Author Topic: Comparative Analysis of Time Steps Distribution in Runge-Kutta Algorithms  (Read 3247 times)

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Author : Salau, T.A.O., Ajide, O.O
International Journal of Scientific & Engineering Research Volume 3, Issue 1, January-2012
ISSN 2229-5518
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Abstract— This study utilized combination of phase plots,time steps distribution and adaptive time steps Runge-Kutta and fifth order algorithms to investigate a harmonically Duffing oscillator.The object is to visually compare fourth and fifth order Runge-Kutta algorithms performance as tools for seeking the chaotic solutions of a harmonically excited Duffing oscillator.Though fifth order algorithms favours higher time steps and as such faster to execute than fourth order for all studied cases.The reliability of results obtained with fourth order worth its higher recorded total computation time steps period.

Keywords— Algorithms, Chaotic Solutions, Duffing Oscillator, Harmonically Excited, Phase Plots, Runge-Kutta and Time Steps

1   INTRODUCTION                                                                     
Extensive 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. According 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 ordinary 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 problems. It is usual when investigating the dynamics of a continuous-time system described by an ordinary differential equation 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 systems with knowledge of caveats involved. Detailed explanation 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 differentiation formulae methods or possibily implicit Runge-Kutta methods.

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 equations 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 robot 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 corresponding approximate solutions at different intervals.  The results and comparison showed that the efficiency of numerical integration algorithm based on the absolute error between the exact and approximate solutions. The implication of this finding 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 single 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 do-main. 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 developed (Mohamed et al, 2011). Due to lower number of function evaluations, the scheme proposed herein has a lower computational cost than the classical third order Runge-Kutta method while maintaining the same order of local accuracy. Bernardo 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 authors combined a special class of Runge-Kutta time discretizations that allows the method to be nonlinearly stable regardless of its accuracy with a finite element space discretization by discontinuous approximations that incorporates the idea of numerical fluxes and slope limiters coined during the remarkable development of high reso-lution 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 applications in Navier-Stokes equations and Hamilton-Jacobian equations. This study no doubt has brought a relief in computational fluid dynamics.This technique has been mostly employed 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) investi-gated the dynamical behaviour of a Duffing oscillator using bifurcation diagrams. The authors employed fourth order Runge-Kutta method in solving relevant second order differential equations. While the bifurcation diagrams obtained revealed the dynamics of the Duffing oscillator, it also shows that the dynamics depend strongly on initial conditions. Salau and Oke (2010) showed how Duffing equation can be applied in predicting the emission characteristics of sawdust particles. The paper explains the modeling of sawdust particle motion as a two dimensional transformation system of continous time series. The authors employed Runge-Kutta algorithm in providing solution to Duffing’s model equation for the sawdust particles. The solution was based on displacement and velocity perspec-tive. The findings of the authors showed a high profile feasi-bility 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 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 excited Duffing oscillator. 

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

In equation (1);  represents 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 respectively ,  and t.  Francis (1987), Dowell (1988) and Narayanan and Jayaraman (1989b) proposed that the combination of  , = 0.21  and    or   , = 0.09  and   parameters leads to chaotic behaviour of harmonically excited Duffing oscilla-tor.This study utilized adaptive time steps Runge-Kutta algorithms to investigate equation ( 1) over one hundred and fifty excitation starting with a time step of ( Excitation Period/1000 ). The phase plot was made with the stable solutions from the last fifty (50) excitation period calculations.

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