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Volume 17,     Number 3,     Fall 2009

 

ON FIFTH AND SIXTH ORDER EXPLICIT
RUNGE-KUTTA METHODS: ORDER
CONDITIONS AND ORDER BARRIERS
J. C. BUTCHER

Abstract. Although Runge-Kutta methods up to order 4 satisfy exactly the same conditions in the case of a single scalar equation as for a general high-dimensional system, the two order theories start to diverge above this order. For example, for order 5, two of the 17 "elementary differentials" from which the Taylor expansions for both the exact and numerical solutions are constructed, coincide and this means that there are only 16 conditions for order 5 in the case of a scalar problem. A method will be presented which exhibits different order behaviours for scalar and high-dimensional problems. The paper will also give a new proof of the first of the Runge-Kutta order barriers.

 

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