Analysis of error control strategies for continuous Runge-Kutta methods

There has been considerable recent progress in the analysis and development of interpolation schemes that can be associated with discrete Runge–Kutta methods. With the availability of these schemes it can now be asked that a numerical method provide a continuous approximation to the solution. Thispaper, rather than view such a continuous method as an interpolant superimposed on a standard discrete method, considers how the interpolant and its associated defect can be effectively used in the underlying error and stepsize control mechanism.In particular four error control strategies are considered that can be used in methods based on Runge–Kutta formula pairs with interpolants. An asymptotic and a nonasymptotic analysis of each strategy are presented. It is shown that a strategy based on direct defect control can provide significant advantages over existing strategies with only a modest increase in cost.