A New Phase-and Amplification-Fitted Sixth-Order Explicit RKN Method to Solve Oscillating Systems

Abstract

An optimization of the sixth-order explicit Runge-Kutta-Nyström method with six stages derived by El-Mikkawy and Rahmo using the phase-fitted and amplification-fitted techniques with constant step-size is constructed in this paper. The new adapted method integrates exactly the common test: y = −w 2 y. The local truncation error of the new method is computed, showing that the order of convergence is maintained. The stability analysis is addressed, showing that the developed method is "almost" P-stable. The numerical experiments demonstrate the high performance of the proposed scheme compared to other existing explicit RKN codes with six stages and same order.
MSC: 65L05; 65L06

Full text

ISSN 1686-0209
Thai Journal of Mathematics
Volume 21 Number 1 (2023)
Pages 219–236
http://thaijmath.in.cmu.ac.th
A New Phase- and Amplification-Fitted Sixth-Order
Explicit RKN Method to Solve Oscillating Systems
Musa Ahmed Demba1,3,4,5, Higinio Ramos2,∗, Wiboonsak Watthayu1and Idris Ahmed6
1Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi,
Bangkok 10140, Thailand
e-mail : musdem2004@gmail.com (M. A. Demba); wibonsak.wat@mail.kmutt.ac.th (W. Watthayu)
2Department of Applied Mathematics, Faculty of Sciences, University of Salamanca, Salamanca 37008,
Spain
e-mail : higra@usal.es (H. Ramos)
3KMUTTFixed Point Research Laboratory, KMUTT-Fixed Point Theory and Applications Research Group,
Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT),
126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand
4Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Science Laboratory Building,
King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru,
Bangkok 10140, Thailand
5Department of Mathematics, Faculty of Computing and Mathematical Sciences, Kano University of Science
and Technology, Wudil, P.M.B 3244 Kano State, Nigeria
e-mail : demba@kustwudil.edu.ng (M. A. Demba)
6Department of Mathematics, Sule Lamido University, P.M.B 048 Kafin-Hausa, Jigawa State, Nigeria
e-mail : idrisahamedgml1988@gmail.com (I. Ahmed)
Abstract An optimization of the sixth-order explicit Runge-Kutta-Nystr¨om method with six stages de-
rived by El-Mikkawy and Rahmo using the phase-fitted and amplification-fitted techniques with constant
step-size is constructed in this paper. The new adapted method integrates exactly the common test:
y00 =−w2y. The local truncation error of the new method is computed, showing that the order of
convergence is maintained. The stability analysis is addressed, showing that the developed method is
“almost” P-stable. The numerical experiments demonstrate the high performance of the proposed scheme
compared to other existing explicit RKN codes with six stages and same order.
MSC: 65L05; 65L06
Keywords: phase-fitted and amplification-fitted schemes; RKN method; oscillatory systems;
initial-value problems
Submission date: 02.08.2021 / Acceptance date: 03.03.2022
*Corresponding author. Published by The Mathematical Association of Thailand.
Copyright c
2023 by TJM. All rights reserved.

220 Thai J. Math. Vol. 21 (2023) /M. A. Demba et al.
1. Introduction
It is the purpose of this paper to effectively solve the special second-order initial-value
problem of the form
y00 =f(x, y), y(x0) = y0, y0(x0) = y0
0,(1.1)
assuming that their solutions are oscillatory, where y∈ <dand f:< × <d→ <dare
sufficiently differentiable functions. In recent and past years, the search of new numerical
algorithms to effectively solve (1.1) has brought the attention of many researchers due
to the great role this problem played in so many areas of applied sciences. To solve
(1.1) directly, the class of Runge-Kutta-Nystr¨om (RKN) methods has been largely used.
Regarding the widespread use of these methods, some RKN methods of sixth-order with
six stages have been developed in [1], [2], and [3]. A lot of adapted RKN methods have
been developed, which are of less algebraic order than the method constructed in this
paper. To mention a few, we cite those in [4–9]. Recently, Demba et al. [10,11] derived
two new explicit RKN methods trigonometrically adapted for solving the kind of problems
in (1.1).
This work aims at the development of a new phase- and amplification-fitted sixth order
explicit RKN method with six stages based on the sixth order method of the RKN6(4)6ER
pair given in [3] for solving the problem in (1.1). The constructed method solves exactly
the test equation y00 =−w2y. The numerical experiments reveal the effectiveness of the
obtained method compared to standard RKN codes of sixth order with six stages.
The remaining part of this paper is organized as follows: the basic theory of explicit
RKN methods, the definitions of phase-lag and amplification error, and the definitions
regarding the stability analysis are addressed in Section 2. Section 3 is devoted to the
construction of the new code, to determine the order and error analysis, and to bring some
details about the periodicity property of the derived code. Some numerical examples are
presented in Section 4, showing the good performance of the proposed scheme. Comments
on the obtained results are given in Section 5, and finally, Section 6 gives a conclusion.
2. Fundamental Concepts
2.1. Explicit Runge-Kutta-Nystr¨
om Methods
An explicit RKN method with rstages is generally expressed by the formulas:
yn+1 =yn+hy0
n+h2
r
X
l=1
blf(xn+clh, Yl),(2.1)
y0
n+1 =y0
n+h
r
X
l=1
dlf(xn+clh, Yl),(2.2)
Yl=yn+clhy0
n+h2
l−1
X
j=1
alj f(xn+cjh, Yj), l = 1, . . . , r, (2.3)
where as usual, yn+1 and y0
n+1 denote approximations for y(xn+1) and y0(xn+1), re-
spectively, and the grid points on the integration interval [x0, xN] are given by xj=
x0+jh, j = 0,1, . . . , N , with ha fixed step-size.
The above method may be formulated compactly using the Butcher array in the form

