Sains Malaysiana 39(6)(2010): 1049–1054

Embedded Pair of Diagonally Implicit Runge-Kutta Method for  Solving Ordinary Differential Equations

(Pasangan Terbenam Kaedah Runge-Kutta Pepenjuru Tersirat  untuk Menyelesaikan Persamaan Pembezaan)

Fudziah Ismail*, Raed Ali Al-Khasawneh, Mohamed Suleiman & Malik Abu Hassan

Jabatan Matematik, Fakulti Sains, Universiti Putra Malaysia , 43400 UPM Serdang

Selangor, Malaysia

Received: 9 April 2009 / Accepted: 26 June 2009

ABSTRACT

Improvements over embedded diagonally implicit Runge-Kutta pair of order four in five are presented. Method of higher stage order with a zero first row and the last row of the coefficient matrix is identical to the vector output is given. The stability aspect of it is also looked into and a standard test problems are solved using the method. Numerical results are tabulated and compared with the existing method.

Keywords: Diagonally implicit; Runge-Kutta; stiff equations; stability

ABSTRAK

Penambahbaikan pasangan terbenam kaedah Runge-Kutta pepenjuru tersirat dipersembahkan. Kaedah dengan peringkat tahap yang lebih tinggi dengan baris pertama sifar dan baris terakhir matriks pekali sama dengan vektor output diberikan. Aspek kestabilannya dikaji dan beberapa masalah piawai diselesaikan menggunakan kaedah tersebut. Keputusan berangka diberikan dan dibandingkan dengan kaedah sedia ada.

Kata kunci: Kestabilan; Pepenjuru tersirat; persamaan kaku; Runge-Kutta

REFERENCES

Butcher J.C. 1987. The Numerical Analysis of Ordinary Differential Equations, Runge-Kutta and General Linear Methods, New York: John Wiley and Sons.

Butcher, J.C. & Chen, D.J.L. 2000. A new Type of Singly-implicit Runge-Kutta method, Applied Numerical Mathematics 34: 179-188.

Dormand J.R. & Prince P.J. 1981. High order embedded Runge-Kutta formula. J. Comput. Appl. Math 7: 67-75.

Enright. W.H., Hull, T.E. & Lindberg B, 1974. Comparing Numerical Methods for stiff Systems of ODEs, Technical Report No 69, Department of Computer Science, University of Toronto, Canada.

Ismail, F. & Suleiman M.B. 1998. Embedded Singly Diagonally Implicit Runge-Kutta method (4,5) in (5,6) for the integration of stiff systems of ODEs, International Journal of Computer Math 66: 325-341.

Kvaerno A, 2004. Singly Diagonally Implicit Runge-Kutta Methods with an explicit first stage, BIT 44: 489-502.

Norsett, S.P. & Thompsen, P.G. 1984. Embedded SDIRK methods of basic order three, BIT 24: 634-464.

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

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