Journal of Mathematics
Volume 2020 (2020), Article ID 5108482, 7 pages
https://doi.org/10.1155/2020/5108482

Research Article

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

Faieza Samat¹ and Eddie Shahril Ismail²

¹GENIUS@Pintar National Gifted Centre, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia
²Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia

Correspondence should be addressed to Faieza Samat; faiezasamat@ukm.edu.my

Received 29 August 2019; Revised 22 March 2020; Accepted 27 March 2020; Published 1 May 2020

Academic Editor: Nasser Saad

Copyright © 2020 Faieza Samat and Eddie Shahril Ismail. 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.

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 of and , where is the known frequency. Numerical results show the advantage of the new method for solving oscillatory problems.

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

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:

where h is the step-size while and represent and , respectively. The associated Butcher tableau for this class of methods is given by

where

and .

Kalogiratou et al. [5] have modified each stage of the explicit hybrid methods and the improved version is

The coefficients and are functions of  = wh, where is the known frequency of the second-order problems. The Butcher tableau for the modified hybrid method is given by

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

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

Next, we choose the free parameter 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 Section 2.1. Consider the modified four-stage explicit hybrid method represented by this tableau:

Associate each formula stage of the modified four-stage explicit hybrid method with the following linear operators:

Using the coefficients from the sixth-order explicit hybrid method and imposing the linear operators to exactly integrate the set { and }, we get

and , where  = wh.

For small , coefficients and 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 that H = λh, e = (1, 1, … ,1)T, σ() = (0, 0, σ₃, σ₄, …, σs)T, and µ() = (0, 0, µ₃, µ₄, … , µs)T. Employing the hybrid methods defined by (5) to the standard equation

gives us

where

and the symbol “×” denotes component-wise multiplication. The characteristic polynomial which determines the solution (12) is

Definition 1 (see [5]). For the hybrid methods corresponding to the characteristic equation (13) and  = wh, the region in the H- plane, such that

is called the region of absolute stability of the method.
The region of stability of the new method is shown in Figure 1.
It is observed that, if  = 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].

Tables 1–5 show the numerical results of the new and existing methods for solving several second-order problems.

Table 1: Numerical results for Problem 1.

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
0.0657106

Table 2: Numerical results for Problem 2.

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
0.0919229

Table 3: Numerical results for Problem 3.

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
0.0218682

Table 4: Numerical results for Problem 4.

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
0.0544479

Table 5: Numerical results for Problem 5.

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
1.73938

Problem 1 (the two-body problem).

with e being the eccentricity of the orbit. The theoretical solution of this problem is

where R satisfies the Kepler’s equation . In this paper, the eccentricity value is chosen to be e = 0. For all codes,  = h is used.

Problem 2.

with and

Solution 1. . For all codes, we use  = 10h for the first component while  = 5h for the second component.

Problem 3 (linear oscillatory problem).

Solution 2.

For MEHM6 and TRIEFW, we choose  = h while for TRIMHLI,  = 5h as given in [7].

Problem 4 (nonlinear oscillatory problem).

Solution 3.
For all codes,  = h.

Problem 5 (the almost periodic problem).

Solution 4. . For all codes, we use  = h.

It is observed from Table 1 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 of and . 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.

Data Availability

The maximum global error and execution time data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors acknowledge the financial support by Universiti Kebangsaan Malaysia through grant GGPM-2017-074.

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