A New Phase- and Amplification-Fitted Sixth-Order Explicit ... 221
c A
bT
dT
being A= (aij )r×ra matrix of coefficients, c= (c1, c2, . . . , cr)Tthe vector of stages,
and b= (b1, b2, . . . , br)T,d= (d1, d2, . . . , dr)Ttwo vectors containing the remaining
coefficients of the method. For short, this can be denoted as (c, A, b, d).
Definition 2.1. ([12]) An explicit Runge-Kutta-Nystr¨om method as given in the equa-
tions (2.1)−(2.3) is said to have algebraic order kif at any grid point xn+1 it holds
(yn+1 −y(xn+h) = O(hk+1),
y0
n+1 −y0(xn+h) = O(hk+1).
(2.4)
2.2. Analysis of Phase-Lag, Amplification Error and Stability
Applying the RKN method in (2.1)−(2.3) to the test equation y00 =−w2y, the phase-
lag, amplification error and the linear stability analysis are derived. In particular, letting
˜
h= (wh)2, the approximate solution provided by (2.1)−(2.3) verifies the recurrence
equation:
Ln+1 =E(˜
h)Ln,
where
Ln+1 ="yn+1
hy0
n+1 #, Ln="yn
hy0
n#, E(˜
h) = "1−˜
hbTN−1e1−˜
hbTN−1c
−˜
hdTN−1e1−˜
hdTN−1c#,
N=I+˜
hA, with A= (aij )r×r, b, c, d the corresponding matrix and vectors of coefficients,
Ithe identity matrix of dimension r, and e= (1,1,...,1)T∈Rr.
For enough small values of µ=wh, it can be assumed that the matrix E(˜
h) possesses
conjugate complex eigenvalues [13]. Under this assumption, an oscillatory numerical
solution should be provided by the method. The oscillatory character depends on the
eigenvalues of the stability matrix E(˜
h). The characteristic equation of this matrix can
be expressed as:
λ2−λT r(E(˜
h)) + Det(E(˜
h)) = 0.(2.5)
Theorem 2.2. ([3]) If we apply to the common test equation y00 =−w2ythe Runge-
Kutta-Nystr¨om scheme in (2.1)−(2.3), we get the formula for calculating directly the
phase-lag (or dispersion error) Ψ(µ)given by:
Ψ(µ) = µ−cos−1

T r(E(˜
h))
2qDet(E(˜
h))
.(2.6)
If Ψ(µ) = O(µl+1), then it is said that the method has a phase-lag of order l. For an
explicit RKN method, T r(E(˜
h) and Det(E(˜
h) are polynomials in µ(we note that in case
of an implicit RKN method these would be rational functions).

222 Thai J. Math. Vol. 21 (2023) /M. A. Demba et al.
Definition 2.3. An explicit Runge-Kutta-Nystr¨om method as given in the equations
(2.1)−(2.3) is said to be phase-fitted, if the phase-lag is zero.
Definition 2.4. ([3]) For the Runge-Kutta-Nystr¨om method given in the equations
(2.1)−(2.3), the value β(µ) = 1 −qDet(E(˜
h)) is known as the amplification error (or
dissipative error). If β(µ) = O(µs+1 ), then it is said that the method has an amplification
error of order s.
Definition 2.5. An explicit Runge-Kutta-Nystr¨om method as given in the equations
(2.1)−(2.3) is said to be amplification-fitted if the amplification error is zero.
We further study the stability property of the developed method when applied to the
test equation, y00 =−w2y.
Definition 2.6. ([3]) The interval I= (0,˜
hb), ˜
hb∈R+∪ {+∞}, so that µ∈(0,˜
hb) is
called
(1) the interval of stability of the RKN method, if ˜
hbis the highest value for which
|λ|<1. In this case, if ˜
hb=∞then the RKN method is called A-stable.
(2) the interval of periodicity of the RKN method, if ˜
hbis the highest value for
which |λ|= 1, and [tr(E(˜
h))]2−4det(E(˜
h)) <0 (the eigenvalues are complex
conjugate). In this case, if ˜
hb=∞then the RKN method is called P-stable.
The adapted method developed is “almost” P-stable, but has no interval of stability.
Using the Mathematica software we have found that the eigenvalues of the developed
method are: λ1=e−i√H, λ2=ei√H, and also the trace and the determinant of the matrix
E(˜
h) for the developed method are, tr(E(˜
h)) = 2 cos(√H) and det(E(˜
h)) = 1. From the
very construction we have that Ψ(µ) = µ−arccos(2 cos(µ)
2√1) = 0, β(µ)=1−qdet(E(˜
h)) =
0, which implies that the method is dispersive of order infinity and dissipative of order
infinity.
Theorem 2.7. The new phase- and amplification-fitted method developed in this paper
is ”almost” P-stable, that is, the conditions for P-stability are verified except for the set
δ:= {υ2∈R:υ26= (nπ)2, n ∈N}.
The theorem can be proved by considering the fact that the adapted method developed
here has respectively the following, tr(E(˜
h)) = 2 cos(µ) and det(E(˜
h)) = 1. According to
[14], an adapted method is “almost” P-stable, if and only if for every µ > 0, det(E(˜
h)) = 1
and |tr(E(˜
h))|<2. Since for the proposed method it is |tr(E(˜
h))|=|2 cos(υ)|<2, for
υ6=nπ, n ∈N,this gives the desired result.
3. Development of the New Scheme
In this section, we will obtain a sixth order explicit phase- and amplification-fitted RKN
scheme based on the higher-order method in the RKN6(4)6ER embedded pair derived by
El-Mikkawy and Rahmo in [2], which we named as RKN6-6ER. The coefficients of the
sixth order RKN method in [2] are shown in Table 1with the correct value of a54 as given
in [3].

