of Case 28 2 .Case 3 0 13: 6 32 = 0D In this case, if we assume a 2=â then we

need the condition (1.11) with 642 = 643 = 0 3 j= 3 9 4 9 5 9 which implies< 24= — 19 08 I f we assume a2 = a ̂ then w e need the condition (1 . 10)with 632 = 0 which implies a3= —19 00 If we assume < 23 = < 24 9 then weneed the condition (L 9) with which implies a 2 = — 1 9 0 0 Therefore

w e can prove the non-existence of solutions of (L 8) in a way similarto that of Case 2a 20

Case 3. 14: a 2 = a ̂= a4a In this case, we can prove the non-existenceof solutions (1. 8) in a way similar to that of Case 2e 2a Consequentlyw e can conclude the fo l lowing result,,

Theorem, The attainable order is 49 5 and 6 for the explicit pseudo-Runge-Kutta method (08 5) o f 29 3 and 4 stage respectively.


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§2. Implicit MethodIn this section, we consider the r-stage implicit pseudo-Runge-

Kutta method ( 00 7 ) . The functions kt ( i = 2, 3 , • • • , r) are no longerdefined explicitly but b y the set of r— 1 implicit equations. W eassume that the solution o f each o f r — l implicit equations. W eassume that the solution of each function k{ may be expressed in thefo rm:

l) (i = 2, 3, »., r),

and

§ 2 1 Non-Existence of Order 6 with r= 2

In [16], we have alrea dy proved the existence of order 5 with

2-stage. We now check whether there exists order 6 with 2-stage,

by investigating the order conditions in the two cases when w 2 is

equal zero and otherwise.

Case I: w2=£Q. We need the following order conditions:

Let us define

A=

A=

i2

~ yi4

i213

15

i

- 1

1

-1

l

, U 7 =\ w 0 1 and F 7 = 1 0 2

J_

2J_3

J_4 /

1 \2 5

, A=

13

±

16


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A = ( A , F T ) .

Then the equation (2. 1) can be expressed as

D7U7= V7, DaUz=0 and D9C/g= 0.

A n d w e have

( 2 . 2 ) det(A) = -^fl»(«« + l) (5«S-fl,-2),

det (A) = - ̂ 1 fa + 1 ) (6«i - a, - 3) .

Therefore, in order to have th e solution of (20 1 ) we havefl2=-l,0.

However, for a2= — l ,Q we have

Rank (A) =2, and Rank (A) =3.

Thus the eq uation (2. 1) has no solution,

Case II: w2 = Q. I n this case, w e need the condition (2 . 1) with^2 = 0, as we have seen in Case 1. 19 th e equat ion (2. 1) has no solution.

§ 2 * 2 , Non-Exlstenee of Order 7 with r = 3

In [15], w e have already proved th e existence of order 6 with3-stage. Proceeding as before, w e check whether there exists order 7with 3-stage, by investigating the order conditions on the two caseswhen w3 is equal zero and otherwise.

Case I: ^3^0. In this case, we need the following orderconditions :

(2 .3 ) ( zz

( 2 . 4 ) (~iyJ

with

+ai-1*. /} ( i=2, 3, •», 7, j= 2, 3),

?»=(-DI

'-1

,?a = 0.Define


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L 2/ i

1 3

14

1

\ 5

\l

2 3

I 9 3

1 3 30 2 03

1 4 409 03

— 1 01 03 //

' DU

-1 -2 2a 2 1a 3 \ + ̂~i\j «31

1 0 Q 2 Q 2O O0 2 303 —:4

= — 1 — 4 40 | 401 -=-+-^ — £ e > o5 5

1 R R 4 R 4 _ »11 O »30 2 v)03 >-. "7̂ ~0 D

1 l— 1 — 6 60 2 60 3 -=- + -=- — W QI

/ ~ ~ 3 ~

1415

i

ry

1 02 ^11

~~ 1 02 a l

1 4 40 2 0 3

-1 a al

1 a| flS|

, £ / n =

M l

-' 10 ~

2 '

13

1415

1 /

/ 1 \

-' 12 ~

6/

r i415

16

1

T/

U*= Vi , WQ, W 2 , W — 1 ) , Ao=( Ao f/io), A2=(-Dl2

Then the equations (2. 3) and (2 e 4) can be expressed as

D1 0U= Q9 A i C A i = 0 and D^U=Q.

