J. Álvarez and J.
An improved class of generalized
Runge-Kutta methods for stiff problems. Part I: The scalar case
Math. Comput., 130 (2002), pp. 537-560.
A new family of $p$-stage methods for the numerical integration of
some scalar equations and systems of 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. We will show in this
first part that, at the cost of losing linearity in the formulas, it is
possible to obtain explicit A-stable and L-stable methods for the numerical
integration of scalar autonomous ODEs. Scalar autonomous ODEs are of very
little interest in current applications. However, be begin studying this
kind of problems because most of the work can be easily extended to a more
general situation. In fact, we will show in a second part (entitled `The
separated system case`), that it is possible to generalize our methods
so that they can be applied to some non-autonomous scalar ODEs and systems.
We will obtain linearly implicit L-stable methods which do not require
Jacobian evaluations. In both parts, 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 scalar.pdf