Stability and error bounds for numerical
methods for stiff initial value problems
Abstract: This article is based on a plenary lecture presented at the
1993 Conference on Scientific Computing and Differential Equations (SCADE 93),
in Auckland, New Zealand, in honor of the 60th birthday of John Butcher. The
article is a survey of the theory of stability and error bounds for numerical
methods for stiff initial value problems. We begin with problems in singular
perturbation form, where the distinction between ``fast'' and ``slow''
variables is clearest. Next we explain Kreiss's characterization of stiff
problems that includes singular perturbation problems as a special case. The
requirement of uniform stability for a class of problems provides an effective
characterization of linear multistep methods appropriate for stiff problems.
We describe the recent extension of these results to Runge-Kutta methods, and
conclude with a brief discussion of error bounds.
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