# A Two-Step Modified Explicit Hybrid Method with Step-Size-Dependent Parameters for Oscillatory Problems

Authors: Faieza Samat; Eddie Shahril Ismail
Journal: Journal of Mathematics
 (2020)

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## Abstract

A new two-step modified explicit hybrid method with parameters depending on the step-size is constructed. This method is derived using the coefficients from a sixth-order explicit hybrid method with extended interval of absolute stability and then imposed each stage of the modified formula to exactly integrate the differential equations with solutions that can be expressed as linear combinations ofsinwx and coswx, where w is the known frequency. Numerical results show the advantage of the new method for solving oscillatory problems.


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## Body

## 1. Introduction

Second-order ordinary differential equations are important tools for modelling physical phenomena in science and engineering. This paper is concerned with the numerical solution of the second-order ordinary differential equations of the form(1)y″x=fx,yx,yx0=y0,y′x0=y0′,having oscillatory solutions. These problems can be numerically solved by general-purpose methods or any other methods specially adapted to the structure of the intended problem. In the case of adapted numerical methods, particular algorithms have been proposed by several authors, see [1–3] to solve these classes of problems.Franco [4] has established the following class of explicit hybrid methods:(2)Y1=yn−1,Y2=yn,Yi=1+ciyn−ciyn−1+h2∑j=1i−1aijfxn+cjh,Yj,i=3,…,s,yn+1=2yn−yn−1+h2b1fn−1+b2fn+∑i=3sbifxn+cih,Yi,where h is the step-size while fn−1 and fn represent fxn−1,yn−1 and fxn,yn, respectively. The associated Butcher tableau for this class of methods is given by(3)where(4)A=aij,c=−1,0,c3…csT,and bT=b1,b2,…,bs−1bs.Kalogiratou et al. [5] have modified each stage of the explicit hybrid methods and the improved version is(5)Y1=yn−1,Y2=yn,Yi=σi1+ciyn−μiciyn−1+h2∑j=1i−1aijfxn+cjh,Yj,i=3,…,s,yn+1=σs+12yn−μs+1yn−1+h2b1fn−1+b2fn+∑i=3sbifxn+cih,Yi.The coefficientsσi and μi are functions of v = wh, where w is the known frequency of the second-order problems. The Butcher tableau for the modified hybrid method is given by(6)We develop a sixth-order explicit hybrid method with extended interval of absolute stability based on the class of explicit hybrid methods (2). Using the coefficients from the sixth-order hybrid method, we develop a new modified hybrid method. The construction of hybrid methods is described in Section 2. In Section 3, we give the stability analysis of the class of modified hybrid method (5). Numerical results are presented in Section 4 for several second-order problems.


## 2. Construction of Hybrid Methods

In this section, we derive the new method with four stages.


### 2.1. Sixth-Order Explicit Hybrid Method

Consider the explicit hybrid methods (2). The associated Butcher tableau for a class of four-stage explicit hybrid methods is given by(7)The sixth-order explicit hybrid method must satisfy the order conditions for a sixth-order hybrid method as stated in [6]. Solving the order conditions, we obtain(8)b1=5c32−260c3−1c3+1,b2=25c32−330c32,b3=−120c3−1c3+1c32,b4=−120c3−1c3+1c32,b5=5c32−260c3−1c3+1,a31=−c3c3−1c3+16,a32=c3c32+3c3+26,a41=c3c32+c3+1c3−16c3+1,a42=−c3c3−3c3−16,a43=−c3c3−16c3+1,a51=−3c3−36c3+15c32−2,a52=30c34−11c32−3c3+26c325c32−2,a53=c33−2c32+2c3−16c325c32−2c3+1,a54=c3−1c3+16c325c32−2,c4=−c3,c5=1.Next, we choose the free parameterc3=7731/10000 to maximize the interval of absolute stability. For detail explanation on stability properties of hybrid methods, refer [4]. The resulting method has a phase-lag of order 6 and a dissipation error of order 7. The interval of absolute stability of this method is (0, 4.54).


