implicit pseudo-runge-kutta processes

Publ. RIMS, Kyoto Univ.20 (1984), 39-56

Implicit Pseudo-Runge-Kutta Processes

By

Masaharu NAKASHIMA*

§ 1. Introduction

The present paper is concerned with the numerical solution ofthe initial value problem ;

//=/(*, y),( 1 - 1 )

> ~ o =3;o-

Of all computational one step methods for the numerical solutionof this problem, the easiest formula is Euler's rule.

It is linear and explicit, but it is of low order.Higher order methods have been achieved by sacrificing the

linearity. These methods were first proposed by Runge [13] andsubsequently developed by Heun [7] and Kutta [8].

Runge-Kutta methods need some functional evaluation at eachstep. And one looked for other methods to decrease the functionalevaluation. These works were due to Ceschino, Kuntzman [4], Byrne,Lambert [1] and Rosen [12].

Now Byrne and Lambert [1] have defined 2-step Runge-Kuttamethods and Costabiles [5] has proposed Pseudo-Runge Kutta me-thods. In [9], the present author has studied Pseudo-Runge Kuttamethods and has proposed some new ones.

In [1], [5], [9] and [10], the methods are explicit forms, however,no implicit methods have been proposed as yet.

It is possible to consider implicit Pseudo-Runge Kutta methods,and in this paper, we shall give implicit Pseudo-Runge Kutta methodsbased on the methods developed in [10].

It will be seen that they are equivalent to certain implicit Runge-

Communicated by S. Hitotumatu, March 19, 1982.Department of Mathematics, Faculty of Science, Kagoshima University, Kagoshima 890,Japan.

40 MASAHARU NAKASHIMA

Kutta methods in some special cases.Their advantage over implicit Runge-Kutta methods lies in the

fact they are of less expensive in terms of the functional evaluationsfor given order.

Let m(p) be the highest order which can be attained by a^>-stage method. In [3], Butcher proved, on his method, the followingresults

m(p)=2p (p = l, 2, 3,-),

whereas our methods have

m(p)=p + 3 (P = 2, 3).

In order to apply implicit Runge-Kutta methods, it is necessary,at each step, to solve non-linear equations by some means. By mak-ing clear the algorithm of our methods of r-stage, we observe thatthe number of non-linear equations is r-1.

Let n(r) be the highest order that can be attained by non-linearequations of order r. Then by our methods

n ( r )= r+4 (r-1, 2),

and by Runge-Kutta methods

«(r)=2r (r=l, 2,-)-

§ 2. Derivation of the Methods

We consider ^-stage implicit Pseudo -Runge-Kutta methods:

(2. 1) yn+i=y»+v(yn-i-yn) + h®(xn_1) xn) yn_l9 yn\K)9

whereP

$(xn-l9 xn, yn_l9 yn\ h}=^ w, ki9»=o

Pki =f(xn +aih, (1+6,0 yn - biyn-i + h 2

(f = 2, 3,-, p) (0<a^l ; i = 2, 3,-, #).

In the above formula (2. 1), the value yn is to be an approxima-tion to the value y(xn} of the solution of (1. 1) for xn— xQ + nh. Coeffi-cients ai9 bi(i = 2, 3,--- , p] and 6£ j-(i=2, 3, • • • , p9 j = Q9 1, • • • , p] arereal constants to be determined.

IMPLICIT PSEUDO-RUNGE-KUTTA PROCESSES 41

We consider only two and three stage methods obtained by sett-ing p = 2, 3 in (2. 1). Throughout the paper, the coefficients areconstrained by

(2.2) ai = bi+ S bis (i = 2, 3,-, />)•

The functions &z-(z^;2) are no longer defined by explicitly butby the set of p — 1 implicit equations. The special case V = WQ = w1 =bi = biQ = Q in (2.1) is implicit Runge-Kutta method. The derivationof the method is rather complicated. The treatement is similar asthose in Butcher [3].

Let D be the differential operator defined by

7^ 9 i £f \ 9

and put

„ :y,) = T£(i = l, 2,-, 5), #/,(*« y»)=5 i(i=l, 23 3),

Jy\^ny 3W /y? Jyy \^ny yn) /yj>"

We also introduce an abbreviation,3 3

S = E5 Z — 2«i=2 ^ ^=2

We assume that the solutions of the functions &»(z'^2) may beexpressed in the form:

(2.3) fe^. + ScyV + OCA6),3 = 1

and

%-i=y(^»-i)-

From (2. 1), we have the following equation:

+D10f*T'+Dllf}T+Daf,,f,Q+Daf,T&+D1J',TS+Daf,T'S)


The constants {A,-}, {BJ, {Q} and {DJ are

-0

C3 =

b

Dg = - - - - W 0 + S £*«>,•,

As = 4- - wo + S "siWi ,

=- -w0 + -Wi B, =~ -

C2 =

C5 =-^H. + Wo + 2 /^w,, C6 = -

C7 =-

A =1 -n + S flf^,-, A = lo-J- - n + S

S affewj), A


• = 5!

u2j = 5 !

