Third Order Accelerated Runge-Kutta Nystrom

Method for Solving Second-Order OrdinaryDifferential Equations

Faranak Rabiei1, Fudziah Ismail2, Norihan Arifin3, and Saeid Emadi4

1,2,3 Mathematics Department, universiti Putra Malaysia43400 UPM Serdang, Selangor, Malaysia

4 University of Derby, UK{rabiei,fudziah,norihan}@math.upm.edu.my, s.emadi60@gmail.com

Abstract. In this paper, the explicit Accelerated Runge-Kutta Nystrom(ARKN) method of order three with two stages for the numerical integra-tion of second-order ordinary differential equations are developed. Themethod is two step in nature. Algebraic order conditions of the methodare obtained and third order ARKN method is derived. Numerical ex-amples are carried out to illustrate the efficiency of the proposed methodcompared to the existing Runge-Kutta Nystrom (RKN) method.

Keywords: Accelerated Runge-Kutta Nystrom method, Two stepmethod, Second-order ordinary differential equations, Order conditions.

1 Introduction

Consider the special second-order ordinary differential equations of the form

y′′ = f(x, y), y(x0) = y0, y′(x0) = y′0. (1)

Such problem often arise in science and engineering fields such celestial me-chanics, molecular dynamics , semi-discretization of wave equations, electronicsand so on. The second-order equations can be directly solved by using Runge-Kutta Nystrom (RKN) methods or multistep methods. Phohomsiri and Udwa-dia [1] constructed the third order Accelerated Runge-Kutta (ARK3) methodfor solving autonomous first-order ordinary differential equations with the formof y′ = f(y). In this paper, we developed the Accelerated Runge-Kutta Nystrommethod of order three (ARKN3) for solving autonomous second-order equa-tion (1) with the form of y′′ = f(y). The method we presented here used onlytwo stages while the existing RKN3 method used three stages.

In section 2, we presented the ARKN3 method and the order conditions ofthe method are obtained in section 3. In section 4, ARKN3 are derived andnumerical examples to illustrate the efficiency of the method compared with theexisting RKN3 are presented in the last section.

A. Abd Manaf et al. (Eds.): ICIEIS 2011, Part III, CCIS 253, pp. 204–209, 2011.c© Springer-Verlag Berlin Heidelberg 2011

Third Order Accelerated Runge-Kutta Nystrom Method 205

2 Construction of ARKN3 Method with Two Stages

Consider the ARK3 method from [1] as follows

yn+1 = yn + h(b1k1 − b−1k−1 + b2(k2 − k−2)), (2)k1 = f(yn), k−1 = f(yn−1),k2 = f(yn + ha1k1), k−2 = f(yn−1 + ha1k−1).

where h is the fixed step size. Here based on the ARK3 method we developedthe ARKN3 method for solving the autonomous second-order equation directlyfollowing the same approach as Dormand [2]. We define:

yn+1 = yn + h(b1f1 − b−1f−1 + b2(f2 − f−2)), (3)

y′n+1 = y′

n + h(b1g1 − b−1g−1 + b2(g2 − g−2)), (4)

g1 = g(yn), g−1 = g(yn−1),

f1 = y′n, f−1 = y′

n−1,

g2 = g(yn + ha1f1), g−2 = g(yn−1 + ha1f−1)

f2 = y′n + ha1g1, f−2 = y′

n−1 + ha1g−1.

by substituting the f−1, f1, f−2, and f2 into g−2 and g2 we have

g2 = g(yn + ha1(y′n + ha1g1)) = g(yn + ha1y

′n + h2a2

1g1),

g−2 = g(yn−1 + ha1(y′n−1 + ha1g−1)) = g(yn−1 + ha1y

′n−1 + h2a2

1g−1),

now substitute the f−1, f1, f−2, f2, g−1, g1, g−2, g2. into (3) obtain

yn+1 = yn + h(b1y′n − b−1y

′n−1 + b2

((y′

n + ha1g1) − (y′n−1 + ha1g−1)

)),

= yn + h(b1 + b2)y′n − h(b−1 + b2)y′

n−1 + h2a1b2(g1 − g−1). (5)

consider the following order conditions of ARK3 (see [1])

b1 − b−1 = 1,

b1 + b2 =12.

simplifying the equation (5) we have

yn+1 = yn +3h

2y′

n +h

2y′

n−1 + h2b (g1 − g−1).

where b = a1b2.By writing Eqs. (3) and (4) in standard form in terms of ki and function f , wedefine the ARKN3 method as follows :

206 F. Rabiei et al.

yn+1 = yn +3h

2y′

n +h

2y′

n−1 + h2b (k1 − k−1),

y′n+1 = y′

n + h(b1k1 − b−1k−1 + b2(k2 − k−2)),

k1 = f(yn), k−1 = f(yn−1),k2 = f(yn + ha1k1), k−2 = f(yn−1 + ha1k−1).