Therefore, in order to have the solutions of (2 B 3) and (2. 4) we need

det(Ao)=0, det(Ai)=0 and det(5 12 )=0. And we have

det ( Ao) = ~ ^2^3 ( f l2 + 1 ) (fl3 + 1 ) (02 - 03) fl O* ^3) ,

det ( Ai) = ~ a\al (a2 + 1 ) (a* + 1 ) (a 2 - a 3 ) det (As) ,

det ( A 2) = ~ ^fl§ te + 1 ) Os + 1 ) (02 - 03) & fe 3 03) ,

where

(2. 5) gi (a2, fl 3) = 1 S f l f f li - 3 (22< 23 (fl2 + 03 ) — 6 (a2 + 03) 2

(2. 6) & (fla, 03) = 42a|a| - 7< 22f l3 (a 2 + < 23) - 2 1 (01 + 01)


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1 0 3

-0.4 4 0.8 0.8As- 9

1 5(a 2 + a3— 1) — 1— 5a2a3 — -y

12 6(al + a2a3 + a| fi , , n 6 8--S- —


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Thus, the equations (2. 3) and (2. 4) have no solution,,

Case II: ^ 3 = 0. In this case, we need the order conditions (2 9 3)with ^3 = 0. Therefore , as we have seen in §2. 1, the equation (2.3)

has no solution. Thus, we can conclude the following results.

Theorem. Th e attainable order is 5 and 6 for the implicit pseudo-Runge-Kutta method (0. 7) o f 2 and 3 stage respectively.

§ 3. Stability Analysis

In this section we attempt to derive -4-stable methods. We definethe stability of the numerical method (0. 7) in the following way.Let us apply the method (0.7) to the test function y' = ly where Xis a complex constant with negative real, we have the followingdifference equation:

(3 . i) A0yn + l - AI y n - A * y n - i =o ,

where A0, A I and A2 are the function of Xh , involving th e coefficientsfl.-j b

{, b

u, v

h Wi. The numerical method (0. 7) is called

.4-stable if

( 1 ) | n l < l i = l ,2.(2) the root | 7* , - |=1 is simple.

where 7- are the characteristic roots of the difference equation (3. 1).If the method satisfies the conditions (1) and (2) for any negative

real ,̂ it is said to be 4̂ 0-stable. If we impose the [̂-stability on the

method (0.7), we can obtain the highest orders of the ^4-stability

fo r each stage r, using some results due to Wanner, Hairer andNorsett [23].

Proposition. In the method (0. 7) with A-stability imposed on, thehighest possible order of the A-stability is of order 2(r — 1) for each stager.

Proof. The proof goes as follows (see [23, §6]):

Let

be a characteristic algebraic equation of multistep method, satisfying


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the following conditions;

( a) Q,(z,R) is irreducible, Q,o(0) ^ 09 |£L ( 0 , 1) ^ = 0 ,( b) deg(Q, r ) 3 £ )8 Then G , Wanner, E , H airer

and S . Po N orsett [239 Theorem 2] showed that if Q,fe # ) i s 4-stablethen the highe st possible orde r is 2/. In our case5 as we will seelater In Section 4. 1 and 4 . 23 th e characteristic equation of ( 0 . 7 )satisfies the conditions (a) and (b ) , thus we get the above proposition,,

§ 3*1. ^-stable Method of Orde r 2 with 2-Stage

W e apply the method (0 .7) wi th r = 2 to the test function jp ' = .̂This yields

3 .2)

where

AI = -— — = [v2 + (w 1 — b22v21 — 6 22 /Z

2 IK .

We see that the method (0B 7) is of order 2 if2 1

(3.3) (-1)'-

and fur thermore we add the following conditions (3. 4) and (3. 5) ,under wh ich the method (0. 7) becomes of order 3 if 5 = 0:

(3 .4) (-D +̂Z>X- = 4-+̂3 j=0 6

— 0. 5b2 —

(3. 5) — ̂ 22^1 + £21^2 = 0,

H ere we note that the condition (3. 5) is nece ssary for J.0-stability0From (3.3), (3 .4 ) and ( 385)3 w e have

(3 .6 ) V1 = 5


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With these values plugged in ( 382 ) , th e characteristic polynomial of(3. 2) becomes

(3. 7) (1 -0. 5*%,, +1 - (6^-4-6(5)

(2-2. 5̂ + 33) %„_ = (),

If we take £ = 1 and d= — ? the difference eq uation (3. 7) has charac-

teristic roots:

r a = ( l + 0 . 5A ) / ( l - 0 . 5 A ) .

