Discrete Dynamics in Nature and Society
Volume 2018 (2018), Article ID 2393015, 10 pages
https://doi.org/10.1155/2018/2393015

Research Article

Efficient Two-Derivative Runge-Kutta-Nyström Methods for Solving General Second-Order Ordinary Differential Equations

T. S. Mohamed,1,2 N. Senu,1,3 Z. B. Ibrahim,1,3 and N. M. A. Nik Long1,3

¹Institute for Mathematical Research, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia
²Department of Mathematics, Faculty of Science, Misrata University, Misrata, Libya
³Department of Mathematics, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia

Correspondence should be addressed to N. Senu; norazak@upm.edu.my

Received 17 September 2017; Revised 2 January 2018; Accepted 5 February 2018; Published 20 March 2018

Academic Editor: Ciprian G. Gal

Copyright © 2018 T. S. Mohamed 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.

Abstract.

This paper proposes and investigates a special class of explicit Runge-Kutta-Nyström (RKN) methods for problems in the form including third derivatives and denoted as STDRKN. The methods involve one evaluation of second derivative and many evaluations of third derivative per step. In this study, methods with two and three stages of orders four and five, respectively, are presented. The stability property of the methods is discussed. Numerical experiments have clearly shown the accuracy and the efficiency of the new methods.

1. Introduction

In this article, we are interested in initial value problems (IVPs) of second-order ordinary differential equations (ODEs):

where are continuous vector valued functions. This type of problems arises naturally in many applied science fields such as the Kepler problems in celestial mechanics, quantum physics, and Newton’s second law in classical mechanics (see Dormand [1], Hairer et al. [2], and Kristensson [3]).

Problems (1) in which the first derivative does not appear explicitly are an important subclass of second order (ODEs). Thus, several numerical methods for directly solving this subclass have been presented (see Dormand [1], Hairer et al. [2], Butcher [4], Lambert [5], and Senu [6]). In the case of direct solutions for the general second order (IVPs), some numerical methods have been proposed (see Chen et al. [7], Franco [8], Jator [9], Awoyemi [10], Wu et al. [11], Wu and Wang [12], and Chawla and Sharma [13]). The objective of this paper is to design STDRKN methods with a minimal number of function evaluation. This paper is organized as follows: In Section 3, we construct STDRKN methods; the stability analysis of STDRKN methods is discussed in Section 4; and numerical results are given in Section 5.

2. The Formulation of STDRKN Methods

In many problems in applications the third derivative

is available and easy to obtain. This derivative can be computed but the evaluation of requires the evaluation of , , . Therefore, in the scalar case (differential systems of dimension one), an evaluation of the third derivative can be as expensive as four evaluations of the second derivative and at least as two -evaluations. An -stage two-derivative Runge-Kutta-Nyström (TDRKN) method for (1) is defined by the formula (see Chen et al. [7])

where

where , are real numbers. This method can also be written in Butcher’s tableau of coefficients as given in Table 1.

Table 1: Butcher tableau for TDRKN methods.

In this paper, a special part of TDRKN method is studied that has the form

where

An alternative expression of formula (5) is given as follows:

where

This STDRKN method can be written in Butcher’s tableau as shown in Table 2. The STDRKN methods are explicit methods if for and are implicit method if for STDRKN methods involve only one evaluation of and many evaluations of per step.

Table 2: Butcher tableau for STDRKN methods.

3. Construction of STDRKN Methods

In this section, our effort is to determine the coefficients of the STDRKN methods as given in (7). Hence, using the Taylor series expansion in (3) with the Taylor series expansion of and and by comparing the coefficients of the power of , we obtained the order conditions of STDRKN methods for and as in (10)–(18), while the rooted trees for STDRKN methods up to order five based on [7] are given in Table 3.

Table 3: Root trees for STDRKN methods up to order five.

Order  

Tree 

Density 

Elementary weight 

Elementary differential 

0

1
1

1

1
1

2

1
2

3

1
6

3

1
6

4

1
12

4

2
12

4

1
12

4

1
24

4

1
24

4

1
24

5

1
20

5

3
20

5

3
20

5

1
20

5

3
40

5

1
40

5

3
40

5

1
40

5

1
40

5

1
60

5

1
60

5

1
60

5

2
120

5

1
120

5

1
120

5

2
120

5

2
120

The following simplifying assumption is suggested in practice:

The following are the order conditions for explicit STDRKN.

The order conditions for :

Third Order

Fourth Order

Fifth Order

Sixth Order

The order conditions for :

Second Order

Third Order

Fourth Order

Fifth Order

Sixth Order

3.1. Two-Stage STDRKN Method of Order Four

To derive the fourth-order STDRKN method, we use the algebraic order conditions up to order four in the equations and , that is, (10)-(11) and (14)–(16), respectively. However, we get a system of equations which consists of 5 nonlinear equations with 6 unknown variables. Solving these equations with the simplifying assumption , the resulting system has one free parameter ; this free parameter can be selected by minimizing error equations. The truncation error norms and global error of the fifth-order condition are defined as follows:

The error equations have a minimum value at which produces

where and are error terms of the fifth-order conditions for and , respectively. The coefficients of the new method can be given in Butcher tableau and denoted by STDRKN4 as seen in Table 4.