### 2.2. The New Method with Parameters Depending on Step-Size

This method is derived using the coefficients from the sixth-order explicit hybrid method in Section2.1. Consider the modified four-stage explicit hybrid method represented by this tableau:(9)Associate each formula stage of the modified four-stage explicit hybrid method with the following linear operators:(10)Lh,ayx=yx+cih−σi1+cihyx+μiciyx−h−h2∑j=1i−1aijy″x+cih,i=3,4,5,Lh,byx=yx+h−2σiyx+μiyx−h−h2∑i=15biy″x+cih.Using the coefficients from the sixth-order explicit hybrid method and imposing the linear operators to exactly integrate the set {sinwx and coswx}, we get(11)σ3=2000000000000cosvsin7731/10000v+2000000000000sinvcos7731/10000v+1267106676297vsinv3546200000000sinv,μ3=2000000000000sin7731/10000v−103676933703v2sinv1546200000000sinv,σ4=584721300000000cosvv2sin7731/10000v−35462000000000000cosvsin7731/10000v+35462000000000000sinvcos7731/10000v−230871636632107v2sinv+584721300000000sinvv2cos7731/10000v8046327800000000sinv,μ4=−584721300000000v2sin7731/10000v−35462000000000000sin7731/10000v+1386247674457893v2sinv27415672200000000sinv,σ5=5262491700000000000cosvv2sin7731/10000v+125697456581527785906cosvsinv+67784748079673892953v2sinv−9004452122180000000sinvv2cos7731/10000v125697456581527785906sinv,μ5=−−226900000000000v2sin7731/10000v−2709814525536321sinv+175416390000000v2sinv2709814525536321,σ6=cosv+7961393523907344v2+5000000000000002404579123373679v2cos7731/10000v−19768361482779668v2cosv,and μ6=1, where v = wh.For smallv, coefficients σi and μi may cause heavy calculations which lead to inaccuracy; hence, it is often convenient to use Taylor expansions for the coefficients. The resulting method is denoted by MEHM6.


## 2.1. Sixth-Order Explicit Hybrid Method

Consider the explicit hybrid methods (2). The associated Butcher tableau for a class of four-stage explicit hybrid methods is given by(7)The sixth-order explicit hybrid method must satisfy the order conditions for a sixth-order hybrid method as stated in [6]. Solving the order conditions, we obtain(8)b1=5c32−260c3−1c3+1,b2=25c32−330c32,b3=−120c3−1c3+1c32,b4=−120c3−1c3+1c32,b5=5c32−260c3−1c3+1,a31=−c3c3−1c3+16,a32=c3c32+3c3+26,a41=c3c32+c3+1c3−16c3+1,a42=−c3c3−3c3−16,a43=−c3c3−16c3+1,a51=−3c3−36c3+15c32−2,a52=30c34−11c32−3c3+26c325c32−2,a53=c33−2c32+2c3−16c325c32−2c3+1,a54=c3−1c3+16c325c32−2,c4=−c3,c5=1.Next, we choose the free parameterc3=7731/10000 to maximize the interval of absolute stability. For detail explanation on stability properties of hybrid methods, refer [4]. The resulting method has a phase-lag of order 6 and a dissipation error of order 7. The interval of absolute stability of this method is (0, 4.54).


## 2.2. The New Method with Parameters Depending on Step-Size

This method is derived using the coefficients from the sixth-order explicit hybrid method in Section2.1. Consider the modified four-stage explicit hybrid method represented by this tableau:(9)Associate each formula stage of the modified four-stage explicit hybrid method with the following linear operators:(10)Lh,ayx=yx+cih−σi1+cihyx+μiciyx−h−h2∑j=1i−1aijy″x+cih,i=3,4,5,Lh,byx=yx+h−2σiyx+μiyx−h−h2∑i=15biy″x+cih.Using the coefficients from the sixth-order explicit hybrid method and imposing the linear operators to exactly integrate the set {sinwx and coswx}, we get(11)σ3=2000000000000cosvsin7731/10000v+2000000000000sinvcos7731/10000v+1267106676297vsinv3546200000000sinv,μ3=2000000000000sin7731/10000v−103676933703v2sinv1546200000000sinv,σ4=584721300000000cosvv2sin7731/10000v−35462000000000000cosvsin7731/10000v+35462000000000000sinvcos7731/10000v−230871636632107v2sinv+584721300000000sinvv2cos7731/10000v8046327800000000sinv,μ4=−584721300000000v2sin7731/10000v−35462000000000000sin7731/10000v+1386247674457893v2sinv27415672200000000sinv,σ5=5262491700000000000cosvv2sin7731/10000v+125697456581527785906cosvsinv+67784748079673892953v2sinv−9004452122180000000sinvv2cos7731/10000v125697456581527785906sinv,μ5=−−226900000000000v2sin7731/10000v−2709814525536321sinv+175416390000000v2sinv2709814525536321,σ6=cosv+7961393523907344v2+5000000000000002404579123373679v2cos7731/10000v−19768361482779668v2cosv,and μ6=1, where v = wh.For smallv, coefficients σi and μi may cause heavy calculations which lead to inaccuracy; hence, it is often convenient to use Taylor expansions for the coefficients. The resulting method is denoted by MEHM6.


