A Zero-Dissipative Phase-Fitted Fourth Order Diagonally Implicit Runge-Kutta-Nyström Method for Solving Oscillatory Problems
Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2014, Article ID 985120, 8 pages
http://dx.doi.org/10.1155/2014/985120
Research Article
A Zero-Dissipative Phase-Fitted Fourth Order
Diagonally Implicit Runge-Kutta-Nyström Method
for Solving Oscillatory Problems
K. W. Moo,1 N. Senu,2 F. Ismail,2 and M. Suleiman2
1
2
Department of Mathematics, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia
Department of Mathematics and Institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor,
Malaysia
Correspondence should be addressed to K. W. Moo; mk
Received 22 March 2014; Revised 7 May 2014; Accepted 7 May 2014; Published 25 May 2014
Academic Editor: Mohamed Abd El Aziz
Copyright © 2014 K. W. Moo et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A new diagonally implicit Runge-Kutta-Nyström (DIRKN) method is constructed for solving second order differential equations
with oscillatory solutions. The method is originally based on existing DIRKN method derived by Senu et al. which is three-stage
and fourth algebraic order. The new derived method has a variable coefficient with phase-lag of order infinity. The numerical
experiments are carried out and the results show the efficiency and accuracy of the new method in comparison with the other
DIRKN methods in the literature.
1. Introduction
In this paper, we are dealing with the initial value problems
(IVPs) related to second order ordinary differential equations
(ODEs) in the form:
𝑦 = 𝑓 (𝑡, 𝑦) ,
𝑦 (𝑡0 ) = 𝑦0 ,
𝑦 (𝑡0 ) = 𝑦0 ,
(1)
where 𝑦 (𝑡) does not appear explicitly and know that their
solutions are periodic. These problems arise in many fields
of applied sciences such as astronomy, quantum mechanics,
physical chemistry, structural mechanics, and electronics.
For solving this kind of problems, one of the numerical
methods is Runge-Kutta-Nyström (RKN) method. RKN
method is frequently used due to its computational advantage
(see [1]). RKN method consists of two main classes; they
are explicit method and implicit method. Generally, it is
quite difficult to handle oscillatory problems of the form (1)
with classical RKN methods. The term phase-lag was first
introduced by Brusa and Nigro [2]. Phase-lag is the angle
between the analytical solution and the numerical solution.
For solving oscillatory problems, phase-lag is an important
property. In [3], van der Houwen and Sommeijer implemented the phase-lag theory to Runge-Kutta (RK) and RKN
methods. They presented a few explicit RK and RKN methods
with reduced phase errors. Based on the minimal phaselag theory, Senu et al. constructed a zero-dissipative RKN
method in [4]. In [5], Simos derived a Runge-Kutta-Fehlberg
method with phase-lag of order infinity. In [6], Papadopoulos
et al. extended the idea of phase-lag of order infinity to RKN
method and presented a phase-fitted RKN method. Based
on idea of phase-lag of order infinity, Papadopoulos and
Simos introduced a new methodology to construct optimized
RKN methods in [7]. Since then, Kosti et al. constructed
two optimized RKN methods with fifth algebraic order in
[8, 9]. Notice that all methods mentioned above are explicit
methods.
In [10], Sommeijer presented two DIRKN methods with
nonempty interval of periodicity. van der Houwen and Sommeijer extended the phase-lag theory to DIRKN methods
and presented a few DIRKN methods with high phase-lag
order for solving oscillatory problems in [11]. Sharp et al. also
presented a few two-stage and three-stage DIRKN methods
with high phase-lag order in [12]. Senu et al. extended the
DIRKN methods to phase-lag order up to order eight and
higher dissipative order in [13, 14]. However, there is no
DIRKN method with phase-lag of order infinity being done
2
Mathematical Problems in Engineering
yet. This motivates us to develop the DIRKN method with
phase-lag of order infinity. In this paper, we will construct a
three-stage phase-fitted DIRKN method which is based on a
three-stage method of algebraic order four derived by Senu et
al. [15].
where
𝜎1,2 =
(𝑖𝑦0 )
1
[𝑦0 ±
],
2
V
or 𝜎1,2 = |𝜎| exp (±𝑖𝜒) .
(9)
Substituting (9) into (8) yields
2. Phase Properties of RKN Method
The general form of a 𝑚-stage implicit RKN method for (1) is
given by
𝑚
+ ℎ2 ∑𝑏𝑖 𝑓 (𝑡𝑛−1 + 𝑐𝑖 ℎ, 𝑌𝑖 ) ,
𝑦𝑛 = 𝑦𝑛−1 + ℎ𝑦𝑛−1
𝑖=1
𝑦𝑛
=
𝑦𝑛−1
+
𝑚
ℎ∑𝑏𝑖 𝑓 (𝑡𝑛−1
𝑖=1
+ 𝑐𝑖 ℎ, 𝑌𝑖 ) ,
𝑌𝑖 = 𝑦𝑛−1 +
2
𝑐1 = −
+ ℎ ∑𝑎𝑖𝑗 𝑓 (𝑡𝑛−1 + 𝑐𝑗 ℎ, 𝑌𝑗 ) ,
𝑗=1
(3)
𝑖 = 1, . . . , 𝑚.
