J. Álvarez and J. Rojo
An improved class of generalized Runge-Kutta methods for stiff problems
Technical Report, (2000)

A new family of explicit $p$-stage methods for the numerical integration of scalar
autonomous ODEs is proposed. These methods can be seen as a generalization of the explicit
$p$-stage Runge-Kutta ones, while providing better order and stability results. In fact, we
show that it is possible to obtain A-stable and L-stable formulas of order three and
five, with only two and three evaluations per step respectively, and without losing
the explicitness of the formulas. It is also possible to generalize the methods to get
formulas for some non-autonomous scalar ODEs and systems. We obtain linearly implicit
A-stable methods which do not require Jacobian evaluations. Some numerical examples are
discussed in order to show the good performance of the new schemes.
AMS(MOS) subject classification: 65L05, 65L07, 65L20 

Preprint file as PDF improv.pdf