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Journal of Computational and Applied Mathematics 103 (1999) 297–300

Comment on: a two-stage fourth-order “almost” P-stable
method for oscillatory problems
T. Van Hecke1;∗ , G. Vanden Berghe, M. Van Daele, H. De Meyer 2
Vakgroep Toegepaste Wiskunde en Informatica, R.U.G.,
Universiteit-Gent, Krijgslaan 281-S9, B9000 Gent, Belgium
Received 15 June 1998; received in revised form 14 October 1998

Abstract
In Chawla and Al-Zanaidi (J. Comput. Appl. Math. 89 (1997) 115–118) a fourth-order “almost” P-stable method for
y′′ =f(x; y) is proposed. We claim that it is possible to retrieve this combination of multistep methods by means of
the theory of parameterized Runge–Kutta–Nystrom (RKN) methods and moreover to generalize the method discussed by
c 1999 Elsevier science B. V. All rights reserved.
Chawla and Al-Zanaidi (J. Comput. Appl. Math. 89 (1997) 115–118).
Keywords: ODE; Second-order IVP; Runge–Kutta–Nystrom methods; P-stability

We rst introduce a Runge–Kutta–Nystrom method in parameterized form (1)–(3) [5]
yn+1 =yn + hyn′ + h2

s

X

b i f(xn + ci h; Yi );

(1)

i=1

′
=yn′
yn+1

+h

s
X

bi f(xn + ci h; Yi )

(2)

i=1

with
′
Yi = (1 − vi )yn + vi yn+1 + (ci − vi − wi )hyn′ + wi hyn+1
+ h2

s
X

xij f(xn + cj h; Yj ):

j=1

∗

Corresponding author. E-mail: tanja.vanhecke@rug.ac.be.
Research assistant of the University of Gent.
2

Research Director of the Fund for Scientic Research (F.W.O. Belgium).
1

c 1999 Elsevier Science B.V. All rights reserved.
0377-0427/99/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 2 6 6 - 0

(3)

298

T. Van Hecke et al. / Journal of Computational and Applied Mathematics 103 (1999) 297–300

An RKN method is therefore also characterized by the tableau
cvw X
T
b
bT

(4)

whereby v and w are vectors with, respectively, the entries vi and wi (i = 1; 2; : : : ; s), and whereby X
is an (s × s) matrix with elements xij . The relation with the classical denition of an RKN method
[2] is A = X + v b T + wbT .
The method introduced in [1]
h2
h ′
yn+1 = yn + (yn+1
+ yn′ ) − (f(xn+1 ; yn+1 ) − f(xn ; yn ));
2
12




h
h
′
yn+1
f(xn+1 ; yn+1 ) + 4f xn + ; yn+1=2 + f(xn ; yn ) ;

= yn′ +
6
2
1
h ′
− yn′ )
yn+1=2 = (yn+1 + yn ) − (yn+1
2
8

(5)
(6)
(7)

for computing oscillating solutions of special second-order initial-value problems, is of the type of
extended single-step methods, where a combination is made of an extended trapezoidal formula,
Simpson’s rule and a three-point formula for the mid-point. This can be rewritten in the form of the
RKN method (1)–(3) in several ways, such as
0

0

1
2

1
2

1

0

0
0
0

0
0
0
1

1
− 48 − 12 − 481
1
1
0
6
3
1
6
1
6

1
3
2
3

;

(8)

0
1
6

or as
0
1

0
1

1
2

1
2

0
0

− 18

0 0 0
0 0 0
0 0 0

:

(9)

1
0 31
6
1 1 2
6 6 3

This method can be retrieved by imposing conditions on the order and stability of an s-stage RKN
method in parameterized form [5].
In order to reconstruct the method of Chawla in form (8), we start by choosing
cT = (0; 21 ; 1)T :

(10)

T. Van Hecke et al. / Journal of Computational and Applied Mathematics 103 (1999) 297–300

299

As the rst and last stage correspond, respectively, to the previous and current integration point, the
following can be motivated:
v1 = v3 = 0;

x11 = x12 = x13 = 0

and x31 = b 1 ; x32 = b 2 ; x33 = b 3 :