## 3. Stability Analysis

In this section, we present the stability analysis of the modified hybrid method. Assume thatH = λh, e = (1, 1, … ,1)T, σ(v) = (0, 0, σ3, σ4, …, σs)T, and µ(v) = (0, 0, µ3, µ4, … , µs)T. Employing the hybrid methods defined by (5) to the standard equation(12)y″x=−λ2yx,λ>0,gives us(13)yn+1−SH2,vyn+PH2,vyn−1=0,where(14)SH2,v=2σs+1−H2bTI+H2A−1σv×e+c,PH2,v=μs+1−H2bTI+H2A−1μv×c,and the symbol “×” denotes component-wise multiplication. The characteristic polynomial which determines the solution (12) is(15)πζ=ζ2−SH2,vζ+PH2,v.Definition 1 (see [5]).
For the hybrid methods corresponding to the characteristic equation (13) and v = wh, the region in the H-v plane, such that(16)PH2,v<1,SH2,v<1+PH2,v,is called the region of absolute stability of the method.
The region of stability of the new method is shown in Figure1.
It is observed that, ifv = 0, then the region of absolute stability collapses into the interval of absolute stability.Figure 1
Region of absolute stability of method MEHM6.


## 4. Numerical Results

The new and existing methods are coded using Microsoft Visual C++ version 6.0 software and applied to some special second-order problems to provide numerical comparisons of the accuracy and execution time of the methods. The accuracy of the methods is measured by maximum global errors, while execution time (in seconds) is measured after the computation of the starting values. For all codes, the starting values are computed using the exact solution formula of each problem. The abbreviations of the codes are as follows:(i)
MEHM6: the modified sixth-order explicit hybrid method with four stages derived in this paper.(ii)
TRIMHLI: trigonometrically fitted multistep hybrid method proposed in [7].(iii)
TRIEFW: two-step trigonometrically fitted explicit hybrid method with four stages derived in [8].Tables1–5 show the numerical results of the new and existing methods for solving several second-order problems.Table 1
Numerical results for Problem1.
Step-size
Method
Maximum global error
Execution time
0.5
TRIMHLI
1.80695E − 014
0.0033931
TRIEFW
4.37497E − 009
0.0051322
MEHM6
7.75494E − 014
0.0047059
0.25
TRIMHLI
4.55112E − 014
0.0067984
TRIEFW
2.07511E − 013
0.0097257
MEHM6
2.94982E − 013
0.0092222
0.125
TRIMHLI
9.27542E − 014
0.0122101
TRIEFW
4.98745E − 013
0.0191737
MEHM6
1.59362E − 013
0.015831
0.0625
TRIMHLI
2.76669E − 013
0.0271642
TRIEFW
3.01395E − 013
0.0369165
MEHM6
1.32373E − 012
0.0370568
0.03125
TRIMHLI
7.09740E − 013
0.0502737
TRIEFW
1.06570E − 012
0.0799963
MEHM6
3.52752E − 012
0.0657106Table 2
Numerical results for Problem2.
Step-size
Method
Maximum global error
Execution time
0.2
TRIMHLI
3.54557E − 002
0.0039083
TRIEFW
1.01259E − 001
0.006824
MEHM6
6.25211E − 006
0.0068333
0.1
TRIMHLI
1.45864E − 002
