Stability and error bounds for numericalmethods for sti initial value problems

This lecture is a survey of the theory of stability and error bounds for numerical methods for stii 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 stii problems that includes singular perturbation problems as a special case. The requirement of uniform stability for a class of problems provides an eeective characterization of linear multistep methods appropriate for stii problems. We describe the recent extension of these results to Runge-Kutta methods, and conclude with a brief discussion of error bounds.