A New Phase- and Amplification-Fitted Sixth-Order Explicit ... 223
Table 1. Coefficients of the RKN6-6ER method in [2]
0
1
77
1
11858
1
3−7189
17118
4070
8559
2
3
4007
2403 −589655
355644
25217
118548
13
15 −4477057
843750
13331783894
2357015625 −281996
5203125
563992
7078125
117265
2002 −1886451746
212088107
22401
31339
2964
127897
178125
5428423
−341
780
386683451
661053840
2853
11840
267
3020
9375
410176 0
−341
780
29774625727
50240091840
8559
23680
801
3020
140625
820352
847
18240
In order to get the new adapted scheme, we equate to zero the phase-lag Ψ(µ) and the
amplification error β(µ), and we get the system:
(Ψ(µ) = 0
β(µ)=0.(3.1)
We solve this system considering the coefficients in Table 1except two of them which
are taking as unknowns. Specifically, we take b5and d5as unknowns. After solving the
system in (3.1) we obtain the following values:
b5=2503125
410176M −1258632233707368303463680000000 + 524994684043706387148025080000 µ2
+38027832783293925493906168800 µ4
−42305110040020986855472545000 µ6
+6389496350903753079525017100 µ8
−396360945814751886526623990 µ10
+12393674919826270714885995 µ12
−163757382111950819488686 µ14 + 443880244626070278520 µ16
−3556135517458913619310080000 µ2cos (µ) + 7969295957655526325216985600 µ6cos (µ)
−125718020321097360886329600 µ8cos (µ)−74269315558590948580693708800 µ4cos (µ)
+1258632233707368303463680000000 cos (µ)!,

224 Thai J. Math. Vol. 21 (2023) /M. A. Demba et al.
d5=625
820352M 4882682690886773063720 µ20
−1766435438191731348692196 µ18
+142671286498878012015349560 µ16
−10126226143892166109616015370 µ14
+475493904396311527376632326825 µ12 + 54688197305084078277852710400 µ10 cos (µ)
−10391680199125544879555652445650 µ10
−10381900296589462467492329664000 µ8cos (µ)
+113086760758089573241298829586500 µ8+ 253690204049060105732403398400000 µ6cos (µ)
−381721832459881021063776477195000 µ6
−1775893905546681693988359573660000 µ4
−630550557973482187135177923840000 µ4cos (µ) + 31997530415514051646287158745000000 µ2
−672109612799734674049605120000000 µ2cos (µ) + 75612331439970150830580576000000000 cos (µ)
−75612331439970150830580576000000000!,(3.2)
where
M=µ2 −28803310743425593080234375000000 + 4800551790570932180039062500000 µ2
+240986472782100847395103125000 µ4
−211575854747321234593653037500 µ6
+27693379469414224574322792750 µ8
−1543565245575968927989765335 µ10
+55158851048499641449369350 µ12
−861578557170344748268248 µ14
+2441341345443386531860 µ16!.(3.3)
The corresponding Taylor series expansions in powers of µresult in
b5=9375
410176
−
261461
93847723200 µ6+20361401
369525410100000 µ8
−
177044709462626977
8669779600607821080000000 µ10
+11347558575343312922557
887568686612225683065000000000 µ12
−
101477791160183648432238539
136685577738282755192010000000000000 µ14 +··· ,
d5=140625
820352
−
1
213290280 µ6
−
618923
739050820200 µ8
−
1251344791
93120403345200000 µ10
−
190297638076116325219
7396405721768547358875000000 µ12 +3527694543209273924031679
994076929005692765032800000000000 µ14 +··· .(3.4)
As µ→0, the newly obtained coefficients b5, d5become the coefficients of the coun-
terpart scheme in the original method. The new adapted RKN scheme will be named as
PFAFRKN6-6ER.