I t follows that th e method (0. 7) of order 2 with 2-stage is A-

stable. With th e values z = l and 8=—, the constants (3. 6 ) are given

b y

* > i = 05 v2 = l, w 0= -o--- T, Wl =—-— , i a =i a = o.5,-

2 a 2{a 2+l) 2

§ 3.2. .̂-stable Method of Order 4 with 3-Stage

By the same pro ced ure as in the 2-stage, we discuss the ^ 4-stabilityof order 4 with 3-stage. Let us apply the method (0. 7) with r = 3to the test function y r =ty. Then we obtain th e difference equation:

(3.8)

where

O n + Z 0 i ) A + ^2A2

2 + fei + ô) K + e22h2

31+632(1 +W-^(1 +63)} ̂3,

( ~ ~ 621633 + 623631) w2 + ( — 622631 4- 621632) ^3,— 62^2 + 63̂3) ,

62o + 62633 — 62363) w 2 + 630 + 62263 — 62632) w^


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PSEUDO RUNGE KUTTA PROCESSES 601

First, w e consider th e case when th e method is of order 40 Theconditions of order 4 with r = 3 are given by

3.9) -l)^ + Z«rX---i (i = 1,2, 3, 4)

(-1)1-1K l=Q K

3

In addition 3.9) , w e consider the following conditions where (38 10)and (3 e 11) are the conditions of order 5,

3.11) (_l)»-iA, Z = 0

(3 .12) — U + wo + S ( - t , - + 4 S a^ ;,) ^ = -i— Z (4af + 2a?) z5 i=2 J=0 5 i=2

From 3.9), 3.10), 3.11) and 3.12) we have

/ •Q iQ __ _ 2 ( 5 f l j — flg — 2) ___

2

_10a 2 — 7 — 12a 3 a 3+l) (a2 —

o = 0. S z / i + a2u >2 + a3w3 — 0.5,

A,o= -

«=2,3).

The most direct way for investigating ^4-stability conditions of themethod (0. 7) is usually the root-locus method, however we use the


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60 2 M A S A H A R U N A K A S H IM A

Schur criterion. The stability conditions for the method (0. 7) with

r = 3 are(3.14) (A ) \A 2 \ thefollowing conditions are necessary

(3.15) (W>33 ~~ *23*32) W+ —

(3.16) ( 22*33 — £23*32) w Q +( —(3. 17) \v2 + e2l+wQK +e22

(3. 18) \v l+e ll + w lh+e l2 h2 \


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Case I: J^O.

From th e .4-stability conditions (30 17) and (3. 18)9 we have(3 .22) e1 2 = Q, e22 = Q, |̂ | i = y, d l -e 2l -w Q =-— e

We see that the condition (3, 25) contradicts the condition (3. 17) =Thus the method (0.7) with r = 3 is unstable in the case J^0 0

Case II: J = 0.We use the m axim um m odulus pr incip le for the analysis of stability

conditions (A ) and (B) in (3 . 14), and we u se the following l e m m aawhose proof is not hard and so left for the reader,,

Lemma* Suppose that

diQ in ( 3 . 8 ) ,

then we have

-r- is analytic for Re (X )


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(3. 30) (/l3-/? 3 )/+ (/I 2 -2/ 21 / 2 3-/

and from ( 30 28) we have

(3.31)

where

(3. 32)

•

1 2J 21 —J llj

— C/23 2 — 22gl3 +/12 3 —

/21 2 — /22 fl l

2= — (/23 1 +/21 3 —

The fo rmula (3. 30) may be rewritten as

3. 33)

with

si

If we use the conditions ( 38 9 )5 (3 . 15) and (3. 16), then we see that

(3.34) £ 3 = 0, ̂= 0.

W e used REDUCE III to obtain (3.34), which requires rather

complicated computation. Consequently ^4-stability conditions for themethod (0.7) with (3.13), (3.19) and J = 0 are given by


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3. 35) (/| 3 -/? 3 )/+ (/i 2 -2/ 21 / 2 3-

(/l2

-2/21

/23

)/+/3

/̂ 0 for

We find the suitable parameters a2, a% and &23 by using computer,which satisfy (3. 35) „ These regions, where in particular we takep(=b22b'& — 623632) = 0. 83 and 03 = 0. 74 respectively., are sketched inFigures I and II. If we take a2 = Q.7, a 3 = Q074 and 0 = 0.83, whosepoints are in the stability region, then th e constants in (00 7) aregiven by :

