SciCADE 2013
International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

Contributed Talk

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Runge-Kutta methods satisfying conjugate order conditions

G. Imran and J.C. Butcher

Abstract
It is known that it is impossible to find explicit Runge-Kutta methods with only five stages and with order five. While this is true, a slight modification of the meaning of ''Order'' makes it false. This generalisation is known as ''Effective Order'' and can be explained in terms of group conjugacy. The derivation of fifth effective order methods involves 10 order conditions but for convenience we can use simplifying assumptions to make derivation easier. In particular, we have found some new effective order five methods using the $C(2)$ assumption; this leads to simpler and more direct derivation of these methods. Also, we have investigated other cases and reached some conclusions. Other applications of effective order will be briefly surveyed.

Bibliography
[1] J.C. Butcher, The effective order of Runge-Kutta methods, Lecture Notes in Math, 109 (1969), pp. 133-139.

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