A New Phase- and Amplification-Fitted Sixth-Order Explicit ... 225
3.1. Order of Convergence
This section is devoted to present the local truncation error of the proposed method
and to get the algebraic order of convergence. This is accomplished by using the usual tool
of Taylor expansions. The local truncation errors (LTE) at the point x1of the solution
and the first derivative are given respectively by:
LT E =y(x0+h)−y1,
LT Eder =y0(x0+h)−y0
1.(3.5)
Proposition 3.1. The corresponding LTEs of the formulas to provide the solution and
the derivative with the new RKN method are, respectively:
LT E =h7
213290280 (fy)2(fx+fyy0) + O(h8),
LT Eder =h7
5040 (fxxxxxx + 15(y0)4fyyyyy00 + 60(y0)3fxyyyy y00 + 60y0fxxxyy y00
+90(y0)2fxxyyy y00 + 21fyfyxx y00 + 60y00 fxyy fx+ 15y00 fy y fxx + 18(y00)2fy y fy
+90y0fxyyy (y00 )2+ 45(y0)2fyyyy (y00)2+ 33(y0)2(fy y )2y00 + 48y0fxy fyxx
+10fyfxyfx+ 12(fy)2y0fxy + 60y0fxxy y fx+ 60(y0)2fxyyy fx+ 20(y0)3fyyyy fx
+24fyy0fxxxy + 30y0fxyy fxx + 15(y0)2fyyyfxx + 6y0fy y fxxx + 78(y0)2fxyy fxy
+66(y0)2fyfxxyy + 33(y0)2fyy fyxx + 64(y0)3fyfxy yy + 36(y0)3fyyy fxy
+48(y0)3fyy fxyy + 21(fy)2(y0)2fyy + 21(y0)4fyfyyyy + 21(y0)4fyyy fyy
+15(y00)3fyyy + 45(y00 )2fxxyy + 15y00 fxxxxy + 18y00 (fxy )2+ (fy)3y00
+(y0)6fyyyyyy + 6(y0)5fxyyyyy + (fy)2fxx + 6fxxxfxy +fyfxxxx
+20fxfxxxy + 6y0fxxxxxy + 15fyxxfxx + 15(y0)4fxxy yyy + 15(y0)2fxxxxy y
+20(y0)3fxxxyyy + 10fyy (fx)2+ 81(y0)2fyyy fyy00 + 60y0fyyy fxy00
+102y0fyfxyy y00 + 66y0fyy fxy y00 + 30y0fyy fyfx) + O(h8),
(3.6)
from which we can infer that the PFAFRKN6-6ER method has algebraic order six.
4. Some Numerical Examples
To assess the performance of the new scheme, we have considered the following RKN
codes of the same order and stages to get fair comparisons:
•PFAFRKN6-6ER: The adapted explicit RKN code developed here,
•RKN6-6ER: An explicit sixth-order six stage RKN method presented in [2],
•RKN6-6ER-PFAF: An optimized explicit sixth-order six stage RKN method
derived by Anastassi and Kosti in [3],
•RKN6-6FM: An explicit sixth-order six stage RKN method developed by Dor-
mand et al. in [1].
We will consider different oscillatory problems appeared in the literature to test the
performance of the above methods:

226 Thai J. Math. Vol. 21 (2023) /M. A. Demba et al.
Problem 1. Homogeneous Problem in [15]
y00 =−w2y, y(0) = 1, y0(0) = −2, x ∈[0,4000],
with a known solution given by
y(x) = −1
4sin(8x) + cos(8x).
To use the adapted methods we have taken the parameter value w= 8.
Problem 2. Non-homogeneous Problem in [16]
y00 =−v2y+ (v2−1) sin x, y(0) = 1, y0(0) = v+ 1, v = 10, x ∈[0,4000],
with a known solution given by
y(x) = sin(10x) + cos(10x) + sin(x).
Now, in the adapted methods we have taken the value υ=w= 10.
Problem 3. Non-linear System in [17]
y00
1+w2y1=2y1y2−sin(2wx)
(y12+y22)3
2
, y1(0) = 1, y0
1(0) = 0,
y00
2+w2y2=y12−y22−sin(2wx)
(y12+y22)3
2
, y2(0) = 0, y0
2(0) = w, x ∈[0,4000],
with a known solution given by
y1(x) = cos(wx),
y2(x) = sin(wx).
To use the adapted methods we have taken the parameter value w= 5.
Problem 4. Non-homogeneous System in [18]
y00
1=−m2y1(x) + m2g(x) + g00(x), y1(0) = b+g(0), y0
1(0) = g0(0),
y00
2=−m2y2(x) + m2g(x) + g00(x), y2(0) = g(0), y0
2(0) = mb +g0(0), x ∈[0,4000],
with a known solution given by
y1(x) = bcos(mx) + g(x)
y2(x) = bsin(mx) + g(x).
For the numerical computations we have taken w=m= 20, g(x) = e−0.05x, and b= 0.1.
Problem 5. Non-homogeneous Problem in [19]
y00 =−25y+ 100 cos 5x, y(0) = 1, y0(0) = 5, x ∈[0,4000],
with a known solution given by
y(x) = sin(5x) + cos(5x) + 10xsin(5x).