(6)

3 Order Conditions

To find the order conditions for ARKN3 method we applied the Taylor’s seriesexpansion to equations (6). Here, after using the Taylor’s series expansion weobtained the order conditions for yn and y′

n up to order three as follows:

order condition for yn:

first order b1 − b−1 = 1,

second order b1 + b2 =12,

third order b2a1 =512

.

order condition for y′n:

third order b2 =512

.

4 Analysis of the Method

To find the coefficients of ARKN3 method in equations (6), all the above orderconditions for yn and y′

n must be satisfied. We choose the value of b−1 ∈ [−1 1] asa free parameter, here we set b−1 = − 1

3 and obtained the remaining parametersas follows:

a1 =12, b1 =

23, b2 =

56, b2 =

512

.

We compared the results of the new method with the existing RKN method oforder three with dispersive order six and dissipative order five with three stages(see [3]). The RKN3 is computed by using the following formulas:

yn+1 = yn + hy′n + h2(b1k1 + b2k2 + b3k3),

y′n+1 = y′

n + h(b′1k1 + b′2k2 + b′3k3),

k1 = f(yn),

Third Order Accelerated Runge-Kutta Nystrom Method 207

k2 = f(yn + c2hy′n + h2a21k1),

k3 = f(yn + c3hy′n + h2(a31k1 + a32k2)),

It is convenient to present the coefficients of RKN3 in in Table 1.The coefficients of existing NRK3 are shown in Table 2.

Table 1. Table of coefficients for NRK3 method

0

c2 a21

c3 a31 a32

b1 b2 b3

b′1 b′2 b′3

Table 2. Table of coefficients for existing NRK3 method

0

12

49

1 720

320

13

0 16

14

34

0

5 Numerical Examples

In this section, we tested a standard set of second-order initial value problemsto show the efficiency and accuracy of the proposed method. The exact solutiony(x) and y′(x) are used to estimate the global error as well as to approximate thestarting values of y1 and y′

1 at the first step [x0 x1]. Consider the y(xi) is exactsolution and y∗(xi) is approximated solution on ith iteration which computedusing formulas (6), the global error of ith iteration will calculate as follow:

errori = |y(xi) − y∗(xi)|.The following problems are solved for x ∈ [0 10].

208 F. Rabiei et al.

Problem 1 (see [2])

y′′ = −y, y(0) = 0, y′(0) = 1,

exact solution: y(x) = sin(x).

Problem 2 (see [4])

y′′1 = − y1

(√

y21 + y2

2)3, y1(0) = 1, y′

1(0) = 0,

y′′2 = − y2

(√

y21 + y2

2)3, y2(0) = 0, y′

2(0) = 1,

exact solutions: y1(x) = cos(x), y2(x) = sin(x).Here, the number of function evaluations are computed as follow :

Number of function evaluations= Number of stages × Number of steps.

where number of stages = 2, and

Number of steps = 10−0h , for h = 0.5, 0.1, 0.05, 0.01, 0.005.

The log(maximum global error) versus the number of function evaluations forthe tested problems are shown in Figures 1 and 2.

0 1000 2000 3000 4000 5000 6000−8

−7

−6

−5

−4

−3

−2

−1

0

Function evaluations

Log 10

(max

glo

bal e

rror

)

ARKN3RKN3

Fig. 1. Maximum global error versus number of function evaluations for problem 1

Third Order Accelerated Runge-Kutta Nystrom Method 209

0 2000 4000 6000 8000 10000 12000−5.5

−5

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

Function evaluations

Log 10

(max

glo

bal e

rror

)

ARKN3RKN3

Fig. 2. Maximum global error versus number of function evaluations for problem 2

6 Discussion and Conclusion

From Figs. 1 and 2 we observe that for both problems the new method is moreaccurate and has lower number of function evaluations compared to the existingmethod.

As a conclusion, the third order ARKN method has been developed for nu-merical integration of second-order ordinary differential equations with reducednumber of function evaluations required per step. The order conditions of thenew method are derived up to order three and by satisfying the appropriateorder conditions we obtained the method of order three. The ARKN3 method isalmost two-step in nature and is computationally more efficient compared.

References

1. Phohomsiri, P., Udwadia, F.E.: Acceleration of Runge-Kutta Integeration Schemes.Discrit. Dynamic. Nature. Soci. 2, 307–314 (2004)

2. Dormand, J.R.: Numerical Method for Differential Equations. A ComputationalApproach. CRC Press Inc. (1996)

3. Norazak. B. S.,: Runge-Kutta Nystrom Methods for Solving Osillatory Problems,Ph.D Thesis, Universiti Putra Malaysia (2010)

4. Franco, J.M.: A Class of Explicit Two Step Hybrid Methods for Second Order IVP’s.J. Comput. Appl. Math. 187, 41–57 (2006)