Inte rnatio nal Jo urnal o f Sc ie ntific & Eng inee ring Re se arc h Vo lume 3, Issue 1 , January -2012 1

ISSN 2229-5518

Comparative Analysis of Time Steps Distribution in Runge-Kutta Algorithms

Salau, T.A.O., Ajide, O.O.

Abs tractThis study utilized combination of phase plots,time steps distribution and adaptive time steps Runge -Kutta and f if th order algorithms to investigate a harmonically Duff ing oscillator.The object is to visually compare f ourth and f if th order Runge -Kutta algorithms perf ormance as tools f or seeking the chaotic solutions of a harmonically excited Duff ing oscillator.Though f if th order algori thms f avours higher time steps and as such f aster to execute than f ourth order f or all studied cases.The reliability of results obtained w ith f ourth order w orth its higher recorded total computation time steps period.

Ke ywords Algorithms, Chaotic Solutions, Duff ing Oscillator, Harmonically Excited, Phase Plots, Runge-Kutta and Time Steps

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1 INTRODUCTION

xtensive literature study shows that numerical technique is very important in obtaining solutions of differential equations of nonlinear systems.The most common univer- sally accepted numerical techniques are Backward differential formulae, Runge-Kutta and Adams-Bashforth-Moulton. Ac- cording to Julyan and Oreste in 1992, Runge-Kutta family of algorithms remain the most popular and used methods for integration. In numerical analysis, the Runge-Kutta methods can be classified as important family of implicit and explicit iterative methods for the approximations of solutions of ordi- nary differential equations. Historically, the Runge-Kutta techniques were developed by the German mathematicians C.Runge and M.W. Kutta. The combination of the two names formed the basis of nomenclature of the method known as Runge-Kutta. The relevance of Runge-Kutta algorithms in finding solutions to problems in nonlinear dynamics cannot be overemphasized. Quite a number of research efforts have been made in the numerical solutions of nonlinear dynamic prob- lems. It is usual when investigating the dynamics of a conti- nuous-time system described by an ordinary differential equa- tion to first investigate in order to obtain trajectories. Julyan and Oreste (1992) were able to elucidate the dynamics of the most commonly used family of numerical integration schemes (Runge-Kutta methods). The study of the authors showed that Runge-Kutta integration should be applied to nonlinear sys- tems with knowledge of caveats involved. Detailed explana- tion was provided for the interaction between stiffness and chaos.The findings of this research revealed that explicit Runge-Kutta schemes should not be used for stiff problems mainly because of their inefficiency. According to the authors, the best alternative method is to employ Backward differentia- tion formulae methods or possibily implicit Runge-Kutta me-
thods.

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Dr. Salau is currently a senior lecturer in the department of Mechanical Eng i- neering, University of Ibadan, Nigeria, +2348028644815.

E-mail: tao.salau@mail.ui.edu.ng

Engr. Ajide is currently a lecturer-II in the department of Mechanical Eng i-

neering, University of Ibadan,Nigeria,+2348062687126

E-mail: getjidefem2@yahoo.co.uk
The conclusions drawn from the paper elucidated the fact that dynamics is not only interested in problems with fixed point solutions, but also in periodic and chaotic behaviour.
The application of bifurcation diagrams in the chaotic study
of nonlinear electrical circuits has been demonstrated (Ajide and Salau, 2011). The relevant second order differential equa- tions were solved for ranges of appropriate parameters using Runge-Kutta method.The solutions obtained from this method were employed to produce bifurcation diagrams. This paper showed that bifurcation diagram is a useful tool for exploring dynamics of nonlinear resonant circuit over a range of control parameters. Ponalagusamy 2009 research paper focused on providing numerical solutions for system of second order ro- bot arm problem using the Runge-Kutta sixth order algorithm. The precised solution of the system of equations representing the arm model of a robot has been compared with the corres- ponding approximate solutions at different intervals. The re- sults and comparison showed that the efficiency of numerical integration algorithm based on the abs olute error between the exact and approximate solutions. The implication of this fin d- ing is that STWS algorithm is not based on Taylor’s series and it is an A-stable method. The dynamics of a torsional system with harmonically varying drying friction torque was investi- gated by Duan and Singh (2008). Nonlinear dynamics of a sin- gle degree of freedom torsional system with dry friction is chosen as a case study. Nonlinear system with a periodically varying normal load was first formulated. This is followed by re-formulation of a multi-term harmonic balance method (MHBM). The reason for this is to directly solve the nonlinear time-varying problem in frequency domain. The feasibility of MHBM is demonstrated with a periodically varying friction and its accuracy is validated by numerical integration using fourth order Runge-Kutta scheme. The set of explicit third order new improved Runge-Kutta (NIRK) method that just employed two function evaluations per step has been devel- oped (Mohamed et al, 2011). Due to lower number of function evaluations, the scheme proposed herein has a lower compu- tational cost than the classical third order Runge-Kutta me- thod while maintaining the same order of local accuracy. Ber- nardo and Chi-Wang (2011) carried out a critical review on the development of Ruge-Kutta discontinuous Galerkin (RKDG) methods for nonlinear convection dominated problems. The

