Variable-Stepsize Explicit Two-Step Runge-Kutta Methods

Variable-step explicit two-step Runge-Kutta methods for the numerical solution of ordinary differential equations are studied. Order conditions are derived and the results about the minimal number of stages required to attain a given order are established up to order five. The existence of embedded pairs of continuous Runge-Kutta methods and two-step Runge-Kutta methods of order $p - 1$ and p is proved. This makes it possible to estimate local discretization error of continuous Runge-Kutta methods without any extra evaluations of the right-hand side of the differential equation. An algorithm to construct such embedded pairs is described, and examples of (3, 4) and (4, 5) pairs are presented. Numerical experiments illustrate that local error estimation of continuous Runge-Kutta methods based on two-step Runge-Kutta methods appears to be almost as reliable as error estimation by Richardson extrapolation, at the same time being much more efficient.