Runge-Kutta interpolants based on values from two successive integration steps (Q910164)

The explicit Runge-Kutta method is one of the most popular techniques for solving non-stiff initial value problems of the form \(y'(x)=f(x,y(x))\), \(x\geq x_ 0\), \(y\in {\mathbb{R}}^ 4\), \(y(x_ 0)=y_ 0\). To get an efficient method for problems requiring dense output, one constructs an interpolant based on sufficient number of approximations \(y_ n\) to \(y(x_ n)\), \(x_ n=x_{n-1}+h_{n-1}\), \(n\geq 1\) and the corresponding derivatives \(y'_ n.\) In the last few years several authors have been working on the idea of producing interpolants for Runge-Kutta methods. The authors of the present note deal with the construction of various new interpolants based on values of the solution and its derivative from two successive integration steps. They also make contributions to quantify the effect of variable step size on the magnitude of the error of the constructed interpolants. There are numerical tests for the efficiency of the authors' results.