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authors combined a special class of Runge-Kutta time discreti- zations that allows the method to be nonlinearly stable regard-
spectively P0 , and t. Francis (1987), Dowell (1988) and Na- rayanan and Jayaraman (1989b) proposed that the combina-
less of its accuracy with a finite element space discretization
tion of
 0.168 , P0 = 0.21 and

 1.0 or

by discontinuous approximations that incorporates the idea of
 0.0168 , P0 = 0.09 and

 1.0

parameters leads to
numerical fluxes and slope limiters coined during the remark-
able development of high resolution finite difference and finite volume schemes. This review revealed that RKDG methods are stable, high-order accurate and highly parallelizable schemes that can easily handle complicated geometries and boundary conditions.The review showed its immense applica- tions in Navier-Stokes equations and Hamilton -Jacobian equa- tions. This study no doubt has brought a relief in computa- tional fluid dynamics.This technique has been mostly em- ployed in analyzing Duffing oscillator dynamics.The Duffing oscillator has been described as a set of two simple coupled ordinary differential equations to solve . Runge-Kutta method has been extensively used for numerical solutions of Duffing oscillator dynamics. Salau and Ajide (2011) investigated the dynamical behaviour of a Duffing oscillator using bifurcation diagrams. The authors employed fourth order Runge-Kutta method in solving relevant second order differential equa- tions. While the bifurcation diagrams obtained revealed the dynamics of the Duffing oscillator, it also shows that the dy- namics depend strongly on initial conditions. Salau and Oke (2010) showed how Duffing equation can be applied in pre-
dicting the emission characteristics of sawdust particles . The
chaotic behaviour of harmonically excited Duffing oscilla-
tor.This study utilized adaptive time steps Runge-Kutta algo- rithms to investigate equation ( 1) over one hundred and fifty

excitation starting with a time step of ( t Excitation Pe-

riod/1000 ). The phase plot was made with the stable sol u- tions from the last fifty (50) excitation period calculations.

2.2 Time Step Selection

Steven and Raymond (2006) argued that employing a con- stant step size to seek solutions of ordinary differential equa- tions of some dynamical systems that exhibits an abrupt change could pose serious limitation.In such engineering problems (chaotic dynamics) of interest,the choice of adaptive time step size becomes inevitable. The formula used for in-
creasing and decreasing the time step ( t ) in this study is
given by (2) and (3) respectively.The tolerance (t ) was fixed at 10-6 for all computation steps while the error ( ) compares
predicted results taking two half-steps with taking a full step called module-1. Similarly module-2 compares predicted re- sults taking three one third with taking a full step. Equation
(2) is used when < t and equation (3) is used when >t .
paper explains the modeling of sawdust particle motion as a two dimensional transformation system of continous time s e- ries. The authors employed Runge-Kutta algorithm in provid- ing solution to Duffing’s model equation for the sawdust pa r- ticles. The solution was based on displacement and velocity perspective. The findings of the authors showed a high profile feasibility of modeling sawdust dynamics as emissions from band saws. The conclusion drawn from this work is that the finding no doubt provides advancement in the knowledge of

t  t0.95(t

t  t0.95(t

)1/ 4

)1/ 5

(2)
(3)
sawdust emission studies.
Despite this wide application of Runge-Kutta method as a
numerical tool in nonlinear dynamics , there is no iota of doubt
that a research gap exists. Available literature shows that a research which compares the performance of different order (Second, Third, Fifth, Sixth e.t.c.) of Runge-Kutta has not been carried out. The objective of this paper is to visually compare fourth and fifth order Runge-Kutta algorithms performance as tools for seeking the chaotic solutions of a harmonically ex- cited Duffing oscillator.