_ 125 _ 644 50662 _ 98044 __ 500 ? V 2 ? Wo

~ _76 9 ? 2 76 9 ? o~ 1 137351 ? l~ 199171 3 2~ 9 1 5 11 '_ 1562500 , __ 1235187911 , _ 3895148891

3 5 2 9 2 02475411 5 2 442175000 9 2 0 247618000 J

33

8668547 , _ 25847383643 , __ 11907931606721 ' 22 J 23

_910951248 22 19234612500 J 23 114523325000

1019782373527 , __ 54331704350559 , _ 3381538061 _31

_ __3 386903125000 30 3683317750000 3 31 247618000

4156771908011 , _ 275520080942732

3288676562500 ^ 33 2708321875000

1 O r

0.5

p=O.83

J0.5 1.0

Figure (1) : The region (a2, «a) which satisfy ^4-stable condition (4.32).


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1.0 r

0.5

3 =0.743

0.5 1.0

a aFigure (2) : The region (a 2 , p) which satisfy ,4-stable condition (4. 32) .

§ 4. Numerical Examples

In this section, w e presen t some num erical results for the e qua tionswhich have been often taken on in the li terature of the numericalanalysis. W e use the following initial-value problems :

= -0.0001, *(*) = -0.001 exp(-1000*)+*.

-0. 0001*) +200 00 exp(-*)'=-*+(0.9999)exp(-0.0001*),

y(x) =2 exp(-*) -exp(- 10000* )*(*) = -exp(-*) +exp(-0. 0001*) .

The eigenvalues of Jacobian m atrix of problems I and II are(-1000, 0) and (-10000, -1) respectively. In using the method( 0 . 7 ) , it is necessary to solve a set of algebraic equations at eachstep to calculate an implicit function k{. O f course, the evaluation of

k{ requires some type of i terative procedure. W e will discuss in detailthose problems in another paper. W e use the N ewton-Raphson


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Iteration method for obtaining an approximate real solution of an

implicit function ki9 setting the initial approximation kf* on theiterative processes by

and we use the quant i ty :

as control of the iteration num ber. The iteration is continued untilE^+i ̂ become smaller than £'*3 where E * is a pre-a ssigned tolerance.The value y i necessary for the evaluation, when w e use the m ethod(0, 7) of order 2, is computed by the implicit R — K method of order4S and the value y± necessary for the method (09 7) of order 4 iscomputed by an implicit R — K m ethod of order 6. In solving animplicit function o f R — K method and the trapezoidal rule, we usethe N ew ton-R aphso n iteration. From Tables we can see that theadvan tage of our m ethod over other m ethods lies In its accura cy,Computations are done in double precision arithmetic on the FA G O MM-230 of Kyushu Universi ty.

Problem I, h=l/2 1 3 5* = 0. I E — 7 , (M: number of iterations).A bsolute Error

(Im) Pseudo-Runge-Method2-stage 3-stage

X

0.51.0

5.0

10.015.020.0

with (4.6)a2 = 0.5, z= l, < 5 = l / 6

yn— y ( x n )

0. 144E-150.0

0.0

0 .00 .00 .0

Z n — z(Xn)

0. 1165-140.2445-140.1175-120.2625-120.4065-120.2255-11

M

33

3

3

33

with (3.13)az= 0. 70, 0 3 = 0.74, p=Q. 83

y * — y (* n )

-0. 328E-8-0.328E-8-0.765E-12-0.815E-10-0.236^-12-0.246E-11

Z a -z(Xn)

0.3285-110.3285-110.1455-120.4075-120.5075-120.2375-11

M

44

4

4

4

4


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60 8 M A S A H A R U N A K A S H I M A

Trapezoidal Rule

X

0.5

1.0

5.0

10.015.02 0 . 0

yn-y(Xn) Z n -Z(x»}

-0.4035-8 0.8075-11-0.1415-8 0.1415-11-0.1505-8 0.3115-12-0.2365-8 0.2495-12-0.6005-8 0.4135-12-0.3455-8 0.1225-12

M5

5

5

5

6

5

( I m ) Runge-Method

Order-4

*-X*.)

0.5 0.2095-9 -0.1.0 0.1515-8 -0.5.0 0.2205-9 -0.

10.0 0.1675-8 -0.15.0 0.3365-9 -0.20 .0 0.2795-8 -0.

