Volume 17, Number 3, Fall 2009
ON FIFTH AND SIXTH ORDER EXPLICIT
RUNGEKUTTA METHODS: ORDER
CONDITIONS AND ORDER BARRIERS
J. C. BUTCHER
Abstract. Although RungeKutta methods up to order
4 satisfy exactly the same conditions in the case of a single
scalar equation as for a general highdimensional 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 highdimensional problems. The
paper will also give a new proof of the first of the RungeKutta
order barriers.
(Subscribers Only)