A New Phase- and Amplification-Fitted Sixth-Order Explicit ... 227
Now, in the adapted methods we have taken the value w= 5.
We have considered the integration interval [x0, xN], with step-length, h=xN−x0
N,
taking different final points, as xN= 100,1000,4000. The obtained maximum absolute
errors, M ax global error , are given in Tables 2to 6, considering different step-sizes h.
Table 2. Maximum absolute errors corresponding to Problem 1
h Methods xN= 100 xN= 1000 xN= 4000
PFAFRKN6-6ER 8.376888(-10) 5.297163(-9) 4.047332(-8)
0.05 RKN6-6ER 1.876489(-6) 1.889563(-5) 7.556011(-5)
RKN6-6ER-PFAF 5.667095(-5) 5.659304(-4) 2.266264(-3)
RKN6-6FM 5.240274(-6) 5.275814(-5) 2.111010(-4)
PFAFRKN6-6ER 4.061289(-8) 2.393823(-7) 9.391945(-7)
0.075 RKN6-6ER 3.238321(-5) 3.236094(-4) 1.296214(-3)
RKN6-6ER-PFAF 3.689401(-2) 3.396261(-1) 1.006993(+0)
RKN6-6FM 6.058584(-5) 6.084909(-4) 2.434357(-3)
PFAFRKN6-6ER 9.005675(-7) 7.208692(-6) 2.830818(-5)
0.1 RKN6-6ER 2.394757(-4) 2.436649(-3) 9.745593(-3)
RKN6-6ER-PFAF 1.153997(+0) 1.153997(+0) 1.153997(+0)
RKN6-6FM 3.445139(-4) 3.481070(-3) 1.392376(-2)
PFAFRKN6-6ER 1.149804(-5) 1.031363(-4) 4.085795(-4)
0.125 RKN6-6ER 1.171023(-3) 1.169972(-2) 4.616601(-2)
RKN6-6ER-PFAF 1.070101(+0) 1.070101(+0) 1.070101(+0)
RKN6-6FM 1.342387(-3) 1.355532(-2) 5.432584(-2)
Table 3. Maximum absolute errors corresponding to Problem 2
h Methods xN= 100 xN= 1000 xN= 4000
PFAFRKN6-6ER 6.087944(-9) 4.514620(-8) 1.029183(-7)
0.05 RKN6-6ER 1.549647(-5) 1.547221(-4) 6.194987(-4)
RKN6-6ER-PFAF 3.434323(-3) 3.429801(-2) 1.353524(-1)
RKN6-6FM 3.452593(-5) 4.472916(-4) 1.390353(-3)
PFAFRKN6-6ER 5.291679(-7) 5.508185(-6) 2.220847(-5)
0.075 RKN6-6ER 2.643734(-4) 2.664580(-3) 1.063403(-2)
RKN6-6ER-PFAF 1.370402(+0) 1.611646(+0) 1.611646(+0)
RKN6-6FM 4.034230(-4) 4.032180(-3) 1.614168(-2)
PFAFRKN6-6ER 1.730785(-5) 1.744420(-4) 6.981273(-4)
0.1 RKN6-6ER 2.008096(-3) 2.001580(-2) 7.876678(-2)
RKN6-6ER-PFAF 1.448751(+0) 1.448751(+0) 1.448751(+0)
RKN6-6FM 2.314834(-3) 2.329405(-2) 9.338362(-2)
PFAFRKN6-6ER 2.430470(-4) 2.523091(-3) 1.017188(-2)
0.125 RKN6-6ER 9.339244(-3) 9.430920(-2) 3.480040(-1)
RKN6-6ER-PFAF 4.595800(+306) 4.595800(+306) 4.595800(+306)
RKN6-6FM 9.020961(-3) 9.181146(-2) 3.698289(-1)

