A new 4th order runge-kutta method for initial value problems with error control

Abstract

A new 4th order Runge-Kutta method for solving initial value problems is derived by replacing the arithmetic means in the formula where etc., by their geometric means i.e. etc. to yield initially a low order accuracy formula. However by re-comparing the Taylor series expansions of k ₁ k ₂ k ₃ and k ₄ in terms of the functional derivatives and the α _ij parameters, a fourth order accuracy formula is obtained which is confirmed by numerical experiments. Then, a new fourth order Runge-Kutta method for solving linear initial value problems of the form y′ = Ay is derived which provides an estimate of the truncation error without any extra function evaluations. The idea follows from the fact that two numerical solutions of similar order can be obtained by using the arithmetic mean (AM) and the geometric mean (GM) averaging of the functional values. The numerical results given confirm that this new method is suitable to be used as an error control strategy.

Keywords:

• Runge-Kutta method
• arithmetic mean
• geometric mean
• error control