Generalized singly-implicit Runge-Kutta methods with arbitrary knots (Q1066624)

This paper generalizes the work of the reviewer on singly-implicit Runge- Kutta methods (methods whose Runge-Kutta matrix has a single real s-fold eigenvalue) to a general family of Nordsieck methods with s internal stages and r external stages (so that r quantities representing y and the first r-1 scaled derivatives of y are carried from step to step). It is shown how to construct singly-implicit Nordsieck methods of order \(s+r-1\) with stage order \(s+r-2\) without restriction on the knots. The similarity transformation is formed which allows efficient implementation when solving the s implicit equations for the s internal approximations, and simple formulas for the r updates at the end of an integration step are established.