228 Thai J. Math. Vol. 21 (2023) /M. A. Demba et al.
Table 4. Maximum absolute errors corresponding to Problem 3
h Methods xN= 100 xN= 1000 xN= 4000
PFAFRKN6-6ER 3.802533(-10) 2.155096(-9) 9.277232(-9)
0.05 RKN6-6ER 4.282131(-8) 1.632977(-7) 1.632977(-7)
RKN6-6ER-PFAF 1.899990(-8) 7.334019(-8) 7.667858(-8)
RKN6-6FM 1.953175(-7) 7.542634(-7) 7.542634(-7)
PFAFRKN6-6ER 9.475666(-9) 3.697725(-8) 3.697725(-8)
0.075 RKN6-6ER 7.249395(-7) 2.799656(-6) 2.804762(-6)
RKN6-6ER-PFAF 1.214422(-5) 4.709903(-5) 4.713322(-5)
RKN6-6FM 2.236286(-6) 8.631087(-6) 8.634346(-6)
PFAFRKN6-6ER 9.349917(-8) 3.600327(-7) 3.600327(-7)
0.1 RKN6-6ER 5.491464(-6) 2.106615(-5) 2.106615(-5)
RKN6-6ER-PFAF 1.223785(-3) 4.727541(-3) 4.727541(-3)
RKN6-6FM 1.261951(-5) 4.880468(-5) 4.881228(-5)
PFAFRKN6-6ER 5.305980(-7) 2.048570(-6) 2.048570(-6)
0.125 RKN6-6ER 2.598789(-5) 1.008581(-4) 1.008581(-4)
RKN6-6ER-PFAF 4.319413(-2) 2.315899(-1) 2.324977(-1)
RKN6-6FM 4.804257(-5) 1.875787(-4) 1.876650(-4)
Table 5. Maximum absolute errors corresponding to Problem 4
h Methods xN= 100 xN= 1000 xN= 4000
PFAFRKN6-6ER 2.826968(-11) 2.890083(-9) 4.628854(-8)
0.0125 RKN6-6ER 1.703302(-8) 1.700487(-7) 6.883544(-7)
RKN6-6ER-PFAF 7.477886(-9) 7.738988(-8) 2.531203(-7)
RKN6-6FM 7.573732(-8) 7.559689(-7) 3.081923(-6)
PFAFRKN6-6ER 1.149865(-9) 7.538132(-9) 9.932046(-9)
0.025 RKN6-6ER 2.182914(-6) 2.190211(-5) 8.758873(-5)
RKN6-6ER-PFAF 4.874585(-4) 4.831134(-3) 1.875728(-2)
RKN6-6FM 4.906952(-6) 4.912734(-5) 1.965290(-4)
PFAFRKN6-6ER 2.578029(-6) 2.484148(-5) 9.901855(-5)
0.05 RKN6-6ER 2.849681(-4) 2.818441(-3) 1.084435(-2)
RKN6-6ER-PFAF 1.052562(-1) 1.052562(-1) 1.052562(-1)
RKN6-6FM 3.285820(-4) 3.298875(-3) 1.320687(-2)
PFAFRKN6-6ER 3.205100(-4) 3.238187(-3) 1.347178(-2)
0.075 RKN6-6ER 4.903214(-3) 4.137844(-2) 9.520212(-2)
RKN6-6ER-PFAF 4.561856(+306) 4.561856(+306) 4.561856(+306)
RKN6-6FM 4.036525(-3) 4.092547(-2) 1.578012(-1)

A New Phase- and Amplification-Fitted Sixth-Order Explicit ... 229
Table 6. Maximum absolute errors corresponding to Problem 5
h Methods xN= 100 xN= 1000 xN= 4000
PFAFRKN6-6ER 2.213611(-7) 1.740843(-5) 9.767030(-4)
0.05 RKN6-6ER 2.111063(-5) 2.131245(-3) 3.396483(-2)
RKN6-6ER-PFAF 9.114216(-6) 9.145261(-4) 1.585036(-2)
RKN6-6FM 9.504147(-5) 9.493622(-3) 1.507729(-1)
PFAFRKN6-6ER 3.713266(-6) 2.999952(-5) 8.333719(-4)
0.075 RKN6-6ER 3.594272(-4) 3.642403(-2) 5.833073(-1)
RKN6-6ER-PFAF 6.123057(-3) 6.095651(-1) 9.779085(+0)
RKN6-6FM 1.088588(-3) 1.082964(-1) 1.737337(+0)
PFAFRKN6-6ER 2.970377(-5) 3.585056(-4) 2.668772(-3)
0.1 RKN6-6ER 2.687021(-3) 2.733629(-1) 4.380288(+0)
RKN6-6ER-PFAF 6.105609(-1) 6.076355(+1) 9.693221(+2)
RKN6-6FM 6.162659(-3) 6.127625(-1) 9.821965(+0)
PFAFRKN6-6ER 1.554099(-4) 2.540054(-3) 2.320075(-2)
0.125 RKN6-6ER 1.295125(-2) 1.302793(+0) 2.094574(+1)
RKN6-6ER-PFAF 2.153911(+1) 1.993890(+3) 2.502423(+4)
RKN6-6FM 2.341523(-2) 2.338821(+0) 3.782324(+1)
To show the efficiency of the developed PFAFRKN6-6ER code, we present in Figures
1 to 5 the efficiency curves for the considered problems. For each problem, the logarithm
of the maximum absolute global error has been plotted versus the logarithm of the total
number of function evaluations. It can be observed the good behavior of the new code.
3.65 3.7 3.75 3.8 3.85 3.9 3.95 4 4.05 4.1
log10(Number of function evaluations)
-10
-8
-6
-4
-2
0
2
log10(Max global error)
HOMOGENEOUS PROBLEM
PFAFRKN6-6ER
RKN6-6ER
RKN6-6ER-PFAF
RKN6-6FM
Figure 1. Efficiency curves corresponding to Problem 1