"3 = 5! - c & y o +-rbJi +3 = -c ryo -rrJi -TTV »J . T" . TT . ^ = 2

The £-stage method (2.1) has order p + 3 (p = 2, 3) if

_n*-i_rL-i- c — 1 ^ k~i '° j- v a^ "it — ai} { } ~

i = l, 2,...., p+3,j = 2, 3,..., A *=2, 3,..., /> + !), (a.^a, for).The case /> = 2 in (2.4) gives the following values for the con-

stants in (2. 1). It is of order 5,

(2.5) w = 77-12c, »b=-i-(45-7c), te;1=^-(33 -5c),

-(37-36'), ^ = - - ( 1 3 9 + 90, 622=-l-(9-c),2 0 ^ — - , - - , 2 2 - -

(c = V4T).

The case ^ = 3 in (2.4) gives the following values for the con-stants in (2. 1). It is of order 6,


„6)

2(5a|-fl,-2)

2(5ai-a2-2)

l(l +a3-) (a2 -a3) w3 -

5— 7^— 6ao(

, -

=6(a2 (a2 + \}bi2 +a, ( 1 +a3) b^ -2a? -3af,

6tl =af- - (6,. + 6z-o + ifc + 6,-2) 0' = 2, 3) ,

where

§ 3. Convergence of Our Methods

In order to apply the method (2. 1), it is necessary to solve, ateach step, non-linear algebraic equations. We seek a solution by aninner iterative procedure and prove the convergence of the method

(2.1).Butcher [3] studied p-stage implicit Runge-Kutta method and

gave the iterations


k?r+»=f(x+aih, y + hZb^kf^+hZ MD (' = 1, 2,..., />),j=l j=r

where k^ denotes the number after AT- times iteration for the implicitnon-linear equations of kr. Similary, we set the iteration in (2, 1)by

(3.1) W+» =f(xn+ath, (1+60%-^-iWs W+1)+Ai;3=0 j=i

Then we have the following Theorem.

Theorem. The iteration process (3. 1) converges provided

where L is the Lipshitz constant of f ( x 9 y) with respect to y and

;IM, ij^i}-The proof is done in a similar way as in Butcher [3] and we will

omit it.

§4. Stability Analysis

In this section, we discuss the stability of our methods. We apply

our method (2. 1) with p = 2 to the test equation y'=Ay which yields

V2= ( -76 + 12V4T) + (-5875. 75 + 926. 25V4T)/*+ (2286. 775-361.225V4l)A2

5

VP3 = 77-12V4l+(4642. 75-733.

The condition for the stability is that the roots of characteristicequation of this difference equation are all of absolute value lessthan 1. This yields that, if /I is negative real, the stability bound is

if /I is complex, the stability region shown in Figure (1).


Let us consider the method (2. 1) with p = 3. In that case wehave

(4.1) 3Wi

where

2h

622631

^23 — { ( ~~ 620633 + 633630) WQ + ( — 622630 + 630632) W3} /L9

L = — (1 — ft (622+633) +fi2 (622633 —623632)}.

Let us consider some specific algorithms in the formula (2.6).If we set

/4 ON h _ _ ^2(Z4+Z5Z10)^. z; 033- ^, r _,„, '

where

V _ (^2

z_ (~2a2 C2^

7 (^^3 7n 7

-2a2 -3a2)Z3 -2a3 -

a2 +a22} w2,

Z7= (al-l. 8a22-0. 8a2}Zl-\-

Z8=(al-L 8*1-0. 8«3) _fe3-1.8ai-a2

Z9= (a|-l. 8*1-0. 8a2)Z3+0. 2(3*1

7 — 1 ^ ^ _ t ; _ i _ w o _ 7 \ —710~^^\~W "30"+"5^ 9W3j 7>

Then we have


^23=0.

If we set, in addition to the condition (4.2),

a2 = Q. 98,

then we have the following stability bound

The stability region is shown in Figure (2).

If we set, in addition to the condition (4.2),

az = Q. 57,

then we have the following stability bound

The stability region is shown in Figure (3).

Next we consider the case where the coefficient 633 varies,

a2 = l,

&33={81(8V3"-14.89) -27(503. 29-272V3~)£-9(2254. 53

(4,3) -1254. 71V3") A2 -(352.8^3" -334. 8)A3}/{27(33. 11V3~

-65. 34)6+9(937. 2V33 -1861. 86) h2- (475. 2V3"

+ 118. 8) A3},

where

h=a + bi,

h=sgn (a^a2 + b2.