Order-6—z(xn) M y*—y(xd Zn-z(Xn) M

2095-12 3 0.3115-10 -0.3065-13 51515-11 3 0.3255-8 0.3255-11 51635-12 3 0.6335-9 -0.5755-12 21555-11 2 -0.2075-8 0.2205-11 21675-12 4 0.6825-8 -0.6615-11 21695-11 2 0.3475-10 0.1115-11 5

Relative Error(Im) Pseudo-Runge-Method

2-stage 3-stage

J.-JMy(xn)

0.5

1.0

5.0

10.015.0

20.0

Zn - z( Xn ) | jfc-X*.)* (* „ ) X*»)

zn -z(x tt }z(Xn)

0.2295-14 0.2025-2 0.6475-110.2425-140.2355-130.2625-130.2705-13

0.1125-12

(Im) Runge-Kutta MethodTrapezoidal Rule O rder 4

yn-y(xj z tt -zy(xn} z(:

(*.) yn-yM *.-*(*.) }O X*.) z(Xn)

0.3525-140.2895-130.4075-130.3385-13

0.1185-12

Order-6

>n—y(xH) z a -z(x n )y(Xn} Z(x a }

0.5 0.4035-11 0.3415+0 0.1505-11 0.2215-9 0.3225-111.0 0.1415-11 0.1505-115.0 0.1555-11 0.3265-13

10.0 0.2495-11 0.1545-12

15.0 0.6195-11 0.1115-1320 .0 0.4595-11 0.8485-13

0.3225-110.1145-120.2205-12

0.4415-120.5595-13


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PSEUDO RUNGE-KUTTA PROCESSES

Computational time

Pseudo-methods2-stage 3-stage

1.60(sec) 1.81 (sec) 1.87 (sec)

6 09

R-K(0.6)

Order-4

1.68 (sec)

Order-6

2. 18 (sec)

Problem II, A = l / 273 5* = 0. IE- 7, (M: number of iterations) .

A bsolute Error(Im) Pseudo-Runge-Method

X

0.5

1.0

5.0

10.015.020.0

2-stage

with (3. 6)02= 0 . 55 z = 1.0, 3=1/6yn-yM Z R -z(Xn) M

-0.2775-1 -0.1535-5 3-0. 1045-2 -0. 1855-5 3-0. 5605-9 -0. 1705-6 3-0.4005-11 -0.2295-8 3-0.2855-13 -0.2315-10 3-0.2025-15 -0.2055-12 3

Trapezoidal Rule

x yn-y(x^) zn

0.5 0.2795-1 -0.1.0 -0.7795-3 -0.5.0 -0.2575-8 -0.

10.0 0.2585-8 -0.15.0 0.2235-8 -0.

3-stage

with (3 .a 2 =0. 70, f la^O .

y.-yM-0.7215-2-0. 6295-4-0.1495-10-0.3435-13-0.5695-14-0.5715-15

-z(*.) M

1545-5 211875-5 211715-6 142305-8 142335-10 13

13)74, p=Q. 83zn—z(xn)

0.1165-90. 1415-90.1295-100.1765-120.3995-140.2275-14

M

5

5

5

5

5

5

20.0 -0.7325-8 -0.2105-12 11

( Im) Runge-Method

X

0.5

1.0

5.0

10.015.020.0

Order-4

yn—y(x ̂ ztt—z(x ̂ M

-0.3965-4 0.1555-11 4-0.6845-7 0.1885-11 4-0.1215-8 0.1735-12 4-0.8185-11 0.3135-14 4-0.4655-11 0.2085-15 4-0.8605-9 0.7355-15 4

O rder-6

y.-yM

-0.2135-80.2955-110.5225-130.3575-15

-0.2045-16-0.1405-11

z « — zC*,)

0.3465-150.5555-150.8885-150.8325-150.8325-150.6665-15

M

5

5

5

5

5

5


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61 0 M A S A H A R U N A K A S H IM A

Relative Error( I m ) Pseudo-Runge-Method

2-stage 3-stage

X y.-yMX * « )

0.5 0.230E-11.0 0. 143E-25.0 0.418E-7

1 0 . 0 0.444£-71 5 . 0 0.4695-72 0 . 0 0.494E-7

za —z(x n }zx a)

0.384E-50.292£-50. 171E-60.229E-80. 2315-100.2055-12

y * — y (x jyM

0.5995-20. 8625-40.1125-80. 3805-90.9835-80. 1395-6

zn— z( xn)z(xn)

0.2925-90.2225-90.1305-100.1765-120. 4005- 140.2285-14

( I m ) Rung e-Kutta MethodTrapezoidal Rule O rder 4 Order-6

y.-y(*3X*.)

