Diagonally Implicit Runge-Kutta Fourth Order Four-Stage Method for Linear Ordinary Differential Equations with Minimized Error Norm


  • Nur Izzati Che Jawias
  • Fudziah Ismail
  • Mohamed Suleiman
  • Azmi Jaafar




Runge-Kutta, Linear ordinary differential equations, Error norm


We constructed a new fourth order four-stage diagonally implicit Runge-Kutta (DIRK) method which is specially designed for the integrations of linear ordinary differential equations (LODEs). The method is obtained based on the
Butcher’s error equations. In the derivation, the error norm is minimized so that the free parameters chosen are obtained from the minimized error norm. Row simplifying assumption is also used so that the number of equations for
the method can be reduced and simplified. A set of test problems are used to validate the method and numerical results show that the new method is more efficient in terms of accuracy compared to the existing method.


Burden R.L., Faires J.D., Numerical Analysis seventh edition, Wadsworth Group. Brooks/Cole, Thomson Learning, Inc, 2001.

Butcher J.C., Numerical Methods for Ordinary Differential Equation, John Wiley & Sons Ltd, 2003.

Dormand J.R., Numerical Methods for Differential Equations, Boca Raton, New York, London and Tokya:CRC Press, Inc, 1996.

Ferracina L., Spijker M.N., Strong stability of Singly-Diagonally-Implicit Runge-Kutta methods. Report no MI 2007-11(2007), Mathematical Institute, Leiden University.

Hairer E., Wanner G., Solving Ordinary Differential Equation II, Springer-Verlag Berlin Heidelberg, 1991.[6] Lambert J.D., Numerical Methods for Ordinary Differential System, John Wiley & Sons Ltd, 1991.

Torrence B.F., Torrence E.A.: How to find the stability regions, The Student’s Introduction to Mathematica, pp 232-264.

Zingg D.W., Chisholm T.T., Runge-Kutta methods for linear ordinary differential equations, Applied Numerical Mathematics, (1999) 31, 227-238.