Table 4: The STDRKN4 method.

0
0

0

1/2

0

0

3.2. Three-Stage STDRKN Method of Order Five

According to the order conditions in (10)–(12) and (14)–(17), we obtain system of equations that consists of 9 nonlinear equations with 14 unknowns variables that need to be solved. Solving the system simultaneously with simplifying assumption conditions yields a solution with three free parameters , and as follows:

Substituting the above solution into , , and gives us

Since the expression error norm (22) is so complicated and not suitable to write and minimizing (22) with respect to the free parameters , and , we get , , and . These values give us

The coefficients of the new method can be given in Butcher tableau and denoted by STDRKN5 as seen in Table 5.

Table 5: The STDRKN5 method.

0
0

0

3/11

0

0

18/25

0

0

4. Stability of the STDRKN Methods

In this part, we study the linear stability of the STDRKN methods. We use the test problem (see [7, 14])

Applying STDRKN method (5) to test problem (24), we obtain

where

with

where

where and . The matrix is called stability matrix, while the stability region of STDRKN method is defined by

are eigenvalues of The stability regions for STDRKN4 and STDRKN5 are shown in Figures 1 and 2, respectively.

Figure 1: The stability region for STDRKN4 method.

Figure 2: The stability region for STDRKN5 method.

5. Numerical Experiments

In this section, we test the effectiveness of the new methods of orders four and five on the same problems for comparison. The numerical methods used for comparison are given as follows:

(i)STDRKN5(3): new special explicit two-derivative RKN method of fifth order derived in this paper(ii)STDRKN4(2): new special explicit two-derivative RKN method of fourth order derived in this paper(iii)TDRKN5(3): three-stage fifth-order two-derivative RKN method derived in [7](iv)TDRKN4(2): two-stage fourth-order two-derivative RKN method derived in [7](v)RKNG5(6): the classical six-stage fifth-order RKN method which is the limit method of ARKNGV5 as the frequency matrix derived in [8](vi)RKNG4: the classical four-stage fourth-order RKN method derived in [2].

Problem 1. Consider the linear homogeneous problem

whose analytic solution is .

Problem 2. Consider the linear nonhomogeneous problem

whose analytic solution is

Problem 3. Consider the nonlinear problem

whose analytic solution is

Problem 4. Consider the famous Van der Pol equation (see [15])

with the initial values

This is a nonlinear equation. Here we take . The problem is integrated on the interval with stepsizes . Since the exact solution of the problem is not available, when estimating the error of each method, we use RKNG4 method given in [2] as a reference numerical solution with a very small stepsize.

Problem 5. Consider the linear homogeneous system (see [7])

where

with the initial values

The problem is integrated on the interval with stepsizes .
The exact solution of the problem is given by

Problem 6. Consider the linear nonhomogeneous system (see [8])

with the initial values

The problem is integrated on the interval with stepsizes .
The exact solution is

Problem 7. Consider the damped wave equation with periodic boundary conditions (see [15])

A semidiscretization in the spatial variable by second-order symmetric differences leads to the following system of second-order ODEs in time:

where with , and

with and and . In this experiment, we take and and the initial conditions as

We choose and integrate the problem in the interval with the stepsizes . The reference numerical solution is obtained by method RKNG4 with a very small stepsize.

6. Conclusion

In this study, the special class of explicit two-derivative Runge-Kutta-Nyström methods of order up to five that involve one -evaluation and minimal number of -evaluations was derived. Figures 3–9 display the efficiency curves showing the common logarithm of the maximum global absolute error throughout the integration versus computational cost measured by time used by each method in the same computation machine. An advantage of the STDRKN methods over the general classical Runge-Kutta-Nyström methods and TDRKN methods is that they can reach higher order with fewer functions evaluations per step and also give us higher stage order than RKN. Some tested problems were performed. From Figures 3, 4, 5, 6, 7, 8, and 9, the numerical results showed that the new methods agreed very well with the existing methods in the literature and required less time compared to the existing methods.

Figure 3: The time curves for all methods for Problem 1.

Figure 4: The time curves for all methods for Problem 2.

Figure 5: The time curves for all methods for Problem 3.

Figure 6: The time curves for all methods for Problem 4.

Figure 7: The time curves for all methods for Problem 5.

Figure 8: The time curves for all methods for Problem 6.

Figure 9: The time curves for all methods for Problem 7.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The authors gratefully acknowledge the financial support of Fundamental Research Grant Scheme (Project no. 01-01-16-1866FR) and Universiti Putra Malaysia.

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