(11)

When all w-entries are chosen zero, the simplied expressions of the stages do no longer depend
′
on the unknown yn+1

. If one requires the method to be of order four, which can be expressed by
means of algebraical equations in terms of the coecients of the method [4], all coecients can be
written in terms of two parameters x22 and x23 .
The linear stability analysis of RKN methods is mainly determined by the notion of P-stability,
which is explained in Denition 1. We start by introducing a matrix M (H ) which arises when
the RKN method is applied to the test equation y′′ = − 2 y,  ∈ R, and setting H = h. Then the
following equation is obtained:
yn+1
′
hyn+1

!

= M (H )

yn
h yn′

!

(12)

;

where
M (H ) =

 + H 2 A)−1 c !
1 − H 2 b(I
:
1 − H 2 b(I + H 2 A)−1 c

 + H 2 A)−1 e
1 − H 2 b(I
−H 2 b(I + H 2 A)−1 e

(13)

Denition 1. An RKN method is called P-stable when
|M (H )| = 1;

2 ± Trace(M (H )) ¿ 0

(14)

is satised for all H ¿0.
It should be remarked that also the original denition of P-stability as given by Lambert and Watson
[3], leaves room for interpretation, in the sense that it is not precisely stated whether the equality sign
in the trace condition in (14) should be withheld or not. When we restrict ourselves to Denition 1,
a distinction has to be made with “almost” P-stable methods where a discrete number of H -values
are excluded in the interval of periodicity.
“Almost” P-stability is only possible for x23 = − 481 , where H 2 = 12 has to be excluded from the
interval of periodicity. Hence the one-parameter family
0
1
2

1

1
4

0
0
− 3x22 0
0
0

1
48

0
0 0
1
+ 2 x22 x22 − 481
1
1
0
6
3
1
6
1
6

1
3
2
3

(15)

0
1
6

consists of fourth-order methods which are all “almost” P-stable. For the particular case of x22 = − 121 ,
Chawla’s method (8) can be found.

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T. Van Hecke et al. / Journal of Computational and Applied Mathematics 103 (1999) 297–300

Alternatively (15) can be written as
0
1
1
2

1
4

0
1
− 2w3

0
0
w3

1
48

0
0
0
0
0
0
1
1
1
+ 6 w3 − 48 − 6 w3 0
1
6
1
6

0
1
6

(16)

1
3
2
3

where (9) is reached for w3 = − 18 .
Denition 2. A Runge–Kutta–Nystrom method has stage-order r i
C(r):

A:cq =

cq+2
;
(q + 2)(q + 1)

q = 0; 1; 2; : : : ; r:

(17)

This means that the order of the internal stages is at least r + 2, i.e.
Yi = y(xn + ci h) + O(hr+3 );

i = 1; 2; : : : ; s:

It is worth remarking that all methods in the generalized family (15) have stage-order (see Denition
2) three.
References
[1] M.M. Chawla, M.A. Al-Zanaidi, A two-stage fourth-order “almost” P-stable method for oscillatory problems,
J. Comput. Appl. Math. 89 (1997) 115 –118.
[2] E. Hairer, S.P. Nrsett, G. Wanner, Solving Ordinary Dierential Equations I, Nonsti Problems, Springer, Heidelberg,
1987.
[3] J.D. Lambert, I.A. Watson, Symmetric multistep methods for periodic initial value problems, J. Inst. Math. Appl. 18
(1976) 189 – 202.
[4] T. Van Hecke, M. Van Daele, G. Vanden Berghe, H. De Meyer, A mono-implicit Runge–Kutta–Nystrom modication
of the Numerov method, J. Comput. Appl. Math. 78 (1997) 161–178.
[5] T. Van Hecke, M. Van Daele, G. Vanden Berghe, H. De Meyer, P-stable Mono-implicit Runge–Kutta–Nystrom
modications of the Numerov Method, in: Lecture Notes in Computer Science 1196, Proc. 1st Internat. Workshop
WNAA’96, Rousse, Bulgaria, 1997, pp. 537 – 545.