2 METHODOLOGY

2.1 Duffing Oscillator

The studied normalized governing equation for the dynamic behaviour of harmonically excited Duffing system is given by equation (1)

2.3 Parameter Details of Studied Case s

Three different cases were studied using the details given in table 1 in conjunction with governing equation ( 1).Common parameters to all cases includes displacement ( x  1.0) , Zero

initial velocity ( x) and excitation frequency ( 1) .

Table 1 : Combined Parameters for Cases

x  x1  x 2 P Sint

(1)
In equation (1);

x, x, xrepresents respectively displace-

ment, velocity and acceleration of the Duffing oscillator about a set datum. The damping coefficient is . Amplitude strength
of harmonic excitation, excitation frequency and time are re-

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3 RESULTS AND DISCUSSION S

Time steps distribution (Case-1)

0.100

0.090

0.080

0.070

0.060

0.050

0.040

0.030

0.020

0.010

0.000

0.000 0.020 0.040 0.060 0.080 0.100 0.120 0.140 0.160

Normalised time step

Order4

Order5

and fifth Order Runge-Kutta Algorithms
Fig.3 can be interpreted qualitatively as figures 1 and 2. How-
ever, the frequency intensities differ and the distributions for the fourth and fifth order algorithms peaked at 0.011 and 0.043 periods respectively.

Fig.1: Comparative Time Steps Distribution (Case-1) of fouth and fifth order Runge-Kutta Algorithms
Fig.1 refers; the time steps distribution range is shorter for the fourth order algorithms and longer for the fifth order algo- rithms.The fourth order algorithms is less tolerant of higher computational time steps than fifth order algorithm.The dis- tributins for the fourth and fifth order algorithms peaked at
0.026 and 0.026 excitation periods respectively.
Fig.4a: Phase Plot Obtained for fourth Order (Case-1)

Time steps distribution (Case-2)

0.400

0.350

0.300

0.250

0.200

0.150

0.100

0.050

0.000

0.000 0.050 0.100 0.150 0.200

Normalise d ste p

Order4

Order5

Fig.4b: Phase Plot Obtained for fifth Order (Case-1)


Fig.2: Comparative Time Steps Distribution (Case-2) of fourth and fifth Order Runge-Kutta Algorithms
Fig.2 can be interpreted qualitatively as figure 1. However the frequency intensities differ drastically. The distributions for the fourth and fifth order algorithms peaked at 0.025 and

0.048 excitation periods respectively.

Time steps distribution (Case-3)

Fig.4c: Phase Plot Obtained for fourth Order (Case-2)

0.500

0.450

0.400

0.350

0.300

0.250

0.200

0.150

0.100

0.050

0.000

0.000 0.050 0.100 0.150

Normalised time step

Order4

Order5

Fig.3: Comparative Time Steps Distribution (Case-3) of fourth

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Fig.4d: Phase Plot Obtained for fifth Order (Case-2)

Fig.4e: Phase Plot Obtained for fourth Order (Case-3)

Fig.4f: Phase Plot Obtained for fifth Order (Case-3)
Fig.4 (a-f) shows the comparison of the phase plots obtained using Runge-Kutta fourth and fifth orders (module-1). Fig.4 (a-f) refers; the phase plots are only similar but not exact for case-1 and case-2 only. A closer observation of the phase plot for case-2 shows that solutions obtained by fourth order algo- rithm are bounded to the negative side of the displacement while the solutions obtained by fifth order algorithm are bounded to the positive side of the displacement.The phase plot in figure 4 (a) compare very well with phase plot obtained by Dowell (1988) . In addition,interpretations of table 2 stron g- ly support higher consistency and reliability of fourth order algorithm results than its fifth order counterpart.Overall com- parative assessment of the phase plots in conjunction with time steps distribution suggest fourth order algorithm results as more reliable than fifth order at th e expense of more com putation steps period(see table 3).Table 3 shows that adaptiv fourth order can be twenty five(25) time fast to execute com paring with its equivalent constant time steps (See case-1 an case-2). Similarly, adaptive fifth order can be fifty (50) tim fast to execute comparing with its equivalent time steps (See all cases). Table 3 further shows that adaptive fifth order can be four times fast to compute as its counterpart fourth order as recorded in case-3. However, reliability of computed results may be doubtful. The ratio of total number of steps taking to seek steady solutions by fourth and fifth order algorithms is
module independent.
Table 2a: Corresponding Phase Plots Referring to fig.4 (a, c, e) for Fourth Order Algorithm
Table 2b: Corresponding Phase Plots Referring to fig. 2(b, d, f)
for Fifth Order Algorithm
Note : A,B,C,D,E,F is the same as fig.4(a),(b),(c),(d),(e) and (f)
respectively.
Table 3a: Total Number of Variable Steps Taken to Obtain the Steady Solutions within Studied 50 Excitation Periods (Fourth Order Runge-Kutta)