In this case, we have the following characteristic roots:

ft = 0.9,

with

51=( -128 + 30V3")/!!

^ = 1 - (b22 +

*2 = — (^22 + ^33) + 2 (622^33 -

Similary, we look for the stability region numerically, which is

contained in the region:

D=[(a,


The stability region is shown in Figure (4). In the case (4. 3) 3

we have &33 >°° as h >0, however the iterations of function k{

are convergent.

The stability intervals for R-K 4, Lawson's method (explicit R-K

type of order 5) and Huta's method (explicit R-K type of order 6)

are shown in Table (1). The stability regions for these methods are

shown in Figure (5).

The stability intervals for R-K 4, Lawson's method andHuta's method.

Method interval of stability

R-K4 (-2. 7, 0)

Lawson (-5. 7, 0)

Huta (-3. 7, 0)

Figure (1). Stability region for the method (2. 1) with p = 2, whose coefficientsare given by (2.5). (


-72.00 -64.00 -56.00

Figure (2). Stability region for the method (2. 1) with /> = 3, whose coefficientsare given by (2.6), (4.2) and ff2 = 0.98. (ReU)^O).

Figure (3). Stability region for the method (2. 1) with ^ = 3, whose coefficientsare given by (2.6), (4.2) and ^ = 0.57.


<35 J

Figure (4). Stability region for the method (2.1) with /> = 3, whose coefficientsare given by (2.6) and (4.3). (Retf)^O).

HUfA'ss.Kth-order formula,

LAWSON i fifth-order

Figure (5). Stability region for R-K method (order 4), Lawson's method(order 5) and Huta's method (order 6).

§5. Computational Results

In Table I and II, we present numerical results for the followinginitial value problems.


-1), y(0)=l ,,y'=-y + z,y(0)=l,y(x]

M - *(0)=0, z(a

' «(0)=5,

Computations are done in double precision arithmetic on FACOMM-200 of Kyushu University.

In Table I, the value yl necessary for the evaluations using theformula (2. 4) is computed by Runge-Kutta methods of order 4 andin Table II, the value yl necessary for the evaluations using theformula (4. 1), (4.2) is computed by Huta's method of order 6.

In order to start the calculations at (xn) 3^), we need to solvethe implicit functions &,-(&' 2). We first caluculate the predictedvalue for the functions by k0 = f(xn, yn}.

Table IError for the solutions to the Problems I, II, III, and IV. Comparison oferrors incurred by using the implicit Runge-Kutta method of order 4 (ImR-K 4), Lawson's method (explicit R-K type of order 5) and the method(2.1) with the coefficients (2.5) (order 5).

Problem I. A=l/24 , M=5 (M : number of iterations),

Lawson's Method Im R-K 4

x function evaluations/step6 11(1+2XAO

2 -0.9970E-105 -0.7074E-11

9 0.7337E-1213 0.5673E-13

0. 6963E-080. 6034E-090.2277E-100.2142E-11

method (2. 1)with (2. 5)

6(l + lxAf)

-0. 1442E-08-0. 1408E-08-0.6463E-11

-0.2671E-12

Problem II. /*=l/24, M=5.

x Lawson's Method

1 0.4180E-084 0. 1169E-098 0. 1375E-11

12 0.2442E-13

Im R-K 4

-0. 1528E-07-0.4188E-10-0.4188E-10-0. 1026E-11

method (2. 1)with (2. 5)

0. 1082E-07-0.2092E-12-0.2092E-12-0.2423E-13


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Acknowledgements

The author wishes to express his hearty thanks to Prof. M. Sakaiof Kagoshima University and Prof. S. Huzino of Kyushu Universityfor helpful discussions and suggestions.

He would also like to express his sincere thanks to Prof. S.Hitotumatu of Kyoto University, Prof. M. Tanaka of YamanashiUniversity and Prof. T. Mitsui of Fukui University for their invaluablesuggestions and advice. In the last, we would like to thank the refereefor numerious fruitful comments.

References

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[3] — _.} Implicit Runge-Kutta Processes, Math Comp., 18 (1964), 50-64.[4] Ceschino, F. and Kuntzmann, J. Numerical Solution of Initial value problems, Prentice-

Hall, Englewood Cliffs, New Jersey, 1966.[5] Costabile, F., Methodi Pseudo-Runge-Kutta di seconda specie, Calcolo., 1 (1970), 305-

322.[6] Henrici, P. Discrete Variable Methods in Ordinary Differential Equations, John Wiley &

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einer unabhangigen Veranderlichen, Z. Math. Physik., 45 (1900), 23-38.[ 8 ] Kutta, W., Beitrag zur naherungsweizen Integration totaler Differential-gleichungen,

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implicit pseudo-runge-kutta processes

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