0.0084681
TRIEFW
2.81845E − 006
0.011891
MEHM6
1.26995E − 007
0.0134972
0.05
TRIMHLI
3.28176E − 004
0.0205792
TRIEFW
1.56159E − 010
0.0262821
MEHM6
1.80522E − 009
0.0224291
0.025
TRIMHLI
3.26199E − 005
0.0366569
TRIEFW
2.18646E − 012
0.0480142
MEHM6
6.55135E − 012
0.0488463
0.0125
TRIMHLI
3.95990E − 006
0.0650158
TRIEFW
8.21140E − 013
0.112478
MEHM6
8.42048E − 013
0.0919229Table 3
Numerical results for Problem3.
Step-size
Method
Maximum global error
Execution time
0.5
TRIMHLI
5.21409E + 000
0.0010963
TRIEFW
8.67167E − 001
0.0015327
MEHM6
8.63619E − 002
0.0015102
0.25
TRIMHLI
2.10044E − 001
0.0021984
TRIEFW
4.86877E − 003
0.0029643
MEHM6
6.83934E − 004
0.0028038
0.125
TRIMHLI
1.11491E − 002
0.0043691
TRIEFW
6.08046E − 005
0.0058454
MEHM6
7.65379E − 006
0.0062404
0.0625
TRIMHLI
1.06673E − 003
0.008595
TRIEFW
9.09300E − 007
0.0137473
MEHM6
1.08162E − 007
0.0160981
0.03125
TRIMHLI
1.25146E − 004
0.0172246
TRIEFW
1.40413E − 008
0.0231007
MEHM6
1.61036E − 009
0.0218682Table 4
Numerical results for Problem4.
Step-size
Method
Maximum global error
Execution time
0.1
TRIMHLI
2.87798E − 002
0.0033725
TRIEFW
2.72234E − 004
0.0041873
MEHM6
3.61139E − 004
0.0039115
0.05
TRIMHLI
2.28247E − 003
0.0057554
TRIEFW
3.96310E − 006
0.0074681
MEHM6
5.69308E − 006
0.007701
0.025
TRIMHLI
2.48592E − 004
0.0113492
TRIEFW
5.84699E − 008
0.015802
MEHM6
8.54033E − 008
0.0132228
0.0125
TRIMHLI
2.96137E − 005
0.0227135
TRIEFW
9.15825E − 010
0.0320138
MEHM6
1.34166E − 009
0.0254876
0.00625
TRIMHLI
3.63541E − 006
0.0449913
TRIEFW
1.44656E − 011
0.0580932
MEHM6
2.18224E − 011
0.0544479Table 5
Numerical results for Problem5.
Step-size
Method
Maximum global error
Execution time
0.5
TRIMHLI
3.20318E − 003
0.0871526
TRIEFW
5.71691E − 006
0.114812
MEHM6
4.36318E − 007
0.112708
0.25
TRIMHLI
3.54962E − 004
0.167776
TRIEFW
8.77577E − 008
0.266239
MEHM6
6.70166E − 009
0.223963
0.125
TRIMHLI
4.31160E − 005
0.356719
TRIEFW
1.36480E − 009
0.475373
MEHM6
1.04656E − 010
0.444169
0.0625
TRIMHLI
5.35121E − 006
0.66339
TRIEFW
2.13321E − 011
0.918823
MEHM6
1.03394E − 012
0.887893
0.03125
TRIMHLI
6.67693E − 007
1.3356
TRIEFW
4.39348E − 013
1.8746
MEHM6
1.93284E − 011
1.73938Problem 1 (the two-body problem).
(17)
y
1
″
=
−
y
1
y
1
2
+
y
2
2
3
/
2
,
y
1
0
=
1
−
e
,
y
1
′
0
=
0
,
y
2
″
=
−
y
2
y
1
2
+
y
2
2
3
/
2
,
y
2
0
=
0
,
y
2
′
0
=
1
+
e
1
−
e
,
0
≤
x
≤
30
,with e being the eccentricity of the orbit. The theoretical solution of this problem is(18)y1x=cosR−e,y2=1−e2sinR,where R satisfies the Kepler’s equation x=R−esinR. In this paper, the eccentricity value is chosen to be e = 0. For all codes, v = h is used.Problem 2.
(19)
y
1
″
+
100
y
1
+
2
y
1
y
2
y
1
2
+
y
2
2
=
f
1
x
,
y
1
0
=
1
,
y
1
′
0
=
ε
,
y
2
″
+
25
y
2
+
y
1
2
−
y
2
2
y
1
2
+
y
2
2
=
f
2
x
,
y
2
0
=
−
ε
,
y
2
′
0
=
5
,