230 Thai J. Math. Vol. 21 (2023) /M. A. Demba et al.
3.65 3.7 3.75 3.8 3.85 3.9 3.95 4 4.05 4.1
log10(Number of function evaluations)
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
log10(Max global error)
NON-HOMOGENEOUS PROBLEM
PFAFRKN6-6ER
RKN6-6ER
RKN6-6ER-PFAF
RKN6-6FM
Figure 2. Efficiency curves corresponding to Problem 2
3.65 3.7 3.75 3.8 3.85 3.9 3.95 4 4.05 4.1
log10(Number of function evaluations)
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
log10(Max global error)
NON-LINEAR SYSTEM
PFAFRKN6-6ER
RKN6-6ER
RKN6-6ER-PFAF
RKN6-6FM
Figure 3. Efficiency curves corresponding to Problem 3

A New Phase- and Amplification-Fitted Sixth-Order Explicit ... 231
3.9 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7
log10(Number of function evaluations)
-12
-10
-8
-6
-4
-2
0
log10(Max global error)
NON-HOMOGENEOUS SYSTEM
PFAFRKN6-6ER
RKN6-6ER
RKN6-6ER-PFAF
RKN6-6FM
Figure 4. Efficiency curves corresponding to Problem 4
3.65 3.7 3.75 3.8 3.85 3.9 3.95 4 4.05 4.1
log10(Number of function evaluations)
-7
-6
-5
-4
-3
-2
-1
0
log10(Max global error)
NON-HOMOGENEOUS PROBLEM
PFAFRKN6-6ER
RKN6-6ER
RKN6-6ER-PFAF
RKN6-6FM
Figure 5. Efficiency curves corresponding to Problem 5

232 Thai J. Math. Vol. 21 (2023) /M. A. Demba et al.
To further demonstrate the efficiency of the constructed PFAFRKN6-6ER code, we
present in Figures 6 to 10 the efficiency curves for the considered problems. Now, the
logarithm of the maximum absolute global error has been plotted versus the CPU time
used. It can be observed the good behavior of the new code.
0 0.05 0.1 0.15 0.2 0.25
Time (s)
-10
-8
-6
-4
-2
0
2
log10(Max global error)
HOMOGENEOUS PROBLEM
PFAFRKN6-6ER
RKN6-6ER
RKN6-6ER-PFAF
RKN6-6FM
Figure 6. Efficiency curves corresponding to Problem 1
0 0.05 0.1 0.15 0.2 0.25
Time (s)
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
log10(Max global error)
NON-HOMOGENEOUS PROBLEM
PFAFRKN6-6ER
RKN6-6ER
RKN6-6ER-PFAF
RKN6-6FM
Figure 7. Efficiency curves corresponding to Problem 2

A New Phase- and Amplification-Fitted Sixth-Order Explicit ... 233
0.1 0.15 0.2 0.25 0.3 0.35 0.4
Time (s)
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
log10(Max global error)
NON-LINEAR SYSTEM
PFAFRKN6-6ER
RKN6-6ER
RKN6-6ER-PFAF
RKN6-6FM
Figure 8. Efficiency curves corresponding to Problem 3
0.16 0.18 0.2 0.22 0.24 0.26 0.28
Time (s)
-12
-10
-8
-6
-4
-2
0
log10(Max global error)
NON-HOMOGENEOUS SYSTEM
PFAFRKN6-6ER
RKN6-6ER
RKN6-6ER-PFAF
RKN6-6FM
Figure 9. Efficiency curves corresponding to Problem 4

234 Thai J. Math. Vol. 21 (2023) /M. A. Demba et al.
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
Time (s)
-7
-6
-5
-4
-3
-2
-1
0
log10(Max global error)
NON-HOMOGENEOUS PROBLEM
PFAFRKN6-6ER
RKN6-6ER
RKN6-6ER-PFAF
RKN6-6FM
Figure 10. Efficiency curves corresponding to Problem 5
5. Discussion
The new PFAFRKN6-6ER code gives minimum error norm, minimum number of func-
tion evaluations, and minimum computational cost (Time(s)). Tables 2−6and Figures
1−10 put an evidence that PFAFRKN6-6ER is a very efficient scheme. Therefore, we can
say that PFAFRKN6-6ER is more appropriate for solving the type of problem in (1.1)
than the other existing RKN methods of order 6 with six stages in the literature.
6. Conclusion
In this study, we have used the methodology for constructing the phase-fitted and
amplification-fitted methods to develop an efficient explicit phase- and amplification-fitted
RKN code based on the RKN6-6ER method due to El-Mikkawy and Rahmo [2]. The new
developed method has two variable coefficients depending on the parameter µ=wh,
which is usually known as the parameter frequency [20,21]. We computed the local
truncation error of the new method, confirming that the algebraic order of convergence
of the underlying code is maintained. In addition, the stability analysis of the new code
revealed that it is “almost” P-stable. The numerical results obtained clearly show that
PFAFRKN6-6ER is more accurate and efficient than other sixth-order six-stage RKN
codes in the literature.
Acknowledgments
The authors appreciate the Center of Excellence in Theoretical and Computational
Science (TaCS-CoE), King Mongkut’s University of Technology, Thonburi (KMUTT),
for the financial support. Moreover, this research work is also supported by the Thailand