The DIRKN method above can be expressed nicely in a
Butcher table as shown as follows:
d
⋅⋅⋅ 𝜅
⋅ ⋅ ⋅ 𝑏𝑚
⋅ ⋅ ⋅ 𝑏𝑚 .
𝑦 (𝑡) = −V2 𝑦 (𝑡) ,
(4)
V ∈ 𝑅.
(5)
Applying method (2) to the test equation (5) yields
[
𝑦𝑛
ℎ𝑦𝑛
] = 𝐷𝑛 [
]
[
𝑦0
ℎ𝑦0
2
],
]
𝛾2 𝑦0 − ℎ𝑦0
,
𝛾1 − 𝛾2
𝑐2 =
𝛾1 𝑦0 − ℎ𝑦0
.
𝛾1 − 𝛾2
(12)
If 𝜌1 and 𝜌2 are complex conjugate, then 𝑐1,2 = |𝑐| exp(±𝑖𝑤)
and 𝜌1,2 = |𝜌| exp(±𝑖𝑝). By substituting both into (10), we
have
𝑦𝑛 = 2 |𝑐| 𝜌 cos (𝑤 + 𝑛𝑝) .
(13)
𝑧 = Vℎ,
(7)
which is the stability polynomial of the RKN method.
It is given that the exact solution of (5) is
𝑛
𝑦 (𝑡𝑛 ) = 𝜎1 [exp (𝑖𝑧)] + 𝜎2 [exp (−𝑖𝑧)] ,
Then, we denote that
𝑅 (𝑧2 ) = trace (𝐷) ,
𝑅 (𝑧2 ) =
(6)
where 𝐷 is the stability matrix of the RKN method and
𝐴, 𝐵, 𝐴 , and 𝐵 are polynomials in terms of 𝑧2 and totally
determined by the parameters of method (2). The characteristic equation of 𝐷 can be written as
𝜉2 − trace (𝐷) 𝜉 + det (𝐷) = 0
Definition 1 (phase-lag and amplification error; see [3]).
Apply the RKN method (2) to the test equation (5). Then we
define the phase-lag as Φ(𝑧) = 𝑧 − 𝑝. If Φ(𝑧) = 𝑂(𝑧𝑞+1 ),
then the RKN method is said to have phase-lag order 𝑞. In
addition, the quantity 𝛼(𝑧) = 1 − |𝜌| is called amplification
error. If 𝛼(𝑧) = 𝑂(𝑧𝑟+1 ), then the RKN method is said to have
dissipation order 𝑟.
𝑄 (𝑧2 ) = det (𝐷) .
(14)
For DIRKN method, let us denote 𝑅 and 𝑄 in the following
form:
2
𝐴 (𝑧 ) 𝐵 (𝑧 )
],
𝐷=[ 2
𝐴 (𝑧 ) 𝐵 (𝑧2 )
[
]
𝑛
(11)
Hence, we have the exact solution (10) and the numerical
solution (13) of (5) in the similar form. From (10) and (13) we
have the following definition.
For diagonally implicit RKN methods, the diagonal elements are equal. We denote the diagonal elements as 𝜅 so that
𝑎11 = 𝑎22 = ⋅ ⋅ ⋅ = 𝑎𝑚𝑚 = 𝜅.
The phase-lag error of method (2) is investigated by using
the homogeneous test equation in the following:
[
𝑦𝑛 = 𝑐1 𝜌1𝑛 + 𝑐2 𝜌2𝑛 ,
where
𝑚
𝑐1 𝜅
𝑐2 𝑎21 𝜅
..
..
..
.
.
.
𝑐𝑚 𝑎𝑚,1 𝑎𝑚,1
𝑏1 𝑏2
𝑏1 𝑏2
(10)
Then, we assume that the eigenvalues of 𝐷 are 𝜌1 , 𝜌2 and they
will be called as the amplification factors of the RKN method.
The consequent eigenvectors are [1, 𝛾1 ]𝑇 , [1, 𝛾2 ]𝑇 , where 𝛾𝑖 =
𝐴 /(𝜌𝑖 − 𝐵 ), 𝑖 = 1, 2. The numerical solution of (5) is
(2)
where
ℎ𝑐𝑖 𝑦𝑛−1
𝑦 (𝑡𝑛 ) = 2 |𝜎| cos (𝜒 + 𝑛𝑧) .
(8)
2 + 𝛼1 𝑧2 + ⋅ ⋅ ⋅ + 𝛼𝑚 𝑧2𝑚
,
𝑚
(1 + 𝜅𝑧2 )
1 + (...truncated)
