Embedded Runge-Kutta formulae with stable equilibrium states

AbstractThe equilibrium theory of Hall and Higham (1988) can be used to determine whether a Runge-Kutta algorithm will perform smoothly when stability restricts the stepsize. In this paper we show that current high quality order 4, 5 pairs do not behave well in this respect, and we determine the extent to which the overall quality must be compromised in order for the equilibrium conditions to be satisfied. Three new formulae are presented and their properties are compared with those of existing formulae