A New Phase- and Amplification-Fitted Sixth-Order Explicit ... 235
Science Research and Innovation (TSRI) Basic Research Fund, for the fiscal year 2021
with project number 64A306000005. The first author also appreciates the support of the
Petchra Pra Jom Klao PhD Research Scholarship from KMUTT with Grant No. 15/2562.
References
[1] J.R. Dormand, M. El-Mikxawy, P.J. Prince, Families of Runge–Kutta–Nystr¨om for-
mulas, IMA Journal of Numerical Analysis, 1987.
[2] M. El-Mikkawy, E. Rahmo, A new optimized non-FSAL embedded Runge–Kutta–
Nystr¨om algorithm of orders 6 and 4 in six stages, Applied Mathematics and Com-
putation 145 (1) (2003) 33–43.
[3] Z.A. Anastassi, A.A. Kosti, A 6 (4) optimized embedded Runge–Kutta–Nystr¨om
pair for the numerical solution of periodic problems, Journal of Computational and
Applied Mathematics 275 (2015) 311–320.
[4] T.E Simos, Exponentially-fitted Runge–Kutta–Nystro¨om method for the numerical
solution of initial-value problems with oscillating solutions, Applied Mathematics
Letters 15 (2) (2002) 217–225.
[5] A.A. Kosti, Z.A. Anastassi, T.E. Simos, An optimized explicit Runge–Kutta–
Nystr¨om method for the numerical solution of orbital and related periodical initial
value problems, Computer Physics Communications 183 (3) (2012) 470–479.
[6] K.W. Moo, N. Senu, F. Ismail, M. Suleiman, New phase-fitted and amplification-
fitted fourth-order and fifth-order Runge–Kutta–Nystr¨om methods for oscillatory
problems, Abstract and Applied Analysis 2013 (2013).
[7] Th. Monovasilis, Z. Kalogiratou, T.E. Simos, Exponentially tted symplectic Runge–
Kutta–Nystr¨om methods, Appl. Math. Inf. Sci. 7 (1) (2013) 81–85.
[8] R. D’Ambrosio, B. Paternoster, G. Santomauro, Revised exponentially fitted Runge–
Kutta–Nystr¨om methods, Applied Mathematics Letters 30 (2014) 56–60.
[9] M.A. Demba, N. Senu, F. Ismail, A four-stage third-order symplectic explicit
trigonometrically-fitted Runge–Kutta–Nystr¨om method for the numerical integra-
tion of oscillatory initial-value problems, International Journal of Pure and Applied
Mathematics 111 (2) (2016) 165–78.
[10] M.A. Demba, N. Senu, F. Ismail, New explicit trigonometrically-fitted fourth-order
and fifth-order Runge–Kutta–Nystr¨om methods for periodic initial value problems,
International Journal of Pure and Applied Mathematics 109 (2) (2016) 207–222.
[11] M.A Demba, N. Senu, F. Ismail, A new fourth-order four stage explicit
trigonometrically-fitted Runge–Kutta–Nystr¨om method for solving periodic prob-
lems, In AIP Conference Proceedings, AIP Publishing 1739 (2016) p. 020041.
[12] E. Hairer, S.P. Nørsett, G. Wanner, Solving Ordinary Differential Equations I, Nonsti
Problems, 2nd Edition, Springer, Berlin, 2008.
[13] P.J.V.D. Houwen, B.P. Sommeijer, Diagonally implicit Runge–Kutta–Nystr¨om meth-
ods for oscillatory problems, SIAM Journal on Numerical Analysis 26 (2) (1989)
414–429.
[14] J.M. Franco, I. G´omez, L. R´andez, Four-stage symplectic and p-stable sdirkn meth-
ods with dispersion of high order, Numerical Algorithms 26 (4) (2011) 347–363.

236 Thai J. Math. Vol. 21 (2023) /M. A. Demba et al.
[15] B. Paternoster, A phase-fitted collocation-based Runge–Kutta–Nystr¨om method,
Applied Numerical Mathematics 35 (4) (2000) 339–355.
[16] M.A. Demba, Trigonometrically-Fitted Explicit Runge–Kutta–Nystrom Methods for
Solving Special Second Order Ordinary Differential Equations with Periodic Solu-
tions, Master’s Thesis, University Putra Malaysia, 2016.
[17] H.V.D. Vyver, A Runge–Kutta–Nystr¨om pair for the numerical integration of per-
turbed oscillators, Computer Physics Communications 167 (2) (2005) 129–142.
[18] N. Senu, M. Suleiman, F. Ismail, N.Md. Arifin, New 4 (3) pairs diagonally implicit
Runge–Kutta–Nystr¨om method for periodic IVPs, Discrete Dynamics in Nature and
Society 2012 (2012).
[19] B.P. Sommeijer, Explicit, high-order Runge–Kutta–Nystr¨om methods for parallel
computers, Applied Numerical Mathematics 13 (1-3) (1993) 221–240.
[20] H. Ramos, J. Vigo-Aguiar, On the frequency choice in trigonometrically fitted meth-
ods, Applied Mathematics Letters 23 (11) (2010) 1378–1381.
[21] J. Vigo-Aguiar, H. Ramos, On the choice of the frequency in trigonometrically-fitted
methods for periodic problems, Journal of Computational and Applied Mathematics
277 (2015) 94–105.