0.5 0.2305-11.0 0. 1055-25.0 0. 1905-6

10. 0 0. 2825-41 5 . 0 0.3645-2

2 0 . 0 0.1775+1

zn-z( Xn )z x n )

0. 3925-50.2965-50. 172£-60.231E-80.233E-10

0.210E-11

y.-y(xn)y x n )

0. 329E-40.937E-70. 9085- 70.908E-70.7665-5

0.2105+0

Zn-z(Xn)

z(xj

0.3915-110.2975-110.1755-120.3135-140.2085-15

0.7365-15

jfc-X*.)X * » )

0. 1775-80.4045-110.3915-110.3965-110.3365-10

0.3445-3

Zn-z(Xn)

ZM

0.8715-150.8745-150.8945-150.8335-150.8335-15

0.6675-15

Computational time

Pseudo-methods R-K (0. 6)Trapezoid

2-stage 3-stage O rder-2 O rder-4

1 . 4 8 ( s e c ) 1 . 7 7 ( s e c ) 2.9 3(sec) 1.55 (sec) 2.78(se c)

Acknowledgments

This work w as done during m y stay at Research Inst i tute forM athematical S ciences, Ky oto Universi ty. I wish to express my heartythanks to Professor S . H itotumatu, Professor M . Yamagut i of KyotoUniversity and Professr T8 Mitsui of N agoya Universi ty for theirinvaluable suggestions and advice. A lso I wish to thank Professor R0Jeltsch o f TH. A a c h e n (Fed. Rep.) for his stimulating discussion andto the referee for his invaluable guidance and criticisms.


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PSEUDO RUNGE-KUTTA PROCESSES 6 11

References

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[2] - , On the attainable order of Runge-Kutta methods, Math. Comp.3 19 (1965),408-417.

[3] - 9 Implicit Runge-Kutta Processes, Math. Comp., 18 (1964), 185-201.[4] - , The non-existence of ten stage eight order explicit Runge-Kutta methods,

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(1965), 408-417.[ 6 ] Byrne, G. D. and Lambert, R. J., Pseudo-Runge-Kutta methods involving two points,

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322.[9] - 9 Metodi Pseudo-Runge-Kutta ottimali, Calcolo., 10 (1973), 101-116.[10] Dahlquist, G., Stability and error bounds in the numerical integration of ordinary

differential equations, Trans. Roy. Inst. Technol. Stockholm, 130 (1959). Stockholm 1958.[11] Henrici, P., Discrete variable methods in ordinary differential equations, John Wiley

and Sons.[12] Heun, K., Neue Methods zur approximative n Integration der Differentialgleichungen

einer unabhangigen Veranderlichen, Z. Math, Pysik., 45 (1900), 23-38.

[13] Kutta, W., Beitrag zur naherungsweisen Integration totater Differentialgleichungen, Z.Math. Pyhs., 46 (1901), 435-453.[14] Nakashima, M. , On a Pseudo-Runge-Kutta Method of order 6 , P roc. Japan Acad., 58

(1982), 66-68.[15] - , On Pseudo-Runge-Kutta Methods with 2 and 3 stages, Publ. RIMS, Kyoto

Univ., 18 (1982), 895-909.- 9 Implicit Pseudo-Runge-Kutta Processes, Publ. RIMS, Kyoto Univ., 20 (1984),39-56.

[17] - , Some implicit four and five order method with optimum processes, Publ.RIMS, Kyoto Univ., 21 (1985), 255-277.

[18] Mitsui, T., Runge-Kutta Type Integration Formulas Including the Evaluation of thesecond Derivative Part 1 Publ. RIMS, Kyoto Univ., 18 (1982), 325-364.

[19] Rosen, J. S., Multi-step Runge-Kutta methods, NASA Technical Note, NASA TND-4400 (1968).

[20] Runge, C., Uber die numerische Auflosung von Differentialgleichungen, Math, Ann.,46 (1895), 167-178.

[21] Tanaka, M., Pseudo-Runge-Kutta methods and their application to the estimation oftruncation errors in 2nd and 3rd order Runge-Kutta methods, Joho Short , 6 (1969) s406-417, in Japanese.

[22] - , On the application of Pseudo-Runge-Kutta methods, Computer-Center Univ.of Tokyo., 4 January-December (1971-1972).

[23] Wanner, G., Hairer, E,. and Norsett, S. P., Order stars and stability theorems, BIT,

18 (1978), 475-489.[24] William, B. G., Pseudo-Runge-Kutta methods of fifth order, J. Assoc. Comput. Mach.,

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