0
≤
x
≤
10
,with ε=10−3 and(20)f1x=2cos10xsin5x+2εsin5xsinx−cos10xcosx−ε2sin2xcos210x+sin25x+2εsinxcos10x−cosxsin5x+ε2+99εsinx,f2x=cos210x−sin25x+2εsinxcos10x+cosxsin5x−ε2cos2xcos210x+sin25x+2εsinxcos10x−cosxsin5x+ε2−24εcosx.Solution 1.
y
1
x
=
cos
10
x
+
ε
sin
x
and
y
2
x
=
sin
5
x
−
ε
cos
x. For all codes, we use v = 10h for the first component while v = 5h for the second component.Problem 3 (linear oscillatory problem).
(21)
y
1
″
=
−
13
y
1
+
12
y
2
+
9
cos
2
x
−
12
sin
2
x
,
y
1
0
=
1
,
y
1
′
0
=
−
4
,
y
2
″
=
12
y
1
−
13
y
2
−
12
cos
2
x
+
9
sin
2
x
,
y
2
0
=
0
,
y
2
′
0
=
8
,
0
≤
x
≤
10.Solution 2.
y
1
x
=
sin
x
−
sin
5
x
+
cos
2
x
and
(22)
y
2
x
=
sin
x
+
sin
5
x
+
sin
2
x
.
For MEHM6 and TRIEFW, we choosev = h while for TRIMHLI, v = 5h as given in [7].Problem 4 (nonlinear oscillatory problem).
(23)
y
1
″
=
−
4
x
2
y
1
−
2
y
2
y
1
2
+
y
2
2
,
y
1
0
=
1
,
y
1
′
0
=
0
,
y
2
″
=
−
4
x
2
y
2
+
2
y
1
y
1
2
+
y
2
2
,
y
2
0
=
0
,
y
2
′
0
=
0
,
0
≤
x
≤
5.Solution 3.
y
1
x
=
cos
x
2
,
y
2
x
=
sin
x
2
.
For all codes,v = h.Problem 5 (the almost periodic problem).
(24)
y
1
″
=
−
y
1
+
0.001
cos
x
,
y
1
0
=
1
,
y
1
′
0
=
0
,
y
2
″
=
−
y
2
+
0.001
sin
x
,
y
2
0
=
0
,
y
2
′
0
=
0.9995
,
0
≤
x
≤
1000.Solution 4.
y
1
x
=
cos
x
+
0.0005
x
sin
x
and
y
2
x
=
sin
x
−
0.0005
x
cos
x. For all codes, we use v = h.It is observed from Table1 that MEHM6 solves Problem 1 with very close accuracy to TRIMHL1. From the results in Tables 3 and 5, MEHM6 gives the best accuracy as compared to the other codes for most of the step-sizes, while in Table 2, MEHM6 is the most accurate for bigger step-sizes. For smaller step-sizes, the accuracy of MEHM6 is close to TRIEFW as shown in Table 2. Table 4 shows that both MEHM6 and TRIEFW codes have the same order of accuracy for all step-sizes.On the other hand, TRIMHLI has the shortest execution time for all problems considered. This is mainly due to the fact that TRIMHLI has more starting values than that for MEHM6. Hence, less number of integration steps is needed by TRIMHLI to advance the computation as compared to MEHM6.


## 5. Conclusions

In this paper, a new two-step modified explicit hybrid method is developed where each stage of the modified method exactly integrates differential equations with solutions that are linear combinations ofsinwx and coswx. From the numerical results, the new method gives the best accuracy when compared with the multistep methods in [7, 8], particularly for linear oscillatory and almost periodic problems. The new method has two starting values, but the execution time is nevertheless acceptable. Hence, the new method is as competitive as the existing methods for solving oscillatory problems.



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*Source: 5108482-2020-05-01.xml*