Cases

Constant

Time Steps

Adaptive

Time

Steps(Module-

1)

Adaptive

Time

Steps(Module-

2)

Case-1

50000

1598

1588

Case-2

50000

1598

1594

Case-3

50000

3667

3704

Table 3b: Total Number of Variable Steps Taken to Obtain the Steady Solutions within Studied 50 Excitation Periods (Fourth Order Runge-Kutta)

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Table 3c: Ratio of Total Number of Steps in Fourth Order to Fifth Order

[13] C. C. Steven and P.C. Raymond, “Numerical Methods for Engineers, Fifth

Edition, McGraw-Hill (International Edition), New York, ISBN 007-124429-

8.2006.

++

4 CONCLUSIONS

This study has visually illustrated the performance of two Runge-Kutta algorithms to seek the chaotic steady solutions of harmonically excited Duffing oscillator. The study has shown that Runge-Kutta fifth order can be four time fast to execute comparing with the corresponding fourth order but at the ex- pense of reliability of the computed results.

REFERENCES

[1] O.O. Ajide and T.A.O. Salau, “Bifurcation Diagrams of Nonlinear RLC Elec- trical Circuits. International Journal of Science and Technology,”Vol. 1, No.3

Pg.136-139, 2011.

[2] C. Bernardo and S. Chi-Wang “, Runge-Kutta Discontinous Galerkin Methods for Convection-Dominated Problems,” Journal of Scientific Computing, Vol.16, No.3, 2001.

[3] E.H. Dowell, “Chaotic Oscillations in Mechanical Sytems ,” Computa-

tional Mechanics, 3,199-216, 1988.

[4] C. Duan and R. Singh, “Dynamics of a Torsional System with Har- monically Varying Drying Friction Torque,”Journal of Physics: Con- ference Series, Vol.96, Ser.96012114. 2008.

[5] C.M. Francis, "Chaotic Vibrations-An Introduction for Applied Scien-

tists and Engineers," John Wiley & Sons, New York, ISBN 0-471-

85685-1, 1987.

[6] L.B.Gregory and P.G. Jerry,”Chaotic Dynamics: An Introduc-

tion,”Cambridge University Press, New York, ISBN 0-521-38258-0

Hardback, ISBN 0-521-38897-X Paperback. 1990.

[7] H.E.Julyan and P. Oreste,"The Dynamics of Runge-Kutta Methods, International Journal of Bifuracation and Chaos," Vol.2, Pg.427-

449.1992.

[8] O. Mohamed, S. Raja and K. Raja, “New Improved Runge-Kutta Method with Reducing Number of Function Evaluations ,” Interna- tional Conference on Software Technology and Engineering, 3rd ICSTE, 2011.

[9] S.Narayanan S. and Jayaraman K., “Control of Chaotic Oscillations

by Vibration Absorber,”ASME Design Technical Conference, 12th Bi- ennial Conference on Mechanical and Noise.DE 18 , 5,391-394.1989.

[10] S. Ponalagusamy and S.Senthilkumar, “System of Second Order Ro- bot Arm Problem by an Efficient Numerical Integration Algorithms,” Archives of Computational Materials Science and Surface Engineer- ing, 1/1, Pg.38-44. 2009.

[11] T.A.O. Salau and O.O. Ajide, “Investigating Duffing Oscillator Using

Bifurcation Diagrams,” International Journal of Mechanics Structural.

ISSN 0974-312X Vol.2, No.2, pg.57-68.International Research Publica-

tion House 2011.

[12] T.A.O. Salau and S.A. Oke, “The Application of Duffing’s Equation in Predicting the Emission Characteristics of Sawdust Particles,” The Kenya Journal of Mechanical Engineering, Vol.6, No.2, Pg.13-32.2010.

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