Sains Malaysiana 42(11)(2013): 1679–1687

Improved Runge-Kutta Methods for Solving Ordinary Differential Equations

(Penambahbaikan Kaedah Runge-Kutta untuk Menyelesaikan Persamaan Pembezaan Biasa)

FARANAK RABIEI¹, FUDZIAH ISMAIL²* & MOHAMED SULEIMAN³

¹Department of Mathematics, Faculty of Science, Universiti Putra Malaysia

43400 UPM Serdang, Selangor, Malaysia

²Department of Mathematics, Faculty of Science and Institute for Mathematical Research,

Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia

³Institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor

Malaysia

Received: 24 February 2012/Accepted: 29 May 2013

ABSTRACT

In this article we proposed three explicit Improved Runge-Kutta (IRK) methods for solving first-order ordinary differential equations. These methods are two-step in nature and require lower number of stages compared to the classical Runge-Kutta method. Therefore the new scheme is computationally more efficient at achieving the same order of local accuracy. The order conditions of the new methods are obtained up to order five using Taylor series expansion and the third and fourth order methods with different stages are derived based on the order conditions. The free parameters are obtained through minimization of the error norm. Convergence of the method is proven and the stability regions are presented. To illustrate the efficiency of the method a number of problems are solved and numerical results showed that the method is more efficient compared to  the existing Runge-Kutta method.

Keywords: Convergence and stability region; improved Runge-Kutta methods; order conditions; ordinary differential equations; two-step methods

ABSTRAK

Dalam artikel ini kami mencadangkan tiga kaedah Runge-Kutta tak tersirat penambahbaikan untuk menyelesaikan persamaan pembezaan peringkat pertama. Kaedah ini adalah dalam bentuk dua langkah dan memerlukan bilangan tahap yang kurang berbanding kaedah Runge-Kutta klasik. Maka kaedah yang baru ini adalah lebih cekap bagi mencapai peringkat kejituan setempat yang sama. Syarat peringkat untuk kaedah ini hingga peringkat kelima diterbitkan menggunakan kembangan siri Taylor dan kaedah peringkat ketiga dan keempat dengan tahap yang berbeza diterbitkan berdasarkan syarat peringkat tersebut. Parameter bebasnya diperoleh melalui norma ralat yang diminimumkan. Penumpuan kaedah ini dibuktikan dan kestabilannya dipersembahkan. Untuk menunjukkan kecekapan kaedah ini, beberapa masalah diselesaikan dan keputusan berangka menunjukkan kaedah ini lebih cekap berbanding kaedah Runge-Kutta sedia ada.

Kata kunci: Kaedah dua langkah; penambahbaikan kaedah Runge-Kutta; penumpuan dan rantau kestabilan; persamaan pembezaan biasa; syarat peringkat

REFERENCES

Butcher, J.C. 2008. The Numerical Methods for Ordinary Differential Equations. John Wiley and Sons.

Goeken, D. & Johnson, O. 2000. Runge-Kutta with higher order derivative approximations. Applied Numerical Mathematics. 34:207- 218.

Hull, T.E., Enright, W.H., Fellen, B.M. & Sedgwick, A.E. 1982. Comparing numerical. Journal of Numerical Analysis 9(4): 603-637.

Phohomsiri, P. & Udwadia, F.E. 2004. Acceleration of Runge- Kutta integeration schemes. Discrete Dynamics in Nature and Society 2: 307-314.

Rabiei, F. & Ismail, F. 2011. Third-order Improved Runge- Kutta method for solving ordinary differential equation. International Journal of Applied Physics and Mathematics 1(3): 191-194.

Udwadia, F.E. & Farahani, A. 2008. Accelerated Runge- Kutta methods. Discrete Dynamics in Nature and Society doi:10.1155/2008/790619.

Xinyuan, W. 2003. A class of Runge-Kutta formulae of order three and four with reduced evaluations of function. Applied Mathematics and Computation 146: 417-432.

*Corresponding author; email: fudziah@science.upm.edu.my